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Migration

My new website is under construction now. Be patient.

Algorithms

---

Miscellaneous

Ostrich Algorithm

Exponential Backoff

Spam

My Devil's Dictionary

Chinese

• 戾拔山兮气盖世: trolls.
• 力排众议: 参议院否决众议院通过的弹劾案.

Polish

• Cześć: CZ, escape!

Interesting

Disaster

• Daallo Airlines Flight 159

Science and Math

• Almost Integer
  • Most notably \((28-\delta^{-1})(\alpha^{-1}-137)=1\).

Languages

• いらっしゃいません
• 反省しる
• 小浜(Obama)市
• Native speakers mistake it's/its because they learn to speak years before they learn to write.

Research

• Another setback for ‘Majorana’ particle as Science paper earns an expression of concern

Ӏ have played Bingo Ρlus foг s᧐me time, and I rｅally "Love" the game. Ӏ am not averse tօ the ads thｅre is јust enough. If it ggot ɑny more, it's the issue. The game is smooth, ɑnd I lovfe Ι am aЬle to control the speed ɑnd the colors on every level аre amazing.. It is aⅼso рossible to tаke part in oother games whіⅼe trｙing to comρlete regular levels, ѡhich is gгeat. Tһanks ffor tһe game! !

漢これ

漢文これくしょん

---

秦

諫逐客書

此四君者，皆以客之功。由此觀之，客何負於秦哉？向使四君卻客而不內，疏士而不用；是使國無富利之實，而秦無強大之名也。

不問可否，不論曲直，非秦者去，為客者逐。然則是所重者，在乎色、樂、珠、玉，而所輕者在乎人民也...

夫物不產於秦，可寶者多；士不產於秦，而願忠者衆。今逐客以資敵國，損民以益讎，內自虛而外樹怨於諸侯，求國之無危，不可得也。

漢

答蘇武書

陵謂足下，當享茅土之薦，受千乘之賞；聞子之歸，賜不過二百萬，位不過典屬國，無尺土之封，加子之勤。而妨功害能之臣，盡為萬戶侯；親戚貪佞之類，悉為廊廟宰。子尚如此，陵復何望哉？且漢厚誅陵以不死，薄賞子以守節，欲使遠聽之臣，望風馳命，此實難矣。所以每顧而不悔者也。陵雖孤恩，漢亦負德。

東晉

蘭亭集序

後之視今，亦由今之視昔，悲夫！

唐

阿房宮賦

使天下之人，不敢言而敢怒。獨夫之心，日益驕固。

秦人不暇自哀，而後人哀之。後人哀之，而不鑑之，亦使後人而復哀後人也。

SFINAE vs Concept

---

A brief introductory example may be found in Concepts versus SFINAE-based constraints.

A Toxic Example

This article is inspired by Item 26 and 27 of Modern Effective C++ by Scott Meyers, 2014.

Universal Reference is Too Greedy

The following code should be short and clear enough.

#include <iostream>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T>
void handle(T&& person)
{
    std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
    std::cout << "Handle number." << std::endl;
}

int main()
{
    Person p;
    SpecialPerson sp;
    int i = 0;
    short s = 0;
    handle(p);
    handle(sp);
    handle(i);
    handle(s);
}

The behavior is, however, puzzling. It prints

Handle person.
Handle person.
Handle number.
Handle person.

Why did the last line print Handle person.? This is because handle(int) requires a conversion in this case while in handle(T&&) the compiler could infer T = short and provide a perfect match.

We may resort to the C++98 approach and declare handle like the following.

void handle(const Person &);
void handle(int);

However, what if we want to retain perfect forwarding? In such case universal reference is inevitable. The problem may be rephrased as follow: const Person& and Person&& are too restrictive as they matches only lvalue or rvalue reference. T&& is too generic as it matches reference to any type. We need something somewhere between.

Detour: How Do C# Handle Such Problem

A simlar problem exists in C#.

using System;
    
class Person
{ }

class SpecialPerson: Person
{ }

public class Program
{
    private static void handle<T>(T person)
    {
        Console.WriteLine("Handle person.");
    }
    
    private static void handle(int number)
    {
        Console.WriteLine("Handle number.");
    }
    
    public static void Main()
    {
        Person p = new Person();
        SpecialPerson sp = new SpecialPerson();
        int i = 0;
        short s = 0;
        handle(p);
        handle(sp);
        handle(i);
        handle(s);
    }
}

The above code prints the following.

Handle person.
Handle person.
Handle number.
Handle person.

What surprises us is the last line, where

short s = 0;
handle(s);

prints

Handle person.

instead of

Handle number.

Imposing constraints on handle solves the problem.

using System;
    
class Person
{ }

class SpecialPerson: Person
{ }

public class Program
{
    private static void handle<T>(T person) where T: Person
    {
        Console.WriteLine("Handle person.");
    }
    
    private static void handle(int number)
    {
        Console.WriteLine("Handle number.");
    }
    
    public static void Main()
    {
        Person p = new Person();
        SpecialPerson sp = new SpecialPerson();
        int i = 0;
        short s = 0;
        handle(p);
        handle(sp);
        handle(i);
        handle(s);
    }
}

Now the program correctly prints

Handle person.
Handle person.
Handle number.
Handle number.

Question: What is the counterpart of where in C++?

Solution: SFINAE vs Concept

The SFINAE Solution

SFINAE stands for substitution failure is not an error. In this case, we make use of std::enable_if and implement handle in the following way.

#include <iostream>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T,
    typename = std::enable_if_t<
        std::is_base_of_v<Person, std::decay_t<T>>
    >
>
void handle(T&& person)
{
    std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
    std::cout << "Handle number." << std::endl;
}

int main()
{
    Person p;
    SpecialPerson sp;
    int i = 0;
    short s = 0;
    handle(p);
    handle(sp);
    handle(i);
    handle(s);
}

It works correctly now and prints

Handle person.
Handle person.
Handle number.
Handle number.

The Concept Solution

Concept is introduced in C++20 and enables the following more straightforward implementation closer to where in C#.

#include <iostream>
#include <concepts>

class Person
{ };

class SpecialPerson: public Person
{ };

template<typename T> requires std::derived_from<std::decay_t<T>, Person>
void handle(T&& person)
{
    std::cout << "Handle person." << std::endl;
}

void handle(int number)
{
    std::cout << "Handle number." << std::endl;
}

int main()
{
    Person p;
    SpecialPerson sp;
    int i = 0;
    short s = 0;
    handle(p);
    handle(sp);
    handle(i);
    handle(s);
}

It works also correctly now and prints

Handle person.
Handle person.
Handle number.
Handle number.

Green's Functions (I)

Definitions and Diagrams

---

This post is not aiming at giving a systematic introduction to Green's functions in condensed matter physics. Rather, it is written in a spaghetti pattern that intertwines many related or unrelated materials. For a systematic approach, see my note on quantum field theory.

Motivation: Why Do We Need Green's Function

The Zero-Temperature Case

In condensed matter physics, it's a common task for us to evaluate the expectation value of a certain operator (for the ground state) \[ \begin{align*} \langle \mathscr{J}(\vb{x}) \rangle &= \frac{\bra{\Omega} \mathscr{J}(\vb{x}) \ket{\Omega}}{\bra{\Omega}\ket{\Omega}} \\ &= \pm i \lim_{t\rightarrow t^+} \lim_{\vb{x}' \rightarrow \vb{x}} \sum_{\alpha\beta} J_{\beta\alpha}(\vb{x}) G_{\alpha\beta}(\vb{x}t, \vb{x}'t') \\ &= \pm i \lim_{t\rightarrow t^+} \lim_{\vb{x}' \rightarrow \vb{x}} \tr[J(\vb{x}) G(\vb{x}t,\vb{x}'t')]. \end{align*} \] for \[ \mathscr{J}(\vb{x}) = \sum_{\alpha\beta} \psi_\beta^\dagger(\vb{x})J_{\beta\alpha}(x) \psi_\alpha(\vb{x}) \] where we have defined the Green's function \[ iG_{\alpha\beta}(\vb{x}t;\vb{x'}t') = \frac{\bra{\Omega} T[\psi_\alpha(\vb{x}t) \psi^\dagger_\beta(\vb{x}'t')]\ket{\Omega}}{\bra{\Omega}\ket{\Omega}}. \]

A few remarks are in order here:

• The field operators are complete in principle: all the observables could be constructed from field operators.
• If the Hamiltonian is time-independent, the Green's function depends only on \((t-t')\). After taking the limit \(t\rightarrow t'\) it doesn't matter at which \(t\) we evaluate the Green's function.
• The limit \(\vb{x}'\rightarrow \vb{x}\) should act before \(J_{\alpha\beta}(\vb{x})\) since \(J\) may contain gradient.

• The number density is given by \[n_{\alpha\beta}(\vb{x}) = \pm i \tr G(\vb{x}t; \vb{x}t^+).\]
• The spin density is given by \[\langle \vb*{\sigma}(\vb{x}) \rangle = \pm i \tr[\vb*{\sigma} G(\vb{xt;\vb{x}t^+})].\]

Surprisingly, we don't need the four-point function in order to evaluated the interaction term. After some calculation we find for \[ V(\vb{x},\vb{x}')_{\alpha\alpha',\beta\beta'} = V(\vb{x}',\vb{x})_{\beta\beta',\alpha\alpha'}. \] that \[ \langle H \rangle = \pm \frac{1}{2}i \int \dd{^3 x} \lim_{t'\rightarrow t^+}\lim_{\vb{x}' \rightarrow \vb{x}} \qty[i\hbar \pdv{}{t} - \frac{\hbar^2 \grad^2}{2m}] \tr G(\vb{x}t; \vb{x'}t'). \]

Useful, right? Now the question becomes how do we evaluate Green's functions. Before we open the Pandora's box of Feynman diagrams, let's see how do we handle the case of finite-temperature.

The Finite-Temperature Case

At non-zero temperature, we have the similar result \[ \begin{align*} \langle J \rangle &= \Tr(\rho_G J) \\ &= \mp \sum_{\alpha\beta} \int \dd{^3 x} \lim_{\vb{x}'\rightarrow\vb{x}} \lim_{\tau'\rightarrow\tau^+} J_{\beta\alpha}(x) \mathscr{G}_{\alpha\beta}(\vb{x}\tau; \vb{x}'\tau') \\ &= \mp \int \dd{^3 x} \lim_{\vb{x}' \rightarrow \vb{x}} \lim_{\tau'\rightarrow \tau^+} \tr[J(x)\mathscr{G}(\vb{x}\tau; \vb{x}'\tau')], \end{align*} \] where we have defined the Matsubara Green's function \[ \mathscr{G}_{\alpha\beta}(\vb{x}\tau;\vb{x}'\tau') = -\Tr[\rho_G T_\tau[\psi_{\alpha}(\vb{x}\tau) \psi^\dagger_{\beta}(\vb{x}'\tau')]], \] the modified Heisenberg picture (i.e. \(\tau = it\)) \[ \begin{align*} \psi_{\alpha}(\vb{x}\tau) &= e^{K\tau/\hbar} \psi_\alpha(\vb{x}) e^{-K\tau/\hbar}, \\ \psi_{\alpha}^\dagger(\vb{x}\tau) &= e^{K\tau/\hbar} \psi_\alpha^\dagger(\vb{x}) e^{-K\tau/\hbar}, \end{align*} \] and the statistical properties \[ \begin{align*} K &= H - \mu N, \\ Z_G &= e^{-\beta\Omega} = \Tr e^{-\beta K}, \\ \rho_G &= Z^{-1}_G e^{-\beta K} = e^{\beta(\Omega - K)}. \end{align*} \]

Example: Degenerate Electron Gas

With average density \(\langle n \rangle = N/V\) fixed, the expectation value of interation energy is given by \[ \langle V \rangle = \frac{1}{2} \int \dd{^3 x} \int \dd{^3 x'} V(\vb{x} - \vb{x}')[\langle \tilde{n}(\vb{x}) \tilde{n}(\vb{x}') \rangle + \langle n(\vb{x})\rangle \langle n(\vb{x}) \rangle - \delta(\vb{x} - \vb{x}')\langle n(\vb{x}) \rangle]. \] where we have defined \(\tilde{n}(\vb{x}) = n(\vb{x}) - \langle n(\vb{x})\rangle\). The last two terms are essentially constants and may therefore be discarded. Now our goal becomes evaluating the polarization propagator (or the four-point function) \[ iD(x, x') = \frac{\bra{\Omega} T[\tilde{n}_H(x') \tilde{n}_H(x)] \ket{\Omega}}{\bra{\Omega} \ket{\Omega}}. \] Its non-interacting counterpart is defined by \[ iD^0(x',x) = \bra{0} T[\tilde{n}_I(x')\tilde{n}_I(x)] \ket{0}. \] Now the expectation value \(\langle V \rangle\) could be divided into the non-interacting contribution and the interating contribution \[ \begin{align*} \langle V \rangle &= \frac{1}{2} \int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD^0(\vb{x}'t, \vb{x}t) + n^2 - \delta(\vb{x} - \vb{x}')n] + \frac{1}{2}\int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD(\vb{x}'t, \vb{x}t) - iD^0(\vb{x}'t, \vb{x}t)] \\ &= \bra{\Phi_0} V \ket{\Phi_0} + \frac{1}{2}\int \dd{^3 x} \dd{^3 x'} V(\vb{x} - \vb{x}')[iD(\vb{x}'t, \vb{x}t) - iD^0(\vb{x}'t, \vb{x}t)]. \end{align*} \]

Category Theory

先輩がうざい後輩の話

---

Definition and Examples

• A category \(C\) consists of the following.
  • A class \(\operatorname{ob} C\) of objects.
  • For each ordered pair of objects \((A,B)\), a set of \(\mathrm{hom}_{C}(A,B)\) whose elements are called morphisms with domain \(A\) and codomain \(B\).
  • For each ordered triple of objects \((A,B,C)\), a map \((f,g)\mapsto gf\) of the product set \(\mathrm{hom}(A,B) \times \mathrm{hom}(B,C)\) into \(\mathrm{hom}(A,C)\).
• The objects and morphisms satisfy the following conditions:
  • If \((A,B) \neq (C,D)\), then \(\mathrm{hom}(A,B)\) and \(\mathrm{hom}(C,D)\) are disjoint.
  • If \(f\in\mathrm{hom}(A,B)\), \(g\in\mathrm{hom}(B,C)\) and \(h\in\mathrm{hom}(B,C)\), then \((hg)f = h(gf)\).
  • For every object \(A\) we have an element \(\mathbb{1}_A\in \mathrm{hom}(A,A)\) such that \(f\mathbb{1}_A = f\) for every \(f\in \mathrm{hom}(A,B)\) and \(\mathbb{1}_A g = g\) for every \(g\in \mathrm{hom}(B,A)\).
• A category \(D\) is called a subcategory of \(C\) if
  • \(\operatorname{ob} C\) is a subclass of \(\operatorname{ob}D\) and
  • for any \(A,B\in \operatorname{ob} D\), \(\mathrm{hom}_D(A,B) \subset \mathrm{hom}_C(A,B)\).
  • \(\mathbb{1}_A\) is defined in the same way as in \(D\).
• A subcategory \(D\) is called full if for every \(A,B\in D\) \[\mathrm{hom}_D(A,B) = \mathrm{hom}_C(A,B).\]

Monoids are identified with categories whose object classes are single-element sets.

• A category \(C\) is called small if \(\operatorname{ob} C\) is a set.
• An element \(f\in \mathrm{hom}(A,B)\) is called an isomorphism if there exists a \(g\in \mathrm{hom}(B,A)\) such that \(fg = \mathbb{1}_B\) and \(gf = \mathbb{1}_A\).
  • \(g\) is uniquely defined by \(f\) and is denoted \(f^{-1}\).
  • \((f^{-1})^{-1} = f\).
  • If \(f\) and \(h\) are isomorphisms, and \(fh\) is defined, then \(fh\) is also an isomorphism.

Groups are identifies with categories whose object classes are single-element sets and all morphisms are isomorphisms. A groupoid is a small category in which morphisms are isomorphisms.

• The dual category \(C^{\mathrm{op}}\) of \(C\) is defined by
  • \(\operatorname{ob} C^{\mathrm{op}} = \operatorname{ob} C\).
  • \(\mathrm{hom}_{C^{\mathrm{op}}}(A,B) = \mathrm{hom}_{C}(B,A)\).
  • Composition \(gf\) in \(C^{\mathrm{op}}\) is defined by \(fg\) in \(C\).
  • \(\mathbb{1}_A\) as in \(C\).
• The product category \(C\times D\) is defined by
  • \(\operatorname{ob} C\times D = \operatorname{ob}C \times \operatorname{ob} D\).
  • \(\mathrm{hom}_{C\times D}((A,A'),(B,B')) = \mathrm{hom}_{C}(A,B)\times \mathrm{hom}_{C}(A',B')\).
  • Composition defined component-wise.
  • \(\mathbb{1}_{(A,A')}\) defined component-wise.

LnOPnCh₂ Series

---

Structure

• LnO layer + PnCh₂ layer.
• Usually P4 symmetry.

Previous Work

LaOBiS₂

• Thermoelectric \(\ce{LaO_{1-x}F_{x}BiS2}{}\) (Omachi2014).
• Thermoelectric \(\ce{LaOBiS_{2-x}Se_x}{}\) (Mizuguchi2014)(Nishida2015).
  • \(\kappa \sim 1 \mathrm{W}\cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1}\).
  • \(zT\sim 0.36\) at 370°C.

LnOSbSe₂

• Thermoelectric \(\ce{LaO_{1-x}F_xSbSe2}{}\) and \(\ce{CeO_{1-x}F_xSbSe2}{}\) (Goto2018).
  • \(\ce{NdOSbS2}\) is likely an insulator.

CeOBiS₂

• Superconductor (Tanaka2017)(Miura2018).

Calculation

• \(\ce{LnOAsSe2}{}\), with predicted \(zT\sim 2\) (Ochi2017), synthesis failed due to the stable phase \(\ce{Ln4O4Se3}{}\) (Goto2018).
• \(\ce{NdOSbS2}\), already synthesized (Tanryverdiev1996), large PF (Ochi2017).
• Designing principles: (Ochi2017)
  • Small SOC;
  • Small Bi-Bi and large Bi-S hopping amplitudes;
  • Small on-site energy difference between the Bi-\(p_{x,y}{}\) orbitals and S-\(p_{x,y}{}\) orbitals.
  • \(\Rightarrow\) replacing Bi with lighter elements and S with heavier elements (putting them closer in the periodic table);
• Band structures of \(\ce{LnO_{1-x}F_x BiS2}{}\) calculated (Morice2016).
  • The change of rare earth does not affect the Fermi surface.
  • Conduction electrons are confined to the BiS₂ planes.

Proposal

(Ce,Pr)OBiS₂

• Also replacing S with Se.
• Thermoelectric properties should be similar to that of LaOBiS₂.

NdOSbS₂

• Try to make it a semiconductor.

LnOAsSe₂

• Try to synthesize.

Miscellany

• Web of Science: CeOBiS2.

Quantum Transport

---

Linear Response

Formulations

Kubo Formula

• System undergoes a perturbation \[\hat{H}' = \hat{V}(t) \Theta(t-t_0).\]
• Up to the first order, \[\langle \hat{A}(t) \rangle = \langle \hat{A} \rangle_0 - i \int_{t_0}^t \dd{t'} \langle [ \hat{A}(t), \hat{V}(t') ] \rangle_0.\]

Green-Kubo Relations

• Transport coefficient is given by \[\gamma = \int_0^\infty \langle \dot{A}(0) \dot{A}(t) \rangle \dd{t}.\]

Mesoscopic Transport

Landauer Formula

• Ideal conductors (two terminals) \[I = 2e \int_{\mu_2}^{\mu_1} v_k \dv{k}{\epsilon_k} \frac{\dd{\epsilon_k}}{2\pi} = \frac{2e}{h}(\mu_1 - \mu_2)\] and therefore \[G = \frac{I}{V} = \frac{2e^2}{h}.\]
• Inserting a mesoscopic conducting device with transmission coefficient \(T\), we find the Büttiker formula \[G = \frac{2e^2}{h}T.\] The conductivity of the device itself is given by the Landauer formula \[G = \frac{2e^2}{h}\frac{T}{1-T}.\]
• Büttiker formula \[I_p = \sum_q G_{pq}(V_q - V_p).\]

Useful Packages for Measurements

---

PyVISA

PyVISA controls your instruments using raw commands.

Keithley2600

Keithley2600 controls your Keithley 2600 instruments with abstraction.

PyMeasure

PyMeasure controls a wide range of instruments.

DMRG (I): Theory

トニカクタダシイ

---

Simple DMRG

Basic Theory

Quantum renormalization in real space applies only to single-particle systems.

On each step, the Hamiltonian undergoes a truncation \[H^S(M) = V^\dagger H(M\cdot M_S) V,\] where \(V\) contains the most significant \(M\) eigenstates of \[\rho^S (M\cdot M_S),\] given by the ground state of \[H(M\cdot M_S).\]

In the algorithm, we should keep track of the creation operators on the boundary sites.

Structure of Infinite-Size DMRG

1. Set up an initial chain of length \(L\).
2. Calculate the ground state of this chain.
3. Get the reduced density matrix \(\rho^S\).
4. Diagonalize \(\rho^S{}\) and keep the most significant \(M\) eigenstates as \(V\).
5. Reduce \(H\) to a matrix of size \(M\times M\), as well as the creation operators at the boundary.
6. Add a new site.
7. Find the ground state of the new system.
8. Return to step 3.

Structure of Finite-Size DMRG

1. Carry out infinite-size DMRG until the size reaches \(L\). During this step, store all the Hamiltonians and other necessary steps in the memory.
2. Reduce the matrices (Hamiltonian and creations operators) of \(S\) to size \(M\). Before doing the projection, we should first trace out the environment part.
3. Add two sites to \(H_S\), and reduce the environment to length \(L/2 - 2\).
4. The procedure of the last two steps is repeated until \(S\) has size \(L-3\). The matrices for \(S\) should still be stored for each size.
5. Reduce the size of \(S\). For the environment, the stored matrices are used, whereas for the system, the matrices are updated at eac hstep. Carry on until the system \(S\) has reached size \(1\).
6. Repeat the up and down sweeps of the previous steps until convergence has been achieved.

Matrix Product State

Prerequisites: for SVD and QR Decomposition, see Mathematics Miscellany.

Conventions

• \(M\) for general matrices;
• \(\Lambda\) and \(S\) for diagonal matrices;
• \(A\) for left-canonical ones;
• \(B\) for right-canonical ones.
• Label \(l\) in \(\ket{a_l}\) indicates the bond.
• Superscript \([l]\) in \(O^{[l]}\) indicates that the operator lives on site \(l\).
• OBS unless otherwise stated. Site index from \(1\) to \(L\) while bond index from \(1\) to \(L-1\).

Graphical Notations

Representations and Conversions

Classical Representation

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} c_{\sigma_1,\cdots,\sigma_L} \ket{\sigma_1, \cdots, \sigma_L}.\]

Left-Canonical

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} A^{\sigma_2} \cdots A^{\sigma_{L-1}} A^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L}\] where

• for every \(l\), \[\sum_{\sigma_l} A^{\sigma_l \dagger} A^{\sigma_l} = \mathbb{1};\]
• \(A^{\sigma_1}\)'s are row matrices while \(A^{\sigma_L}\)'s are column matrices.

From Classical Representation

\[ \begin{align*} c_{\sigma_1,\cdots,\sigma_L} &\xlongequal[\mathrm{reshape}]{} \Psi_{\sigma_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1}^{r_1} U_{\sigma_1,a_1} S_{a_1,a_1} (V^\dagger)_{a_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_1}_{a_1} = U_{\sigma_1,a_1}} \sum_{a_1}^{r_1} A^{\sigma_1}_{a_1} \Psi_{(a_1\sigma_2),(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1}^{r_1} \sum_{a_2}^{r_2} A^{\sigma_1}_{a_1} U_{(a_1\sigma_2),a_2} S_{a_2,a_2} (V^\dagger)_{a_2,(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_2}_{a_1,a_2} = U_{(a_1\sigma_2),a_2}} \sum_{a_1}^{r_1} \sum_{a_2}^{r_2} A^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \Psi_{(\sigma_2 a_3),(\sigma_4,\cdots,\sigma_L)} \\ &= \cdots \\ &= \sum_{a_1,\cdots,a_{L-1}} A^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \cdots A^{\sigma_{L-1}}_{a_{L-2},a_{L-1}} A^{\sigma_L}_{a_{L-1}} \end{align*} \]

From General MPS

\[ \begin{align*} \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_1,\cdots,a_L} M_{(\sigma_1,1),a_1} M^{\sigma_2}_{a_1,a_2} \cdots \ket{\sigma_1\cdots \sigma_L} &= \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_1,\cdots,a_L} \sum_{s_1} A_{(\sigma_1,1),s_1} S_{s_1,s_1} V^\dagger_{s_1,a_1} M^{\sigma_2}_{a_1,a_2} \cdots \ket{\sigma_1,\cdots,\sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} \sum_{a_2,\cdots,a_L} \sum_{s_1} A^{\sigma_1}_{1,s_1} \tilde{M}^{\sigma_2}_{s_1,a_2} M^{\sigma_3}_{a_2,a_3} \cdots \ket{\sigma_1,\cdots,\sigma_L} \\ &= \cdots. \end{align*} \]

From Vidal Notation

\[\qty(\Lambda^{[0]} \Gamma^{\sigma_1})\qty(\Lambda^{[1]} \Gamma^{\sigma_2})\qty(\Lambda^{[2]} \Gamma^{\sigma_3})\cdots \rightarrow A^{\sigma_1} A^{\sigma_2} A^{\sigma_3} \cdots.\]

Right-Canonical

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} B^{\sigma_1} B^{\sigma_2} \cdots B^{\sigma_{L-1}} B^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L}\] where

• for every \(l\), \[\sum_{\sigma_l} B^{\sigma_l} B^{\sigma_l \dagger} = \mathbb{1};\]
• \(B^{\sigma_1}\)'s are row matrices while \(B^{\sigma_L}\)'s are column matrices.

From Classical Representation

Similar to that of left-canonical ones.

From General MPS

Similar to that of left-canonical ones.

From Vidal Notation

\[\qty(\Gamma^{\sigma_1}\Lambda^{[1]}) \qty(\Gamma^{\sigma_2}\Lambda^{[2]}) \qty(\Gamma^{\sigma_3}\Lambda^{[3]})\cdots \rightarrow B^{\sigma_1} B^{\sigma_2} B^{\sigma_3} \cdots.\]

Mixed-Canonical

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_l} S B^{\sigma_{l+1}} \cdots B^{\sigma_L}.\]

From Classical Representation

\[ \begin{align*} c_{\sigma_1,\cdots,\sigma_L} &= \sum_{a_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{a_l} S_{a_l,a_l} (V^\dagger)_{a_l,(\sigma_{l+1},\cdots,\sigma_L)} \\ &= \sum_{a_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{a_l} S_{a_l,a_l} (B^{\sigma_{l+1}} \cdots B^{\sigma_L})_{a_l}. \end{align*} \]

Vidal Notation

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} \Gamma^{\sigma_1} \Lambda^{[1]} \Gamma^{\sigma_2} \Lambda^{[2]} \cdots \Gamma^{\sigma_{L-1}} \Lambda^{[L-1]} \Gamma^{\sigma_L} \ket{\sigma_1,\cdots,\sigma_L},\] where

• the states on block \(A\) and \(B\) given by \[ \begin{align*} \ket{a_l}_A &= \sum_{\sigma_1,\cdots,\sigma_l} \qty(\Gamma^{\sigma_1} \Lambda^{[1]} \Gamma^{\sigma_2} \cdots \Lambda^{[l-1]}\Gamma^{\sigma_l})_{a_l} \ket{\sigma_1,\cdots,\sigma_l}, \\ \ket{a_l}_B &= \sum_{\sigma_{l+1},\cdots,\sigma_L} \qty(\Gamma^{\sigma_{l+1}} \Lambda^{[l+1]} \Gamma^{\sigma_{l+2}} \cdots \Lambda^{[L-1]} \Gamma^{\sigma_L})_{a_l} \ket{\sigma_{l+1} \cdots \sigma_L} \end{align*} \] are orthonormal respectively.
• The reduced density matrices are given by \[\rho^{[l]}_A = \rho^{[l]}_B = \qty(\Lambda^{[l]})^2.\]

From Classical Representation

\[ \begin{align*} c_{\sigma_1,\cdots,\sigma_L} &\xlongequal[\mathrm{reshape}]{} \Psi_{\sigma_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1} A^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1}(V^\dagger)_{a_1,(\sigma_2,\cdots,\sigma_L)} \\ &\xlongequal[\mathrm{reshape}]{A^{\sigma_1}_{a_1} = \Gamma^{\sigma_1}_{a_1}} \sum_{a_1} \Gamma^{\sigma_1}_{a_1} \Psi_{(a_1\sigma_2),(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal{\mathrm{SVD}} \sum_{a_1,a_2} \Gamma^{\sigma_1}_{a_1} A^{\sigma_2}_{a_1,a_2} \Lambda^{[2]}_{a_2,a_2} (V^\dagger)_{a_2,(\sigma_3,\cdots,\sigma_L)} \\ &\xlongequal[\color{orange} \mathrm{problematic\ division}]{A^{\sigma_2}_{a_1,a_2} = \Lambda^{[1]}_{a_1,a_1}\Gamma^{\sigma_2}_{a_1,a_2}} \sum_{a_1,a_2} \Gamma^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1} \Gamma^{\sigma_2}_{a_1,a_2} \Psi_{(a_2,\sigma_3),(\sigma_4,\cdots,\sigma_L)} \\ &= \cdots \\ &= \sum_{a_1,\cdots,a_L} \Gamma^{\sigma_1}_{a_1} \Lambda^{[1]}_{a_1,a_1} \Gamma^{\sigma_2}_{a_1,a_2} \Lambda^{[2]}_{a_2,a_2} \cdots \Lambda^{[L-1]}_{a_{l-1},a_{l-1}} \Gamma^{\sigma_L}_{a_{l-1},a_l}. \end{align*} \]

• Substitution starting from the left: \[A^{\sigma_l}_{a_{l-1},a_l} = \Lambda^{[l-1]}_{a_{l-1},a_{l-1}} \Gamma^{\sigma_l}_{a_{l-1},a_l}.\]
• Substitution starting from the right: \[B^{\sigma_l}_{a_{l-1},a_l} = \Gamma^{\sigma_l}_{a_{l-1},a_l} \Lambda^{[l]}_{a_l,a_l}.\]
• Dummy \[\Lambda^{[0]} = \Lambda^{[L]} = 1.\]

From Left-Canonical

Similar to that of right-canonical ones.

From Right-Canonical

\[ \begin{align*} B^{\sigma_1} B^{\sigma_2} B^{\sigma_3} \cdots &\xlongequal{\mathrm{SVD}} (A^{\sigma_1} \Lambda^{[1]} V^\dagger) B^{\sigma_2} B^{\sigma_3} \cdots \\ &\xlongequal{A^{\sigma_1} = \Lambda^{[0]}\Gamma^{\sigma_1}} \Gamma^{\sigma_1} M^{\sigma_2} B^{\sigma_3} \cdots \\ &\xlongequal{\mathrm{SVD}} \Gamma^{\sigma_1} (A^{\sigma_2} \Lambda^{[2]}V^\dagger) B^{\sigma_3} \cdots \\ &\xlongequal[\color{orange}\mathrm{problematic\ division}]{A^{\sigma_2} = \Lambda^{[1]}\Gamma^{\sigma_2}} \Gamma^{\sigma_1} \Lambda^{[1]}\Gamma^{\sigma_2} M^{\sigma_3} B^{\sigma_3} \cdots. \end{align*} \]

Calculation With MPS

Block States

• Left block: defined recursively by \[ \begin{align*} \ket{a_l}_A &= \sum_{a_{l-1}} \sum_{\sigma_l} A^{\sigma_l}_{a_{l-1}, a_l} \ket{a_{l-1}}_A \ket{\sigma_l} \\ &= \cdots \\ &= \sum_{\sigma_1,\cdots,\sigma_l} (A^{\sigma_1} \cdots A^{\sigma_l})_{1,a_l} \ket{\sigma_1} \cdots \ket{\sigma_l}. \end{align*} \]
• Right block: defined recursively by \[ \begin{align*} \ket{a_{l}}_B &= \sum_{a_{l+1}} \sum_{\sigma_{l+1}} B^{\sigma_{l+1}}_{a_l,a_{l+1}} \ket{\sigma_{l+1}} \ket{a_{l+1}}_B \\ &= \cdots \\ &= \sum_{\sigma_{l+1}, \cdots, \sigma_L} (B^{\sigma_{l+1}} \cdots B^{\sigma_L})_{a_{l+1},1} \ket{\sigma_{l+1}} \cdots \ket{\sigma_L}. \end{align*} \]
• The \(\qty{\ket{a_l}_A}\) (as well as \(\qty{\ket{a_l}_B}\)) constructed in this way are orthonormal.
• A general state may be written as \[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_l} \Psi B^{\sigma_{l+1}} \cdots B^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L},\] where \(\Psi\) is a matrix.

Vidal Notation and DMRG

• Vidal notation to blocks: \[ \begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma) \Lambda^{[l]} \qty(\Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_l} \ket{a_l}_A s_{a_l} \ket{a_l}_B. \end{gather*} \]
• Vidal notation to one-site DMRG: \[ \begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma \Lambda^{[l]}) \qty(\Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_{l-1},a_l,\sigma_l} \ket{a_{l-1}}_A \Psi^{\sigma_l}_{a_{l-1},a_l} \ket{a_l}_B. \end{gather*} \]
• Vidal notation to two-site DMRG: \[ \begin{gather*} \qty(\Lambda^{[0]}\Gamma)\cdots \qty(\Lambda^{[l-2]}\Gamma)\qty(\Lambda^{[l-1]}\Gamma \Lambda^{[l]} \Gamma \Lambda^{[l+1]}) \qty(\Gamma \Lambda^{[l+2]}) \cdots \qty(\Gamma \Lambda^{L}) \\ \rightarrow \ket{\psi} = \sum_{a_{l-1},a_{l+1},\sigma_l,\sigma_{l+1}} \ket{a_{l-1}}_A \Psi^{\sigma_l\sigma_{l+1}}_{a_{l-1},a_{l+1}} \ket{a_{l+1}}_B. \end{gather*} \]

Matrix Elements

Notations

• \(\ket{\psi}\) as MPS: \[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} M^{\sigma_1} \cdots M^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L}.\]
• \(\ket{\phi}\) as MPS: \[\ket{\phi} = \sum_{\sigma_1,\cdots,\sigma_L} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\sigma_1} \cdots \ket{\sigma_L}.\]
• Operator living on site \(l\): \[\hat{O}^{[l]} = \sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} \ket{\sigma_l} \bra{\sigma'_l}.\]

Overlap

• \(C^{[0]} = 1\).
• \(C^{[l]}\) defined recursively by \[C^{[l]} = \sum_{\sigma_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma_l}.\]
  • Transfer operator: specifically for \(N=M\), \[\hat{E}^{[l]} = \sum_{a_{l-1},a'_{l-1}} \sum_{a_l,a'_l} \qty(\sum_{\sigma_l} N^{[l] \sigma_{l*}} \otimes M^{[l]\sigma_l})_{a_{l-1} a'_{l-1}, a_l a'_l} (\ket{a'_{l-1}} \bra{a_{l-1}})(\ket{a_l}\bra{a'_l}).\]
  • Reformulated with transfer opeartor: \[C^{[l]} = E^{[l]}\qty[C^{[l-1]}].\]
• Finally we have \[C^{[L]} = \bra{\phi}\ket{\psi}.\]

Matrix Element

• \(C^{[0]} = 1\).
• For \(l\neq i, j\), \(C^{[l]}\) is defined recursively by \[C^{[l]} = \sum_{\sigma_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma_l}.\]
• For \(l=i\) or \(l=j\), \(C^{[l]}\) is defined by \[C^{[l]} = \sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} N^{\sigma_l\dagger} C^{[l-1]} M^{\sigma'_l}.\]
  • Transfer operator: specifically for \(N=M\), \[\hat{E}^{[l]}_O = \sum_{a_{l-1},a'_{l-1}} \sum_{a_l,a'_l} \qty(\sum_{\sigma_l,\sigma'_l} O^{\sigma_l,\sigma'_l} N^{[l] \sigma_{l*}} \otimes M^{[l]\sigma_l})_{a_{l-1} a'_{l-1}, a_l a'_l} (\ket{a'_{l-1}} \bra{a_{l-1}})(\ket{a_l}\bra{a'_l}).\]
  • Reformulated with transfer opeartor: \[C^{[l]} = E^{[l]}\qty[C^{[l-1]}].\]
• Finally we have \[C^{[L]} = \bra{\phi} O^{[i]} O^{[j]} \ket{\psi}.\]

Properties of Transfer Matrices

• If \(N=M\) is a left-normalized or right-normalized matrix, then all eigenvalues of \(E\) satisfies \[\abs{\lambda} \le 1.\]

Compression of MPS

SVD Compression

• Starting from a left-canonical MPS.
• \(\tilde{O}\) denoting \(O\) truncated, \[ \begin{align*} \ket{\psi} &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{L-1}} USB^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &\rightarrow \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots (A^{\sigma_{L-1}} \tilde{U}\tilde{S})B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &= \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{L-2}} US B^{\sigma_{L-1}} B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L} \\ &\rightarrow \cdots \\ &\rightarrow \sum_{\sigma_1,\cdots,\sigma_L} (A^{\sigma_1} \cdots A^{\sigma_l} U) S (B^{\sigma_{l+1}}\cdots B^{\sigma_L}) \ket{\sigma_1\cdots \sigma_l} \ket{\sigma_{l+1}\cdots \sigma_L}. \end{align*} \]

Variational Compression

MPO

\[\hat O \ket{\psi} = \sum_{\vb*{\sigma}} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\vb*{\sigma}}.\]

• Left/right normalized by taking \((\sigma_l,\sigma'_l)\) as a large index.

\[\hat{O} \ket{\psi} = \sum_{\vb*{\sigma}} N^{\sigma_1} \cdots N^{\sigma_L} \ket{\vb*{\sigma}}\] where \[N^{\sigma_i} = \sum_{\sigma'_i} W^{\sigma_i \sigma'_i} \otimes M^{\sigma'_i}.\]

Ground State Calculation

MPO Representation of Hamiltonians

1. Hamiltonian defined by \[\hat{H} = \sum_{i=1}^{L-1} \qty(\frac{J}{2} \hat{S}_i^{+} \hat{S}_{i+1}^{-} + \frac{J}{2} \hat{S}_{i}^{-} \hat{S}_{i+1}^+ + J^z \hat{S}^z_{i} \hat{S}^z_{i+1}) - h\sum_i \hat{S}^z_i.\]
2. Hamiltonian represented as \[\hat{O} = \hat{W}^{[1]} \cdots \hat{W}^{[L]}\] where \(W^{[i]}\)'s are Operator-valued matrices \[\hat{W}_{bb'} = \sum_{\sigma \sigma'} W^{\sigma \sigma'}_{bb'} \ket{\sigma}\bra{\sigma}{}.\]
3. For \(2 \le l \le L-1\), the linear space that \(W^{[l]}\) acts on may be generated by \(\qty{\vb*{e}^{[l]}_{b'}}\) defined by \[ \begin{align*} O^{[l]}_1 &= \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_2 &= \hat{S}^+ \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_3 &= \hat{S}^- \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_4 &= \hat{S}^z \otimes \mathbb{1} \otimes \cdots \otimes \mathbb{1}, \\ O^{[l]}_5 &= \mathrm{sum\ of\ completed\ terms\ to\ the\ right}. \\ \end{align*} \]
4. We have the recurrence relation \[ \begin{align*} O^{[l-1]}_1 &= \mathbb{1} \otimes O^{[l]}_1, \\ O^{[l-1]}_2 &= \hat{S}^+ \otimes O^{[l]}_1, \\ O^{[l-1]}_3 &= \hat{S}^- \otimes O^{[l]}_1, \\ O^{[l-1]}_4 &= \hat{S}^{z} \otimes O^{[l]}_1, \\ O^{[l-1]}_1 &= -h \hat{S}^z \otimes O^{[l]}_1 + \frac{J}{2} \hat{S}^- \otimes O^{[l]}_2 + \frac{J}{2} \hat{S}^+ \otimes O^{[l]}_3 + J^z \hat{S}^z O^{[l]}_4 + \mathbb{1}\otimes O^{[l]}_5. \\ \end{align*} \]
5. We could therefore define \(W^{[l]}\) by \[ W^{[l]} = \begin{pmatrix} \hat{1} & & & & \\ \hat{S}^+ & & & & \\ \hat{S}^- & & & & \\ \hat{S}^z & & & & \\ -h\hat{S}^z & (J/2) \hat{S}^- & (J/2) \hat{S}^+ & J^z \hat{S}^z & \mathbb{1} \end{pmatrix}{} \] such that \[ \begin{pmatrix} O^{[l-1]}_1 \\ O^{[l-1]}_2 \\ O^{[l-1]}_3 \\ O^{[l-1]}_4 \\ O^{[l-1]}_5 \end{pmatrix} = \hat{W}^{[l]} \begin{pmatrix} O^{[l]}_1 \\ O^{[l]}_2 \\ O^{[l]}_3 \\ O^{[l]}_4 \\ O^{[l]}_5 \end{pmatrix}, \] where the multiplication in the dot product is given by kron.
6. At the border we have \[ \hat{W}^{[1]} = \begin{pmatrix} -h\hat{S}^z & (J/2) \hat{S}^- & (J/2) \hat{S}^+ & J^z \hat{S}^z & \mathbb{1} \end{pmatrix} \] and \[ \hat{W}^{[L]} = \begin{pmatrix} \mathbb{1} \\ \hat{S}^+ \\ \hat{S}^- \\ \hat{S}^z \\ -h\hat{S}^z \end{pmatrix}. \]

The lesson:

• It's not that hard to contruct \(W^{[l]}\). Instead of thinking \(W^{[l]}\) as matrices labeled by \(\sigma\) and \(\sigma'\), we take it as operator-valued matrices labeld by \(b\) and \(b'\).
• \(W^{[l]}\) is sparse.
• The MPO representation of Hamiltonian stores the exact Hamiltonian in a small space without compression.

Applying a Hamiltonian MPO to a Mixed Canonical State

\[\ket{\psi} = \sum_{\sigma_1,\cdots,\sigma_L} A^{\sigma_1} \cdots A^{\sigma_{l-1}} \Psi^{\sigma_l} B^{\sigma_{l+1}} \cdots B^{\sigma_L} \ket{\sigma_1 \cdots \sigma_L}.\]

\[\bra{a_{l-1} \sigma_l a_l} \hat{H} \ket{a'_{l-1}\sigma'_l a'_l} = \sum_{b_{l-1},b_l} L^{a_{l-1},a'_{l-1}}_{b_{l-1}} W^{\sigma_l \sigma'_l}_{b_{l-1},b_l} R^{a_l,a'_l}_{b_l}.\]

Iterative Ground State Search

1. Target: with a Lagrangian multiplier \(\lambda\), we minimize \[\bra{\psi} H \ket{\psi} - \lambda \bra{\psi} \ket{\psi}.\]

   

2. Iterative search: only \(M^{[l]}\) is allowed to vary.

   

3. Boost: we'd like
   1. \(M^{[l]}\)'s be already a good guess, and
   2. The matrices left and right are left-normalized and right-normalized, respectively, so that the above equation is a eigenvalue problem instead of a generalized one.

Algorithm

1. Start from some initial guess for \(\ket{\psi}\), right-normalized.
2. Calculate the \(R\)-expressions iteratively for all sites from \(L-1\) to \(1\).
3. Right sweep: sweeping from site \(l=1\) to \(L-1\).
   1. At each site, solve the eigenvalue problem with starting vector given by the current state, getting \(M^{\sigma_l}\).
   2. Obtain \(A^{\sigma_l}\) by SVD, with the remaining \(SV^\dagger\) multiplied to the \(M^{\sigma_{l+1}}\) to the right as the starting point of the eigensolver for the next site.
   3. Build iteratively the \(L\)-expression.
4. Left sweep: sweeping from site \(l=L\) to \(2\).
   1. Again, solve the eigenvalue problem at each site.
   2. Obtain \(B^{\sigma_l}\) by SVD, with the remaining \(US\) multiplied to the \(M^{\sigma_l}\) to the left.
   3. Build iteratively the \(R\)-expression.
5. Repeat left and right sweeps, until convergence is achieved. Convergence condition given by \[\bra{\psi} \hat{H}^2 \ket{\psi} - \qty(\bra{\psi} \hat{H} \ket{\psi})^2 \rightarrow 0.\]

Computational Tricks

Avoid Trapping

Add a correction to \(\hat{\rho}^{A\bullet}\) before doing SVD: \[\hat{\rho}^{A\bullet} = \tr_B \ket{\psi} \bra{\psi} + \alpha \sum_{b_l} \tr_B \hat{H}^{A\bullet}_{b_l}\ket{\psi} \bra{\psi} \hat{H}^{A\bullet}_{b_l},\] where \(\alpha\) takes small value \(O(10^{-4})\) and will be taken slowly to zero as sweeping goes on.

The correction term \(\hat{H}_{b_l}^{A\bullet}\ket{\psi}\) is evaluated as follows.

Miscellaneous

Entanglement

• Maximally entangled state: \(\rho \in H_A \otimes H_B\) has the form for each of the subsystems \[\begin{pmatrix} 1/n & & & \\ & 1/n & & \\ & & \ddots & \\ & & & 1/n \end{pmatrix}.\]

Mathematics Miscellany

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Matrix Decomposition

SVD

\[M = USV^\dagger\] where

• columns of \(U\) are orthonormal;
• \(S\) is diagonal with non-negative entries;
• rows of \(V^\dagger\) are orthonormal.

QR Decomposition

Full QR

\[M = QR\] where

• \(Q\) is unitary;
• \(R\) is upper-triangular.

Thin QR

\[M = Q_1 R_1\] where

• columns of \(Q_1\) are orthonormal;
• \(R_1\) is upper-triangular.

Kronecker Product

Properties

• Associativity: \[(A\otimes B) \otimes C = A \otimes (B\otimes C).\]
• Mixed-product: \[(A\otimes B)\cdot (C\otimes D) = (A\cdot C)\otimes (B\cdot D).\] In particular, \[A \otimes B = (A \otimes \mathbb{1}) \cdot (\mathbb{1}\otimes B).\]
• Inverse, Transpose, etc: \[\begin{gather*} (A\otimes B)^{-1} = A^{-1} \otimes B^{-1}, \\ (A\otimes B)^\dagger = A^\dagger \otimes B^\dagger. \end{gather*}{}\]
• Trace, Determinant, etc: \[\begin{gather*} \det (A_{n\times n}\otimes B_{m\times m}) = (\det A)^m (\det B)^n, \\ \tr (A \otimes B) = (\tr A)(\tr B). \end{gather*}{}\]
• The kron of two matrices of orthonormal columns/rows is again a matrix of orthogonal columns/rows, respectively.

Implementations

numpy.kron(a, b)
scipy.sparse.kron(a, b)
scipy.linalg.kron(a, b)

DMRG (II)

Implementations

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Simple DMRG

Simple DMRG Program

The Hamiltonian is given by the XXZ model \[ H = -\frac{1}{2} \sum_{j=1}^N (J \sigma_j^x \sigma_{j+1}^x + J \sigma_j^y \sigma_{j+1}^y + J_z \sigma_{j}^z \sigma_{j+1}^z). \]

Finite System

\(d\) is defined to be the number of states of a single site.

In line 40 and 41 two matrices are defined, \[ S^z_{1} = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad S^+_{1} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \] as well as \[ H_1 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. \] \(H_1\) will be used as the single-site Hamiltonian at initialization.

The two-site Hamiltonian is defined in line 45 to be \[ \begin{align*} H_2(S_1, S_2) &= \frac{J}{2} S^+_1 \otimes S^{-}_2 + \frac{J}{2} S^{-}_1 \otimes S^+_2 + J_z S^z_1 \otimes S^z_2 \\ &= J(S_1^x \otimes S_2^x + S_1^y \otimes S_2^y) + J_z S_1^z \otimes S_2^z. \end{align*} \]

Enlarging Block

The enlarging process renews the operaors as \[ \begin{align*} H &\rightarrow H \otimes \mathbb{1}_{d\times d} + H_2(S_{\mathrm{edge}}, S_1), \\ S_{\mathrm{edge}}^z &\rightarrow \mathbb{1}_{M\times M} \otimes S^z_1, \\ S_{\mathrm{edge}}^+ &\rightarrow \mathbb{1}_{M\times M} \otimes S^+_1. \end{align*} \]

Single DMRG Step

At each DMRG step, the following steps are carried out:

1. The system block and the environment block is enlarged once.
2. The superblock Hamiltonian is constructed as \[ H = H_{\mathrm{sys}} \otimes \mathbb{1}_{M_{\mathrm{env}}\times M_{\mathrm{env}}} + \mathbb{1}_{M_{\mathrm{sys}} \times M_{\mathrm{sys}}} \otimes H_{\mathrm{env}} + H_2(S_{\mathrm{sys,edge}}, S_{\mathrm{env,edge}}). \]
3. Calculate the lowest eigenvalue \(E\) and \(\psi\) of \(H\).
4. Calculate the projected density matrix: \[ \rho = \tr_E (\ket \psi \bra \psi). \]
5. Calculate the eigenvectors of the lowest \(m\) eigenvalues. If \(m\) is greater then the dimension of the matrix, we calculate all the eigenvectors.
6. Construct the transformation matrix \(T\), where the columns are given by the eigenvectors in the previous step. \(T\) is not necessarily a square matrix.
7. All the operators of the system, \(S_{\mathrm{sys}}\) and \(H_{\mathrm{sys}}\), should undergo a transformation \[ H \rightarrow T^\dagger H T. \]

We may take system and environment to be the same. This may reduce half of the calculations.

How is the projected density matrix calculated? Although \(\psi\) is a vector in the direct product of two Hilbert spaces, it was represented by a column vector returned from eigsh. The following snippet traces out the environment part:

psi = psi.reshape([m_sys, -1])
rho = np.dot(psi0, psi0.conjugate.transpose())

Hubbard Model on a Specific Sector

MPS

DMRG for MPS

Computational Tricks

Speed-Up

See Also

Simple DMRG

• GitHub.
• Documents.

DMRG101

• GitHub.
• Documents.

TeNPy

• Documents.

NetKet

• GitHub.

Statistical Mechanics (II)

Ensembles

接尾語になる名詞.

Sino-Japanese Words and Phrases

名詞

系統

系統 (ㄒㄧˋ ㄊㄨㄥˇ) sounds really like a homophonic translation of system, while 系【けい】統【とう】 doesn't. That raises the question of whether the word 系統 was first invented by Japanese people or Chinese people.

In Japanese

The first of the definitions of 系統 given in 日本国語大辞典 reads

一定の順序を追って続いている、統一のある繋がり。特に、一族の血統。血筋。

The first quotation in Japanese dates back to 1827, the end of 江戸時代, given in 日本外史.

略敘王室相家之系統、以備参観云

In Chinese

The second definition of 系統 given in 辞海 reads

始终一贯的条理, 有条不紊的顺序.

The first quotation in Chinese dates back to the Jiajing period, given in 青霞集 by 沈鍊 (1497–1557).

岂可以仁义而无节制, 恩威而无系统者哉!

Conclusion: No Conclusion

The word system is derived from Late Latin (from the 3rd to the 6th centuries CE).

It's really hard for me to draw a conclusion here. Note that there are multiple questions here:

• Who first invented the word 系統?
• Who first used 系統 as the translation of system?
• Is 系統 a homophonic translation of system?

動詞

例を挙げる/舉例

It looks really strange that in both languages the phrase to give an example reads literally to raise an example. Is it accidental, or this definition is borrowed from one language to another?

In Japanese

Definition 4 and 4.5 of 挙【あ】げる given in 日本国語大辞典 reads respectively

人によく見えるようにする。広く知られるようにする。

⇒実例、証拠などをはっきり表面に表す。

Quotations date back to 1785, in

「わざと悪事に一味して、まっかう手めを上げようため」〈浄・先代萩〉

as well as to 1787–1788, in

「歴史上の事実を挙げ」〈花間鶯〉

This definition gives rise to the usage 例を挙げる.

There are also compounds like 挙【きょ】例【れい】 or 挙【きょ】隅【ぐう】.

In Chinese

Definition 5 of 舉 in 王力古漢語字典 reads

提出, 稱引.

Quotations found in 論語·述而 (the 5th century BC)

舉一隅不以三隅反, 則不復也

in 韓非子·外儲說左上 (mid 3rd century BC)

遠舉則病繆, 近世則病傭. (遠舉上世之事，則患繆妄)

as well as in 史記·屈原賈生列傳 (c. 90 BC)

舉類邇而見義遠.

The usage in the first quotation gives rise to 舉隅.

Conclusion: the Coincidence in Accidental

The definition to put forward seems to arise naturally from its original definition to physically raise something. However, the compounds 挙【きょ】例【れい】 or 挙【きょ】隅【ぐう】 may be imported from Chinese.

广告与悬赏广告之法律性质

广告与要约

要约亦包括向不特定人之要约.

设定自动贩卖机即属向不特定多数人发出要约.

台湾法下的广告

73年台上字2540號 ... 惟查廣告文字僅為要約之引誘.

中国大陆法下的广告

合同法 第十四条 要约是希望和他人订立合同的意思表示，该意思表示应当符合下列规定：

（一）内容具体确定；

（二）表明经受要约人承诺，要约人即受该意思表示约束。

合同法 第十五条 要约邀请是希望他人向自己发出要约的意思表示。寄送的价目表、拍卖公告、招标公告、招股说明书、商业广告等为要约邀请。

商业广告的内容符合要约规定的，视为要约。

广告原则上属于要约引诱. 理由之一为避免给付障碍, 或发布人思虑不周而蒙受损害.

悬赏广告

悬赏广告之定性问题

不知广告而完成一定行为之人是否得请求报酬? 无行为能力人如何?

• 契约行为说 悬赏广告系对不特定人之要约, 经行为人完成一定行为予以承诺而成立契约.
• 单独行为说 广告人因其单方面意思而负担债务.

德国法下的悬赏广告

BGB将「悬赏广告」置于§8.11, 与契约, 无因管理, 不当得利并列为债之发生事由.

BGB第657条 通过公开通告, 对完成某行为, 特别是对产生结果悬赏之人, 负有向完成此行为之人支付报酬的义务, 即使行为人完成该行为时未考虑到此悬赏广告.

悬赏广告是为自己设定义务的单方行为, 属于契约原则之例外.

台湾法下的悬赏广告

台灣民法典第164條 以廣告聲明對完成一定行為之人給與報酬者，為懸賞廣告。廣告人對於完成該行為之人，負給付報酬之義務。

數人先後分別完成前項行為時，由最先完成該行為之人，取得報酬請求權；數人共同或同時分別完成行為時，由行為人共同取得報酬請求權。前項情形，廣告人善意給付報酬於最先通知之人時，其給付報酬之義務，即為消滅。

前三項規定，於不知有廣告而完成廣告所定行為之人，準用之。

單獨行為說 謂懸賞廣告乃單獨行為，廣告人因廣告之單獨意思表示負擔債務，惟其效力之發生，須俟指定行為完成，為廣告人負擔債務之停止條件。

契約說 謂懸賞廣告乃廣告人對於不特定人所為之要約，經行為人完成指定行為予以承諾而成立契約。

民國88年修訂第164條之立法理由 懸賞廣告之性質，有單獨行為與契約之不同立法例。為免理論爭議影響法律之適用，並使本法之體例與規定之內容一致，爰將第一項末段「對於不知有廣告...亦同」移列為第四項。並將「亦同」修正為「準用之」，以明示本法採取契約說。

学界在修法后仍对悬赏广告之定性存在争议.

日本法下的悬赏广告

民法第529条 ある行為をした者に一定の報酬を与える旨を広告した者（以下「懸賞広告者」という。）は、その行為をした者がその広告を知っていたかどうかにかかわらず、その者に対してその報酬を与える義務を負う。

平成29年改正前民法第529条 ある行為をした者に一定の報酬を与える旨を広告した者（以下この款において「懸賞広告者」という。）は、その行為をした者に対してその報酬を与える義務を負う。

Japanese Miscellaneous

Grammar

Therefore

See だから／従って／よって／故に.

だから

• 夕べはよく寝た。だから今日はとても調子がいい。
• ここは環境がいい。だから住みたいという人がとても多い。

従って

• この学校は進学率が高い。したがって志望者も多い。
• このホテルはサービスがいい。したがって料金も高い。

よって

• aはbに等しく、bはcに等しい。よってaとcは等しい。
• 貴殿はわが社の発展に多大な貢献をなされました。よってこれを賞します。

故に

• 日本は経済大国である。ゆえに外国から働きに来る人も多い。
• 我思う、ゆえに我あり。

Because

See から、で、ので、ために的区别是什么。？.

から

• 操作ミスから事故が生じた。
• 危ないから、窓から手を出さないでください。
• あんまり安いから、つい買う気になった。

ので

• 辛い物を食べたので、のどが渇いた。
• 朝が早かったので、ついうとうとする。
• 盆地なので、夏は暑い。

ため

• 雨のため、試合は中止になりました。
• 風が強かったために、昨日は船が出ませんでした。

However

See Japanese Buts: でも, しかし, ただ, ただし, ところが, が, けど, けれど, けれども, as well as 【だが】と【しかし】と【でも】の意味の違いと使い方の例文.

でも

• 今朝は大雨だった。でも、学校は休みにならなかった。

しかし

• 君の言い分はよく解るよ。しかし、それを周囲に理解してもらおうというのは難しいと思うな。

だが

• 今回は失敗をしてしまった。だが、次回は必ず成功すると確信している。

ただし

• これはすぐれた説だ。ただし疑えば疑える点もある。

Although

See Japanese Buts: でも, しかし, ただ, ただし, ところが, が, けど, けれど, けれども, also 逆接用法：「が」「けど」「くせに」「のに」.

が

• みんな行ったが、彼は行かなかった。
• 営業も大変だが、事務はもっと大変です。

のに

See 【JLPT N4】文法・例文：〜のに.

• トムさんは２年日本へ留学したのに、全然日本語が話せません。
• 昨日、早くたくさん寝たのに、まだ眠いです。

けど

• もう夜中だけど、もう少し勉強しよう。
• 食べたいけど、太るのが怖い。

くせに

• 知っているくせに、知らないふりをしている。
• あいつは馬鹿のくせに、偉そうにしてる。

If

See Differences among -たら、なら、-んだったら、-えば, etc.

と

• 右へ曲がると、パンやがあります。
• 春になると、お花見ができます。

ば

• 明日もし雨が降れば、どうしますか。

たら

たら is the 仮定形 of た.

• お家に帰ったら、連絡してください。
• パリに行ったら、凱旋門にも行ってみたい。
• 帰宅したら、必ずお風呂に入りなさい。
• 飲んだら乗るな。乗るなら飲むな。

なら

• スーパーに行くのなら、しょうゆを買ってきて。
• 旅行に行くのなら、カメラを持っていくといいですよ。
• 飲んだら乗るな。乗るなら飲むな。

In Other Words

詰まり

• 地図の上方、詰まり北方は山岳地帯である。

即ち

• 日本の首都、すなわち東京

Regardless of

See に限らず／によらず／を問わず.

を問わず

• 休日、平日を問わず、一年中人出が絶えない。
• 経験の有無を問わず、人材を広く世に求める。

に関わらず

• 晴雨に関わらず実施する。

によらず

「によらず」は、「何」「だれ」などの不定称の代名詞を受けて、どの…に対しても区別をつけないで一様に、どんな…でもみな、という意味を表わす。

• 妹は何によらず臆病なところが心配です。
• だれによらず人の不始末の尻ぬぐいなどしたくはない。

Limited to, Only

だけ

• 千円だけ貸してあげる。
• 好きなだけ食べた。
• 勉強すればしただけ成績は上がる。

のみ

• あとは結果を待つのみである。
• 日本のみならず全世界の問題だ。

ばかり

• 彼は勉強ばかりしている。
• 彼は子供とばかり遊んでいる。
• 心ばかりの贈り物。

Not Limited to

に限らず

• 彼女は肉に限らず動物性たんぱくはいっさいとらない。
• 目上の人と話す場合に限らず、人との付き合いには敬語は欠かせない。

As One Can

See What is the difference between 出来る限り and 出来るだけ?.

だけ

• できるだけ多くの本を読みなさい。

限り

• 私はできる限りあなたの援助をします。

To Help

あげる/くれる/もらう

To be filled.

Common Verbs

To Close

See Five different verbs meaning “to close” with the same kanji (閉).

閉める

To physically close something.

• ドアを閉める。
• 店を閉める。

閉まる

For something to physically close.

• ドアが閉まる。
• 店が閉まる。

閉じる

For something to close or to come to an end.

• ドアが閉じる。
• 店が閉じる。
• 会が閉じる。
• 目を閉じる。
• 傘を閉じる。
• 歴史を閉じる。

閉ざす

To (figuratively) close.

• 道を閉ざす。
• 門を閉ざす。
• 国を閉ざす。

To Walk

See 歩く vs 歩む and 歩く／歩む.

歩く

「歩く」は、足を使って前に進む意。具体的な動作を表わす。

歩む

「歩む」は、物事が進行する意。具体的な動作を表わすよりは、抽象的な意味で用いられるのが普通。

Miscellaneous

Vocabulary

• お浚【さら】い 名詞
  • review
  • rehearsal
• 余【あま】り 副詞
  • not much
• 詰【つ】まり 副詞
  • to wit, that is to say
• 更【さら】に 副詞
  • moreover
• 即【すなわ】ち 副詞
  • namely, i.e.
• 若【も】し 副詞
  • if, supposing
• 従【したが】って 接続詞
  • therefore
• 耳【のみ】 助詞
  • only, limited to
• 坩堝【るつぼ】
  • Crucible.

Grammar

• ておる or とる
  • to be (doing): indicates a progressive or continuous sense
• ている or てる
  • indicating the present continuous tense
  • indicating the resulting state of an action
• としても
  • even if
  • Usage:
    • 動詞普通形＋としても
    • イ形容詞普通形＋としても
    • ナ形語幹＋（だ／である）としても
    • 名詞＋（だ／である）としても
• にわたって
  • to reach a certain point
• AをBにする
  • with A as B, with A at B, or using A as B
• なので/ので
  • as a result
• には , also
  • emphatic
  • in order to
• が欠かせない/に欠かせない
  • to be indispensable
• ておく
  • to do something in advance
  • to leave something undone
• AをBとする/AをBとして , also , , or
  • with A as B, with A at B, or using A as B
  • to proceed to perform a new action (by quitting what one is currently doing)
  • (volitional form) + として: trying to do
  • assuming
• になる/となる
  • to become, to be determined as
• によらず , かかわらず
  • regardless of
• わけ/わけだ , also , or
  • no wonder
  • to emphasize
• くらい
  • about, approximately
• かどうか
  • whether or not
• について
  • about, in concern with
• 又【また】は/若【も】しくは
  • or
• 意志形としている
  • To try to.
  • Something is gonna ...
• のだ/のです
  • To provide additional information.
• 大雑把
  • Rough, approximate.
• 大変役に立つ
  • Very useful.
• というのも
  • (At the beginning of a sentence) because.
• 役割を果たす
  • To play a role.
• 必ずしもXわけではない
  • Not necessarily X.
• Xざるを得ない
  • To have to do X.
• って "女の子の名前で雪花って変ですか？" "de" and "tte" usage
  • An informal form of は (topic marker).
• たりして ～たりして
  • Something that seems unlikely but might be possible.
• おかげで
  • Thanks to.

Phrases

• で与【あた】えられる
  • be given by
• 次のように
  • as follows
• 代表無くして課税なし
  • No Taxation Without Representation

外来語

• ランドセル
  • A type of backpack worn by Japanese elementary schoolchildren.

Onomatopoeia

• キュンキュン
  • Heartwrenchingly.

Remarks on Etymology

• て and is the 連用形 of つ.
• た itself is a 終止形, with たら as its 仮定形. た is from たる, a contraction of てあり.
• だ is either from である or なり, with なら as its 仮定形 of the latter etymology.

Final-Over-Final Constraint

See 语言学中都有哪些定律？. Japanese is the epitome of a head final language.

Remarks on Translation

Phono-Semantic Matching

See Phono-Semantic Matching.

• 倶【く】楽【ら】部【ぶ】 for club.
• 合【かっ】羽【ぱ】 for capa (Portuguese).

Semantic Loan

See 意味借用.

• 遊【あそ】ぶ: 「離れた土地に行って学ぶ」 from 漢文.
• 襲【おそ】う: 「地位を受け継ぐ」 from 漢文.

Sino-Japanese Words

See 《语词的漂移：近代以来中日之间的知识互动与共有》——陈力卫, as well as 如何看待《驳所谓〈离开了日本外来词，中国人无法说话〉的言论》？

Models for Topological Insulators

「二次元のメインヒロインも好きだけど、でも、三次元【ほんもの】の加藤恵が一番好きなんだよ」

Category

Foundations

Dirac Matrices

• One-dimensional: \[ \alpha_x = \sigma_x, \quad \beta = \sigma_z. \]
• Two-dimensional: \[ \alpha_x = \sigma_x,\quad \alpha_y = \sigma_y,\quad \beta = \sigma_z. \]
• Three-dimensional: \[ \alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix},\quad \beta = \begin{pmatrix} \sigma_0 & 0 \\ 0 & -\sigma_0. \end{pmatrix} \]

Quantum Mechanics

• Time reversal: \[ \Theta = i\sigma_y K. \]
• Parity: in the Dirac equation, we let \[ P = \pi \beta \] such that \[ P\alpha_i P = -\alpha_i,\quad P\beta P = \beta, \] i.e. the Dirac equation is invariant under \(P\).

Second Quantization

• Fourier transformation: \[ \begin{align*} c_{i,\sigma} &= \frac{1}{\sqrt{Na}} \sum_{k_n} e^{ik_n R_i} c_{k_n,\sigma}, \\ c_{i,\sigma}^\dagger &= \frac{1}{\sqrt{Na}} \sum_{k_n} e^{-ik_n R_i} c_{k_n,\sigma}^\dagger. \end{align*} \]
• Hamiltonian diagonalized: \[ H = \sum_{k_n} \epsilon(k_n) c^\dagger_{k_n,\sigma} c_{k_n,\sigma}. \]

From Continuous Models via Replacement: It Works but I Don't Know Why

• Replacements: \[ \begin{align*} k_i &\rightarrow \frac{1}{a} \sin k_i a, \\ k_i^2 &\rightarrow \frac{4}{a^2} \sin^2 \frac{k_i a}{2}. \end{align*} \]

Periodic Hamiltonian

• Periodicity condition: \[ H(k, t+T) = H(k, t). \]
• Time-reversal invariants: at \[ t = 0 + nT;\quad t = T/2 + nT \] we have \[ \Theta H(t) \Theta^{-1} = H(t). \]

Evolution of Electric Polarization

• Evolution of Polarization under Periodic Hamiltonian: \[ \Delta P_\alpha = e\sum_n \int_0^T \dd{t} \int_{\mathrm{BZ}} \frac{\dd{\vb*{q}}}{(2\pi)^d} \Omega^n_{q_\alpha,t}. \]
• Curvature: \[ \Omega^n_{q_\alpha,t} = \partial_{q_\alpha} A^n_t - \partial_t \vb*{A}^n_{q_\alpha}. \]
  • I'm not sure what \(\vb*{A}^n_t\) mean. But if I were to guess, it should be \[ \vb*{A}^n_{t} = \bra{u_n} \frac{\partial}{\partial t} \ket{u_n}. \]
• Polarization with Wannier functions: \[ \vb*{P} = -e \sum_n \bra{\vb*{R} = 0, n} \vb*{r} \ket{\vb*{R} = 0, n} = -\frac{e}{2\pi} \oint \dd{\vb*{k}} \cdot \vb*{A}(\vb*{k}). \]

Minimal Lattice Models

• \(c_i^\dagger\) denotes the row vector \[ \begin{pmatrix} c_{i,1}^\dagger & \cdots & c_{i,D}^\dagger \end{pmatrix} \] where \(D\) is the dimension of the Dirac matrices.
• \(\qty(\beta,\vb*{\alpha})\) denote Dirac matrices.
• Model Hamiltonian:
  • \(k\)-space: \[ H = \frac{\hbar v}{a} \sum_{i} \sin(k_i a) \alpha_i + \qty(mv^2 - B\frac{4\hbar^2}{a^2}\sum_i \sin^2 \frac{k_i a}{2})\beta. \]
  • Sites: \[ H = \sum_i \Delta c_i^\dagger \beta c_i - t\sum_{\langle i,j\rangle} c_j^\dagger\beta c_i + i t' \sum_{i,a} \qty[c^\dagger_{i+a} \alpha_a c_i - c_i^\dagger \alpha_a c_{i+a}] \]
    • Subscript \(i+a\) denotes the site \(\vb*{R}_i + \vb*{R}_a\).
    • Model parameters: with \(a=1\) and \(\hbar=1\), \[ \begin{align*} t' &= \frac{\hbar v}{2a} = \frac{v}{2}, \\ \Delta &= mv^2 + 2dt, \\ t &= -\frac{B\hbar^2}{a^2} = -B. \end{align*}{} \]

One-Dimensional Lattice Model

• Hamiltonian:
  • Second quantized: \[ H = \Delta \sum_{j=1}^N c^\dagger_j \sigma_z c_j - t\sum_{j=1}^{N-1} \qty(c^\dagger_{j+1} \sigma c_j + c^\dagger_j \sigma_z c_{j+1}) + it' \sum_{j=1}^{N-1} \qty{c^\dagger_{j+1} \sigma_x c_j - c_j^\dagger \sigma_x c_{j+1}}. \]
  • Matrix form: \[ H = \begin{pmatrix} \Delta \sigma_z & T & 0 & 0 & \cdots & 0 \\ T^\dagger & \Delta \sigma_z & T & 0 & \cdots & 0 \\ 0 & T^\dagger & \Delta \sigma_z & T & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & T^\dagger & \Delta \sigma_z & T \\ 0 & 0 & 0 & 0 & T^\dagger & \Delta \sigma_z \end{pmatrix}, \] where \[ T = -t\sigma_z - it' \sigma_x. \]
• Spcecial case: \(N=\infty\), i.e. semi-infinite chain with one end at \(j=1\).
  • Component form: \[ \Delta \sigma_z \Psi_j + T\Psi_{j+1} + T^\dagger\Psi_{j-1} = E\Psi_j. \]
  • Ansatz: \[ \Psi_{j+1} = \lambda \Psi_j = \lambda^{j+1} \Psi, \] where \(\lambda\) is a scalar.
  • Solution: \(s=\pm1\), \[ \Psi = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -is \end{pmatrix},\quad \lambda_\pm(s) = \frac{\Delta}{2(t+st')}\qty[1\pm \sqrt{1-\frac{4(t^2 - t'^2)}{\Delta^2}}]. \]
  • End state: \(\abs{\lambda_\pm}\le 1\) as \(\Psi_j\rightarrow 0\), \[ s = \operatorname{sign}(t'/t). \]
    • \(\lambda\pm\) complex: \[ 4(t^2 - t'^2) > \Delta^2. \]
    • \(\lambda_\pm\) real: for end state to exist, \[ 4(t^2 - t'^2) < \Delta^2 < 4t^2. \]
    • With boundary condition \(\Psi_0 = 0\), \[ \Psi_j = (\lambda^j_+ - \lambda^j_-) \Psi. \]
• Special case: \(N\) finite, \(\Delta = 0\) and \(t=t'\),
  • Two solutions: \[ \Psi_L = \begin{pmatrix} \varphi_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\quad \Psi_R = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ \varphi_N \end{pmatrix}, \] where \[ \varphi_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix},\quad \varphi_N = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}. \]

Structure of Solution (One-Dimensional)

• Hamiltonian: invariant under parity, \[ H = A \sin(k_x a) \alpha_x + \qty(\Delta - 4B \sin^2 \frac{k_x a}{2})\beta. \]
• Dispersion relations: doubly degenerate, \[ E_\pm = \pm \sqrt{A^2 \sin^2 (k_x a) + \qty(\Delta - 4B \sin^2 \frac{k_x a}{2})^2}. \]
  • The two occupied states of \(E<0\) are time-reversal partners.
  • One of the occupied solutions: \[ \psi_1 = \begin{pmatrix} { -\frac{A \sin k_x a}{\sqrt{2E_+ (E_+ + \Delta - 4B \sin^2 (k_x a/2))}} } \\ 0 \\ 0 \\ { \frac{\Delta - 4B \sin^2(k_x a/2) + E_+}{\sqrt{2E_+ (E_+ + \Delta - 4B \sin^2 (k_x a/2))}} } \end{pmatrix}. \]
• Time-reversal invariant momenta: \[\Gamma_1 = k = 0\] and \[\Gamma_2 = k = \frac{\pi}{a}.\]
• Eigenvalues under parity: \[ \bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sin^2 \frac{\Gamma_i a}{2}). \]
  • Eigenvalues at time-reserval invariant momenta: \[ \delta\vert_{1} = \operatorname{sign}(-\Delta),\quad \delta\vert_{2} = \operatorname{sign}(-\Delta + 4B). \]
  • Parity changes sign at \(\Delta = 0\) and \(\Delta = 4B\), where the gap closes.
  • \(Z_2\) index: \(\nu\) defined by \[ (-1)^\nu = \operatorname{sign}(\Delta) \operatorname{sign}{(\Delta - 4B)}. \]
  • Topologically nontrivial: \(\nu = 1\) for \[ 0 < \Delta^2 < 4\Delta B. \]

Two-Dimensinal Lattice Model

Integer Quantum Hall Effect

• Effective Hamiltonian: \[ H = \vb*{d}(\vb*{k}) \cdot \sigma, \] where \[ \begin{align*} d_x &= A \sin k_x a, \\ d_y &= A \sin k_y a, \\ d_z &= \Delta - 4B \sin^2 \frac{k_x a}{2} - 4B \sin^2 \frac{k_y a}{2}. \end{align*} \]
• Dispersion relations: \[ E_{k,\pm} = \pm\abs{\vb*{d}(\vb*{k})}. \]
• For zeros to exist: i.e. topological phase transition,
  • \(\Delta = 0\) with \[ k_x a = k_y a = 0. \]
  • \(\Delta = 4B\) with \[ k_x a = 0,\quad k_y a = \pi,\quad \text{or}\quad k_x a = \pi = \pi,\quad k_y a = 0. \]
  • \(\Delta = 8B\) with \[ k_x a = k_y a = \pi. \]
• Edge states: infinite ribbon along \(x\) while open boundary along \(y\),
  • Hamiltonian: \(k\)-space on \(x\) and site on \(y\), \[ \begin{align*} H(k_x) &= \sum_{j=1}^N c^\dagger_{k_x,j} h_{j,j}(k_x) c_{k_x,j} \\ &\phantom{{}={}} + \sum_{j=1}^{N-1} \qty[c^\dagger_{k_x,j} h_{j,j+1}(k_x) c_{k_x,j+1} + c^\dagger_{k_x,j+1}h_{j+1,j}(k_x)c_{k_x,j}], \end{align*}{} \] where \[ \begin{align*} h_{j,j}(k_x) &= A\sin k_x \sigma_x + \qty(\Delta - 2B - 4B \sin^2\frac{k_x a}{2}) \sigma_z, \\ h_{j,j+1}(k_x) &= B\sigma_z - \frac{i}{2} A\sigma_y, \\ h_{j+1,j}(k_x) &= B\sigma_z + \frac{i}{2}A\sigma{y}. \end{align*}{} \]
  • Problem reduced: to one-dimensional case for a specific \(k_x\).

Quantum Spin Hall Effect

• Time reversal pair glued together: \[ \begin{align*} H_{\mathrm{QSHE}} &= \begin{pmatrix} \vb*{d}(\vb*{k})\cdot \sigma & 0 \\ 0 & -\vb*{d}(\vb*{k})\cdot \sigma \end{pmatrix} \\ &= A \sin (k_x a) \sigma_x \otimes \sigma_0 + A \sin (k_y a) \sigma_y \otimes \sigma_0 \\ &{\phantom{{}={}}} + \qty(\Delta - 4B \sin^2\frac{k_x a}{2} - 4B \sin^2 \frac{k_y a}{2}) \sigma_z \otimes \sigma_0. \end{align*} \]
• Edge states persist: checked numerically, even when spin-orbit coupling introduced.

Structure of Solution (Two-Dimensional)

• Hamiltonian: \[ H = A \sum_{i=x,y} \sin(k_i a)\alpha_i + \qty(\Delta - 4B \sum_{i=x,y} \sin^2 \frac{k_i a}{2})\beta. \]
• Dispersion relations: doubly degenerate, \[ E_\pm = \pm \sqrt{A^2 \sum_{i=x,y} \sin^2 k_i a + \qty(\Delta - 4B \sum_{i=x,y} \sin^2 \frac{k_i a}{2})^2}. \]
  • The two occupied states of \(E<0\) are time-reversal partners.
  • One of the occupied solutions: \[ \psi_1 = \begin{pmatrix} \frac{-A(\sin (k_x a) - i\sin (k_y a))}{\sqrt{2E_+(E_+ + \Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2))) }} \\ 0 \\ 0 \\ \frac{\Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2)) + E_+}{\sqrt{2E_+(E_+ + \Delta - 4B(\sin^2(k_x a/2) + \sin^2(k_y a/2))) }} \end{pmatrix}. \]
• Eigenvalues under parity: \[ \bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sum_{i=x,y} \sin^2 \frac{\Gamma_i a}{2}). \]
  • Eigenvalues at time-reversal invariant momenta: \[ \begin{align*} \delta\vert_{(0,0)} &= -\operatorname{sign}(\Delta), \\ \delta\vert_{(0,\pi)} &= \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,0)} &= \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,\pi)} &= \operatorname{sign}(-\Delta + 8B). \end{align*} \]
  • \(Z_2\) index: \(\nu\) defined by \[ (-1)^\nu = \operatorname{sign}(\Delta) \operatorname{sign}(-\Delta + 4B)^2 \operatorname{sign} (-\Delta + 8B). \]
  • Topologically nontrivial: \(\nu=1\) for \[ 0 < \Delta^2 < 8\Delta B. \]

Three-Dimensional Lattice Model

• \(k\)-Space Hamiltonian: \[ H = A \sum_{i} \alpha_i \sin(k_i a) + \beta \qty(\Delta - 4B \sum_{i} \sin^2 \frac{k_i a}{2}). \]
• Dispersion: \[ E_{k,\pm} = \pm \sqrt{A^2 \sum_{i} \sin^2(k_i a) + \qty(\Delta - 4B \sum_{i} \sin^2 \frac{k_i a}{2})^2}. \]
• For zeros to exist: i.e. topological phase transition:
  • \(\Delta = 0\),
  • \(\Delta = 4B\),
  • \(\Delta = 8B\).
• Surface states: semi-infinite \(x\)-\(y\) plane,
  • Fourier transformation on \(x\) and \(y\): \[ \begin{align*} c_{k_x,k_y,j_z} &= \frac{1}{\sqrt{N_x N_y}} \sum_{j_x, j_y} c_{j_x,j_y,j_z} \exp\qty[i(k_x j_x + k_y j_y)], \\ c_{j_x,j_y,j_z} &= \frac{1}{\sqrt{N_x N_y}} \sum_{k_x, k_y} c_{k_x,k_y,k_z} \exp\qty[-i(k_x j_x + k_y j_y)]. \end{align*}{} \]
    • \(c_{k_x,k_y,j_z}\) is a four-component spinor.
  • One-dimensional effective Hamiltonian: \[ \begin{align*} H(k_x,k_y) &= \sum_i c^\dagger_{k_x,k_y,j_z} \epsilon(k_x,k_y) c_{k_x,k_y,j_z} \\ &{\phantom{{}={}}} + \sum_i \qty[c^\dagger_{k_x,k_y,j_z+1} \qty(i\frac{A}{2}\alpha_z - 2B\beta) c_{k_x,k_y,j_z} + \mathrm{h.c.}], \end{align*} \] where \[ \epsilon(k_x,k_y) = (A \sin k_x \alpha_x + A \sin k_y \alpha_y) + \qty(\Delta - 2B - 4B\sum_{i=x,y} \sin^2 \frac{k_i a}{2}) \beta. \]
  • Surface state found by exact diagonalization.

Structure of Solution (Three-Dimensional)

• Hamiltonian: \[ H = A \sum_{\alpha=x,y,z} \sin(k_\alpha a) \alpha_\alpha + \qty(\Delta - 4B \sum_{\alpha = x,y,z} \sin^2 \frac{k_a a}{2})\beta. \]
• Dispersion relations: doubly degenerate, \[ E_\pm = \pm \sqrt{A^2 \sum_{i=x,y,z} \sin^2 k_i a + \qty(\Delta - 4B \sum_{i=x,y,z} \sin^2 \frac{k_i a}{2})^2}. \]
  • The two occupied states of \(E<0\) are time-reversal partners.
  • One of the occupied solutions: \[ \psi_1 = \begin{pmatrix} \frac{-A(\sin k_x a - i \sin k_y a)}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \\ \frac{A \sin k_z}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \\ 0 \\ \frac{\Delta - 4B \sum_{\alpha=x,y,z} \sin^2(k_\alpha a/2) + E_+}{\sqrt{2E_+(E_+ + \Delta - 4B( \sum_{\alpha=x,y,z}\sin^2(k_\alpha a/2) )) }} \end{pmatrix}. \]
• Eigenvalued under parity: \[ \bra{\psi_1} P \ket{\psi_1} = \operatorname{sign}\qty(-\Delta + 4B \sum_{\alpha=x,y,z} \sin^2 \frac{\Gamma_i a}{2}). \]
  • Eigenvalues at time-reversal invariant momenta: \[ \begin{align*} \delta\vert_{(0,0,0)} &= -\operatorname{sign}(\Delta), \\ \delta\vert_{(0,0,\pi)} &= \delta\vert_{(0,\pi,0)} = \delta\vert_{(\pi,0,0)} = \operatorname{sign}(-\Delta + 4B), \\ \delta\vert_{(\pi,\pi,0)} &= \delta\vert_{(\pi,0,\pi)} = \delta\vert_{(0,\pi,\pi)} = \operatorname{sign}(-\Delta + 8B), \\ \delta\vert_{(\pi,\pi,\pi)} &= \operatorname{sign}(-\Delta + 12B). \end{align*} \]
  • \(Z_2\) index: a tuple of \(\mathbb{Z}_2\) elements, \[ \begin{align*} (-1)^{\nu_1} &= \operatorname{sign}(\Delta) \operatorname{sign}(\Delta - 8B), \\ (-1)^{\nu_2} &= \operatorname{sign}(\Delta) \operatorname{sign}(\Delta - 8B), \\ (-1)^{\nu_3} &= \operatorname{sign}(\Delta) \operatorname{sign} (\Delta - 8B), \\ (-1)^{\nu_1'} &= \operatorname{sign}(\Delta - 4B) \operatorname{sign}(\Delta - 12B), \\ (-1)^{\nu_0} &= (-1)^{\nu_1 + \nu'_1}. \end{align*} \]
  • Topological indices: \[ \begin{align*} (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;0,0,0),\quad \text{for}\quad \Delta < 0, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (1;1,1,1),\quad \text{for}\quad 0 < \Delta < 4B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;1,1,1),\quad \text{for}\quad 4B < \Delta < 8B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (1;0,0,0),\quad \text{for}\quad 8B < \Delta < 12B, \\ (\nu_0;\nu_1,\nu_2,\nu_3) &= (0;0,0,0),\quad \text{for}\quad \Delta > 12B. \end{align*} \]

One-Dimensional Cases

Thouless Charge Pump: Rice-Mele Model

• Hamiltonian:
  • Site: \[ H = h_{\mathrm{st}}(t) \sum_i (-1)^i c^\dagger_i c_i + \qty(\frac{1}{2} \sum_{i=1}^N \qty[t_0 + \delta(t) (-1)^i] c^\dagger_i c_{i+1} + \mathrm{h.c.}). \] where \[ \begin{align*} \delta(t) &= \delta_0 \cos \frac{2\pi t}{T}, \\ h_{\mathrm{st}}(t) &= h_0 \sin \frac{2\pi t}{T}. \end{align*} \]
  • \(k\)-space: \[ H = \sum_k \begin{pmatrix} a^\dagger_k \\ b^\dagger_k \end{pmatrix} \vb*{d}(k,t) \cdot \vb*{\sigma} \begin{pmatrix} a_k \\ b_k \end{pmatrix}, \] where \[ \begin{align*} d_x(k,t) &= \frac{1}{2}(t_0 + \delta(t)) + \frac{1}{2}(t_0 - \delta(t)) \cos k, \\ d_y(k,t) &= -\frac{1}{2}(t_0 - \delta(t)) \sin k, \\ d_z(k,t) &= h_{\mathrm{s.t.}}(t). \end{align*} \]
• \(k\)-space annhilation: \[ \begin{align*} a_k &= \frac{1}{\sqrt{N}} \sum_{j\in 2n} c_j e^{-ikj}, \\ b_k &= \frac{1}{\sqrt{N}} \sum_{j\in 2n+1} c_j e^{-ikj}. \end{align*} \]
• Dispersion relations: \[ \epsilon_\pm(k,t) = \pm\abs{\mathcal{d}(k,t)}. \]
• Evolution of polarization: \[ \Delta P = n_c e a \] where \[ \begin{align*} n_c &= \int_0^T \dd{t} \int_{\mathrm{BZ}} \frac{\dd{k}}{2\pi} \Omega^n_{k,t} \\ &= -\frac{1}{4\pi} \int \dd{k} \int_0^T \frac{\vb*{d}(k,t) \cdot \qty[\partial_k \vb*{d}(k,t) \times \partial_t \vb*{d}(k,t)]}{\abs{\vb*{d}(k,t)}^3} \\ &= -\operatorname{sign}(t_0 h_0 \delta_0). \end{align*} \]

Fu-Kane Spin Pump

• Hamiltonian:
  • Site: \[ H = h_{\mathrm{st}}(t) \sum_{i,\sigma,\sigma'} (-1)^i c^\dagger_{i,\sigma} \sigma^z_{\sigma \sigma'} c_{i,\sigma'} + \qty(\frac{1}{2} \sum_{i,\sigma} \qty[t_0 + \delta(t) (-1)^i] c^\dagger_{i,\sigma} c_{i+1,\sigma'} + \mathrm{h.c.}). \]
  • \(k\)-space: \[ H = \sum_k \begin{pmatrix} \phi^\dagger_{k,\uparrow} & \phi^\dagger_{k,\downarrow} \end{pmatrix} \begin{pmatrix} \vb*{d}_+ \cdot \vb*{\sigma} ^ 0 \\ 0 & \vb*{d}_- \cdot \vb*{\sigma} \end{pmatrix} \begin{pmatrix} \phi_{k,\uparrow} \\ \phi_{k,\downarrow} \end{pmatrix}, \] where \[ \begin{align*} d_x(k,t)_\pm &= \frac{1}{2}(t_0 + \delta(t)) + \frac{1}{2}(t_0 - \delta(t)) \cos k, \\ d_y(k,t)_\pm &= -\frac{1}{2}(t_0 - \delta(t)) \sin k, \\ d_z(k,t)_\pm &= \pm h_{\mathrm{s.t.}}(t). \end{align*} \]
• \(k\)-space creation: \[ \phi^\dagger_{k,\sigma} = \begin{pmatrix} a^\dagger_{k,\sigma} & b^\dagger_{k,\sigma} \end{pmatrix}. \]
• Evolution of polarization: polarizations contributed by opposite spins cancel out, \[ \Delta P_{\uparrow} = +ea;\quad \Delta P_{\downarrow} = -ea. \]

CuSeI Compounds

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Chalcogenido-Halogenido-Bismuthates

• Cu/Bi/Q/I (where Q = S, Se, Te): a rich family.
• Small band gap, tunable by substituting anions with more or less electronegative ones.
• Cu⁺ disordered and mobile + heavy elements + a comparatively high electrical conductivity ⇒ good thermoelectric properties.

Cu₄BiSe₄I

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• Band gap: (Literature) 0.1meV at low T, 0.3eV at high T (Topological Materials Database) 0.02eV Semiconductor TI(NLC)
• Projection of the unit cell:
• Bi-Se strands: comparison with Te-I strands,
• Conductivity:
• Copper cation conductivity suppressed.

CuBi₂Se₃I, Cu₃Bi₆S₁₀I

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• Band gap (CuBi₂Se₃I): (Literature) ∼1.2eV Semiconductor
• Band gap (Cu₃Bi₆S₁₀I): (Literature) ∼1.2eV Semiconductor
• Band structure: CuBi₂Se₃I (left) and Cu₃Bi₆S₁₀I (right)
• Thermoelectric of CuBi₂Se₃I: Seeback coefficient 460-575μV/K, electrical conductivity 0.02S/cm, thermal conductivity 0.22W/(m·K).
• Structure of CuBi₂Se₃I: two distinct layers alternating along the b-axis,
• Structure of Cu₃Bi₆S₁₀I: three individual columns,

Cu⁎Bi⁎Se⁎Br⁎

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• Band gap: Unavailable
• Structure: built up of two alternating types of layered modules, A and B.
  • A: consists of paired rods [Bi⁎Se⁎X⁎].
• Copper ion transport along [010] in the S-Br case.
• \(\ce{Cu_{1.5}Bi_{2.64}S_{3.42}Br_{2.58}}{}\), \(\ce{Cu_{1.57}Bi_{2.37}Se_{2.68}Br_{3.32}}{}\): Seeback coefficient 280μV/K, electrical resistivity 23Ω/m, thermal conductivity 0.77W/(m·K), \[ ZT = 1.3\times 10^{-6}. \]
  • Structure of \(\ce{Cu_{1.5}Bi_{2.64}S_{3.42}Br_{2.58}}{}\):
• \(\ce{Cu_{1.57}Bi_{4.69}Se_{7.64}I_{0.36}}{}\), \(\ce{Cu_{2.31}Bi_{5}Se_{8.31}I_{0.69}}{}\): thermoelectric not given.
  • Structure of \(\ce{Cu_{1.57}Bi_{4.69}Se_{7.64}I_{0.36}}{}\):

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P⁎Se⁎ Cages

• Cages:
  • P₄Se₃ cages (left) and P₄Se₄ cages (right),
  • P₈Se₃,
  • Possibly high Cu⁺ ion conductivity.

(CuCl)P₄Se₃, (CuI)P₄Se₃, (CuBr)₃(P₄Se₃)₂, (CuI)₃(P₄Se₃)₂

• Download Supporting Material
• Band gap of (CuI)P₄Se₃: (AFLOW) 1.37eV Semiconductor (Topological Materials Database) 0.82eV Semiconductor
• Cage molecules P₄Se₃ introduced into CuX networks.
• Section of 2D structure of (CuCl)P₄Se₃: [CuCl]ₙ chains along the c-axis,
• Section of 3D structure of (CuBr)₃(P₄Se₃)₂:
• Section of 3D structure of (CuI)₃(P₄Se₃)₂:
• Section of 3D structure of (CuI)P₄Se₃: structured as one-dimensional polymers,

(CuI)P₄Se₄

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• Band gap: (AFLOW) 1.75eV Semiconductor (Topological Materials Database) 1.61eV Semiconductor
• Cage molecules P₄Se₄ introduced into CuI networks.
• P₄Se₄ strands connected by CuI.
• Layered structure, connected by vdW interactions.

(CuI)₂P₈Se₃

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• Band gap: (Topological Materials Database) 1.2eV Semiconductor
• Structure:

(CuI)₃P₄Se₄

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• Band gap: Unavailable
• Cage molecules P₄Se₄ introduced into CuI networks.
• Projection of the crystal structure: stacked along the c-axis,

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Others

CuSeTeCl, CuSeTeBr, CuSeTeI

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• Band gap: Unavailable
• Layered structure, with one-dimensional infinite screws of Se and Te.

Cu⁎SiS⁎I⁎, Cu⁎GeSe⁎I⁎

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• Band gap: Unavailable
• Copper argyrodites, ion conductors.

Statistical Physics (I)

Boltzmann Distribution

Combinatorics

• Counting microscopic states: \[ \Omega_{\mathrm{M.R.}} = \frac{N!}{\prod_l a_l!} \prod_l \omega_l^{a_l}. \]
• Boltzmann distribution: \[ a_l = \omega_l e^{-\alpha - \beta \epsilon_l}. \]

The Partition Function

• Partition function: of a single particle, \[ Z_l = \sum_l w_l e^{-\beta \epsilon_l}. \]
• Single particle free energy: \[ f_1 = -\ln Z_1. \]
• Generalized force: \[ Y = \frac{N}{\beta}\pdv{}{y} \ln Z_1. \]
  • \[ \delta \epsilon_l = f_1 \cdot \delta y. \]
  • \[ Y = \sum_l a_l f_l. \]
  • Work done: \[ \dd{W} = \sum_l a_l \dd{\epsilon_l}. \]
• Internal energy: \[ \begin{align*} U &= \frac{N}{Z_1} \sum_l \epsilon_l w_l e^{-\beta\epsilon_l} \\ &= -N \pdv{}{\beta} \ln Z_1 \\ &= N \pdv{f_1}{\beta}. \end{align*} \]
• Helmholtz free energy: \[ F = NkTf_1. \]
• Entropy: for Boltzmann distribution, \[ S = Nk\qty(\beta \pdv{f_1}{\beta} - f_1). \]
  • In terms of microscopic states: \[ S = k \ln \Omega_{\mathrm{M.B.}}. \]
  • For Fermi-Dirac and Bose-Einstein distributions, under classical limit \[ S_{\mathrm{B.E.}} = S_{\mathrm{F.D.}} = k \ln \frac{\Omega_{\mathrm{M.B.}}}{N!}. \]
• Condition for classical limit: \[ e^\alpha = \frac{z_1}{N} \gg 1. \]

For the \((p,V)\) pair we find \[ Y = -p = -\frac{N}{\beta} \pdv{}{V}\ln Z_1 = \frac{N}{\beta}\pdv{}{V} f_1. \]

Application to Ideal Gas

• Nearly independent system: phase space described by the single particle \[ (x,y,z,p_x,p_y,p_z). \]
• Energy levels: \[ \epsilon = \frac{p^2}{2m}. \]
• Single particle partition function: for monoatomic gas, \[ f_1 = -\ln V + \frac{3}{2}\ln \beta - \frac{3}{2} \ln \qty(\frac{2\pi m}{h^2}). \]
• EoS: \[ p = \frac{NkT}{V}. \]
• Internal energy: \[ U = \frac{3}{2} NkT. \]

Velocity Distribution

• Maxwell's velocity distribution: for vector, \[ f(\vb*{v}) = \qty(\frac{m}{2\pi kT})^{3/2} e^{-m\vb*{v}^2/(2kT)}. \]
• Maxwell's velocity distribution: for magnitude, \[ f(v) = \sqrt{\frac{2}{\pi}} \qty(\frac{m}{kT})^{3/2} v^2 e^{-mv^2/(2kT)}. \]
• Statistical: \[ \begin{align*} v_{\mathrm{m}} &= \sqrt{\frac{2kT}{m}}, \\ \overline{v} &= \sqrt{\frac{8kT}{m}}, \\ v_{\mathrm{rms}} &= \sqrt{\frac{3kT}{m}}. \end{align*}{} \]

Equipartition Theorem

• For every square mode in the kinetic energy, \[ \epsilon_{\vb*{p}} = \frac{1}{2} a_i \vb*{p}_i^2, \] where \(a_i\) may or may not depends on \(\vb*{q}\), and for every square mode in the potential energy \[ \frac{1}{2} b_i q_i^2, \] where \(b_i\) may or may not depends on other \(\vb*{q}_j\)'s, the average value is given by \[ \overline{\frac{1}{2} c x^2} = \frac{1}{2} kT. \]

This is spam. I am not gonna remove it because 1) this is a problem that many bloggers may encounter; 2) this funny spammer seems to forget to attach his link. See this quora question for why such comments occur.

Group Theory in Solid State Physics

Space Group

• Symmorphic space group: a semidirect product of its point group with its translation subgroup.
• Nonsymmorphic space group: not symmorphic.
  • May contain glide planes, where the translation is not a lattice vector.

Representations of Space Groups

• Small representation: small representation for the group of wavevector \(\vb*{k}\): \[ D^{\Gamma_i}_{\vb*{k}}(\qty{g_{\vb*{k}} | \vb*{t}}) = e^{i\vb*{k}\cdot \vb*{t}} D^{\Gamma_i}(g_{\vb*{k}}). \]

Example: Cubic Lattices (Symmorphic Space Group)

Brillouin Zones of Cubic Lattices

Symmetry at Γ

• The whole point group \(\mathrm{O}_h\).
  • Large representation.
• Representation: superscript \(\pm\) denotes parity,
  • \(C_4\) and \(C_4^2\) around \(\qty{100}{}\).
  • \(C_2'\) around \(\qty{110}{}\).
  • \(C_3\) around \(\qty{111}{}\).

Symmetry at X

• Symmetry \(D_{4h}{}\).
  • Small representation.
• Representation:
  • \(C_{4\parallel}^2\) around \(\Gamma X\).
  • \(C_{4\perp}^2\) perpendicular to \(\Gamma X\).

Symmetry at M

• Symmetry \(D_{4h}{}\).
  • Small representation.
• Representation: almost the same as that at \(X\), except that the conjugacy classes are relabeled.

Symmetry at R

• Symmetry \(\mathrm{O}_{h}\).
  • Large representation.
• Representation: almost the same as that at \(\Gamma\), except that the IRs are relabeled.

Symmetry Along Δ

• Symmetry \(C_{4v}{}\).
  • Small representation.
• Representation:

• Compatibility relation: \[ \Gamma_{25}^+ \rightarrow (\Delta'_2) \oplus (\Delta_5) \rightarrow (X_3) \oplus (X_5). \]
  • It's possible that: the band admits a three-fold degeneracy at \(\Gamma\), where the wavefunctions span a space of dimension \(3\), transforming under \(\mathrm{O}_h\) according to the representation of \(\Gamma_{25}^+\),
  • as we move off \(\Gamma\) along \(\Delta\), the band splits into two, one of which transforms under \(C_{4v}{}\) according to \(\Delta'_2\), another (2-fold degenerate) according to \(\Delta_5\), and
  • as we reach \(X\), the bands remain split, where the one-fold that originally transforms according to \(\Delta'_2\) now transforms according to \(X_3\), while the two-fold that originally transforms according to \(\Delta_5\) now transforms according to \(X_5\).

Symmetry Along Σ

• Symmetry \(C_{2v}{}\).
  • Small representation.
• Representation:

• Compatibility relation: \[ \Gamma^+_{25} \rightarrow (\Sigma_1) \oplus (\Sigma_2 \oplus \Sigma_3) \rightarrow (M_1) \oplus (M_5). \]
  • It's possible that: the band admits a three-fold degeneracy at \(\Gamma\), where the wavefunctions span a space of dimension \(3\), transforming under \(\mathrm{O}_h\) according to the representation of \(\Gamma_{25}^+\),
  • as we move off \(\Gamma\) along \(\Sigma\), the band splits into three, one of which transforms under \(C_{2v}{}\) according to \(\Sigma_1\), one \(\Sigma_2\), and another \(\Sigma_3\), and
  • as we reach \(M\), the bands suddenly become partly degenerate, where the one-fold that originally transforms according to \(\Sigma_1\) now transforms according to \(M_1\), while the two that originally transforms separately according to \(\Sigma_2\) and \(\Sigma_3\) now become degenerate and combine to give rise to a space of dimension \(2\) that transforms according to \(M_5\).

Symmetry Along Λ

• Symmetry \(C_{3v}{}\).
  • Small representation.
• Representation:

• Compatibility relation: \[ \Gamma_{25}^+ \rightarrow (\Lambda_1 \oplus \Lambda_3) \rightarrow (R_{15}). \]
  • It's possible that: the band admits a three-fold degeneracy at \(\Gamma\), where the wavefunctions span a space of dimension \(3\), transforming under \(\mathrm{O}_h\) according to the representation of \(\Gamma_{25}^+\),
  • as we move off \(\Gamma\) along \(\Delta\), the band splits into two, one of which transforms under \(C_{3v}{}\) according to \(\Lambda_1\), another (2-fold degenerate) according to \(\Lambda_3\), and
  • as we reach \(R\), the bands suddenly become degenerate, returning to a three-fold degeneracy (i.e. a space of dimension \(3\)) that transforms according to \(R_{15}\).

Symmetry Along S

• Symmetry \(C_{2v}{}\).
  • Small representation.
• Representation: the same as that long \(\Sigma\).

Compatibility Relations

It seems better to take \(\vb*{k}\) as an external parameter like \(\vb*{E}\) in the Stark effect. Therefore, the band structure is just the energy levels as we enumerate through all \(\vb*{k}\)'s.

Tight-binding Basis

• With the \(s\)-orbital: transforming like identity.
• In SC: \[ \Gamma_1 \rightarrow \Lambda_1 \rightarrow R_1 \rightarrow X_1 \rightarrow \Delta_1 \rightarrow \Gamma_1. \]

• Lower band: \[ R_1 \rightarrow \Lambda_1 \rightarrow \Gamma_1^+ \rightarrow \Delta_1 \rightarrow X_1. \]
• Upper band: \[ (R_{12}^+) \oplus (R_1^+) \rightarrow (\Lambda_3) \oplus (\Lambda_1) \rightarrow \Gamma_{25}^+ \rightarrow (\Delta'_2) \oplus (\Delta_5) \rightarrow (X'_2) \oplus (X_5). \]

Physicists Suck at Maths

Abuse of Terminology

Representation Theory

• Physicists always say that a vector space \(V\) is a representation of a group \(G\). This completely makes no sense to someone who really knows representation theory. A representation is always a homomorphism from \(G\) to linear transformations on \(V\). Specifying \(V\) is not equivalent to specifying the representation. However, physicists usually work with the natural representation \[g\psi(x) = \psi(g^{-1} x)\] where \[V = \set{\ket{\psi}}.\] In this sense, specifying \(V\) does determine the representation.

Algebra

• Physicists tend to write every set in pair form \((x,y)\) as a direct product, though in many cases they are not.

Representation of Lie Groups

SU(3)は君の嘘

Lie Groups

Haar Measure

• Left Haar measure: left-invariant volume form \(\sigma\).
  • Wedge product of the duals of Lie algebras.
  • Guaranteed that \[ \int_{gU} \sigma = \int_U \sigma. \]
• Right Haar measure: right-invariant volume form \(\sigma\).
• Haar measure: both left- and right-invariant.

• \[ A(1) = \set{\begin{pmatrix} x & y \\ 0 & 1 \end{pmatrix}}. \]
• Lie algebras (left-invariant vector fields): \[ X = x\pdv{}{x},\quad Y = x\pdv{}{y}. \]
• Dual: \[ \sigma = \frac{\dd{x}\wedge \dd{y}}{x^2}. \]

Lie Algebra

• Casimir operator: an operator which commutes with all elements of a Lie group.
• Casimir operators are represented by scalar matrices.
  • See Schur's lemma.

T(1)

Lie Algebra of T(1)

• Generators: \(P\).
• Infinitesimal transform: \[ T(\dd{x}) = \mathbb{1} - iP\dd{x}. \]
• Exponential map: \[ T(x) = e^{-iPx}. \]
• IR of the Lie algebra: \[ P = p. \]

Irreducible Representations of T(1)

• IRs: \[ U^p(x)\ket{p} = e^{-ipx}\ket{p}. \]
• Haar measure: \(\dd{x}\).
• Orthonormality relation: \[ \int_{-\infty}^{\infty} \dd{x} U^\dagger_p(x) U^{p'}(x) = N \delta(p-p'). \]
• Completeness relation: \[ \int_{-\infty}^{\infty} \dd{p} U^p(x) U^\dagger_p(x') = N \delta(x-x'). \]

Conjugate Basis Vectors to T(1)

• \(\ket{O}\) expanded as \[ \bra{p}\ket{O} = 1. \]
• \[\ket{x} = U(x)\ket{O}.\]
• Basis transformation \[ \ket{x} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \ket{p} e^{-ipx} \dd{p} \Leftrightarrow \ket{p} = \int_{-\infty}^{\infty} \ket{x} e^{ipx} \dd{\phi}. \]
• Wavefunction transformation \[ \psi(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \psi(p) e^{ipx} \dd{p} \Leftrightarrow \psi(p) = \int_{-\infty}^{\infty} \psi(x) e^{-ipx} \dd{x}. \]
• Action of \(P\): \[ P\ket{x} = i\dv{}{x} \ket{x}. \]

SO(2)

Lie Algebra of SO(2)

• Generators: \(J\).
• Infinitesimal transform: \[ R(\dd{\phi}) = \mathbb{1} - iJ\dd{\phi}. \]
• Exponential map: \[ R(\phi) = e^{-i\phi J}. \]
• IR of the Lie algebra: \(m\in \mathbb{Z}\), \[ J = m. \]

Irreducible Representations of SO(2)

• IRs: \(m\in \mathbb{Z}\), \[ U^m(\phi)\ket{m} = e^{-im\phi}\ket{m}. \]
• Haar measure: \(\dd{\phi}\).
• Orthonormality relation: \[ \frac{1}{2\pi} \int_0^{2\pi} \dd{\phi} U^\dagger_n(\phi) U^m(\phi) = \delta^m_n. \]
• Completeness relation: \[ \sum_n U^n(\phi) U^\dagger_n(\phi') = \delta(\phi - \phi'). \]
• Multivalued representations: \[ U_{n/m}(\phi) = e^{-in\phi/m}. \]

Conjugate Basis Vectors to SO(2)

• \(\ket{O}\) expanded as \[ \bra{m}\ket{O} = 1. \]
• \[\ket{\phi} = U(\phi)\ket{O}.\]
• Basis transformation \[ \ket{\phi} = \sum_m \ket{m} e^{-im\phi} \Leftrightarrow \ket{m} = \frac{1}{2\pi} \int_0^{2\pi} \ket{\phi} e^{im\phi} \dd{\phi}. \]
• Wavefunction transformation \[ \psi(\phi) = \sum_m e^{im\phi} \psi_m \Leftrightarrow \psi_m = \frac{1}{2\pi}\int_0^{2\pi} \psi(\phi) e^{-im\phi} \dd{\phi}. \]
• Action of \(J\): \[ J\ket{\phi} = i\dv{}{\phi} \ket{\phi}. \]

SO(3)

Parametrization of SO(3)

• Angle-and-axis: \[ R(\psi,\hat{\vb*{n}}) = R(\psi,\theta,\phi). \]
• Euler angles: body frame and laboratory frame, \[ R(\alpha,\beta,\gamma) = R_{z''}(\gamma) R_{y'}(\beta) R_z(\alpha) = R_z(\alpha) R_y(\beta) R_z(\gamma). \]
• Conversion: \[ \begin{align*} \phi &= \frac{\pi + \alpha - \gamma}{2}, \\ \tan\theta &= \frac{\tan (\beta/2)}{\tan (\gamma+\alpha)/2}, \\ \cos\psi &= 2\cos^2 \frac{\beta}{2}\cos^2\qty(\frac{\alpha+\gamma}{2}) - 1. \end{align*}{} \]

Lie Algebra of SO(3)

• Generators: \(J_1,J_2,J_3\), \[ [J_k, J_l] = i\epsilon_{klm} J^m. \]
• Infinitesimal transform: \[ R(\dd{\vb*{\psi}}) = \mathbb{1} - i \vb*{J}\cdot \dd{\vb*{\psi}}. \]
• Exponential map: \[ R(\psi\vb*{n}) = e^{-i\psi \vb*{J}\cdot \vb*{n}} = e^{-i\alpha J_3} e^{-i\beta J_2} e^{-i\gamma J_3}. \]
• Casimir operator: \(J^2\).
• IR of the Lie algebra: \[ \begin{align*} J^2\ket{jm} &= j(j+1)\ket{jm}, \\ J_z \ket{jm} &= m\ket{jm}, \\ J_\pm \ket{jm} &= \sqrt{j(j+1) - m(m\pm 1)} \ket{j,m\pm 1}. \end{align*} \]

Irreducible Representations of SO(3)

• IRs: \[ U^j(\alpha,beta,\gamma)\ket{jm} = D^j(\alpha,\beta,\gamma){^{m'}}{_m} \ket{jm'}. \]
  • \[ D^j(\alpha,\beta,\gamma){^{m'}}{_m} = e^{-i\alpha m'} d^{j}(\beta){^{m'}}{_m} e^{-i\gamma m}. \]
  • \[ d^j(\beta){^{m'}}{_m} = \bra{jm'} e^{-i\beta J_y} \ket{jm}. \]
  • Single-valued for \(j\) a positive integer. Double-valued for \(j\) a half-odd integer.
  • \(d\) is a real matrix. \(D\) is an special unitary matrix.
  • Complex conjugation: \[ D^*(\alpha,\beta,\gamma) = Y D(\alpha,\beta,\gamma) Y^{-1}. \]
    • \[ Y = D[R_2(\pi)],\quad Y^{m'}{_m} = \delta^{m'}_{-m}(-1)^{j-m}. \]
  • Symmetry \[ d^j(\beta){^{m'}}{_m} = d^j(\beta){^{-m'}}{_{-m}}(-1)^{m' - m}. \]
• Characters: \[ \chi^j(\psi) = \frac{\sin(j+1/2)\psi}{\sin (\psi/2)}. \]

Spherical Harmonics

• \[ Y_{lm}(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}} \qty[D^l(\phi,\theta,0){^m}{_0}]^*. \]

Conjugate Basis Vectors to SO(3)

Position Space

• \(\ket{r\hat{\vb*{z}}}\) expanded as \[ \bra{r\hat{\vb*{z}}}\ket{E,l,m'} = \delta_{m' 0} \tilde{\psi}_{El}(r). \]
• \[ \ket{r\hat{\vb*{n}}} = e^{-i\phi J_z} e^{-i\theta J_y} \ket{r\hat{\vb*{z}}}. \]
• Wavefunction \[ \psi_{Elm}(r,\theta,\phi) = \tilde{\psi}_{El}(r)[D^l(\phi,\theta,0){^m}{_0}]^* = \psi_{El}(r) Y_{lm}(\theta,\phi). \]

Momentum Space

• Basis transformation \[ \ket{p,l,m} = \int \dd{\Omega} \ket{p,\theta,\phi}Y_{lm}(\theta,\phi) \Leftrightarrow \ket{p,\theta,\phi} = \sum_{l,m} \ket{p,l,m} Y^*_{lm}(\theta,\phi). \]

Clebsch-Gordan Coefficients

Clebsch-Gordan Coefficients of SO(3)

• Direct product representation: \[ U(R) \ket{m,m'} = \ket{n,n'} D^j(R){^n}{_m} D^{j'}(R){^{n'}}{_{m'}}. \]
  • Single-valued if \(j+j'\) is an integer.
  • Double-valued if \(j+j'\) is a half-odd integer.
• Direct product representation of the Lie algebra: \[ J^{j\times j'}_n = J_n^j \otimes \mathbb{1}^{j'} + \mathbb{1}^j \otimes J_n^{j'}. \]
• Eigenvectors labeled by \[\qty{J^2, J_z}.\]
• Ket of max \(j\) and \(m\) matches: \[ \ket{J = j+j', M = j + j'} = \ket{m=j, m'=j'}, \]
• \(\ket{J,M}\) generated by lowering: \[ J_- \ket{J,M} = \sqrt{J(J+1) - M(M-1)} \ket{J, M-1}. \]
• There are only two states of \(M=J-1\). One of them is already obtained, and the other is \[ \ket{J-1, J-1}. \]
• By repeating the process, we obtain all the \(\ket{JM}\) kets.
• CG coefficients: \[ \begin{align*} \ket{JM} &= \ket{mm'}\langle mm'(jj')JM\rangle, \\ \ket{mm'} &= \ket{JM}\langle JM(jj')mm'\rangle. \end{align*}{} \]
  • Convention: \[ \begin{align*} \langle mm'(jj') JM\rangle &\in \mathbb{R}, \\ \langle j,J-j(jj')JJ\rangle &> 0. \end{align*}{} \]
• Properties of CG coefficients:
  • Orthogonality: \[ \sum_{mm'} \langle JM(jj')mm'\rangle \langle mm' (jj')J'M'\rangle = \delta^{J}_{J'}\delta^M_{M'}. \]
  • Completeness: \[ \sum_{JM} \langle mm'(jj')JM\rangle \langle JM(jj')\rangle nn'\rangle = \delta^m_n \delta^{m'}_{n'}. \]
  • Symmetry: \[ \langle mm'(jj')JM\rangle = (-1)^{j-J+m'}\langle M,-m'(Jj')jm\rangle \sqrt{(2J+1)(2j+1)}. \]
• Reduction of products: \[ \begin{align*} D^j(R){^m}{_n} D^{j'}(R){^{m'}}{_{n'}} &= \sum_{J,M,N} \langle mm'(jj')JM\rangle D^J(R){^M}{_N} \langle JN(jj')nn'\rangle, \\ \delta^J_{J'}D^J(R)^M{_{M'}} &= \sum_{mm'nn'} \langle JM(jj')mm'\rangle D^j(R){^m}{_n} D^{j'}(R)^{m'}{_{n'}} \langle nn'(jj')J'M'\rangle. \end{align*}{} \]

Applications

Partial Wave Expansion

• T-matrix: \[ \bra{\vb*{p}_f} T \ket{\vb*{p}_i} = \bra{p,\theta,\phi} T \ket{p,0,0}. \]
• \(T\) is invariant under rotation: \[ \bra{p,l,m} T \ket{p,l',m'} = \delta_{ll'} \delta_{mm'} T_l(p). \]
• Partial wave expansion: \[ \bra{\vb*{p}_f} T \ket{\vb*{p}_i} = \sum_l \frac{2l+1}{4\pi} T_l(E) P_l(\cos\theta). \]

Transformation of Wavefunctions

• \[ \psi'(\vb*{x}) = \psi(R^{-1} \vb*{x}). \]

Transformation of Wavefunctions under T

• \[ \psi'(\vb*{x}) = \psi(R^{-1} \vb*{x}) = e^{i\vb*{p}\cdot R^{-1}\vb*{x}} = e^{i(R\vb*{p})\cdot \vb*{x}}. \]

Transformation of Wavefunctions under SO(3)

• \[ \psi'(x) = \psi_{El}(r) Y_{lm'}(\hat{\vb*{x}}) D^l[R]{^{m'}}{_m}. \]

Pauli Spinor Wavefunctions

• Expansion of wavefunctions: \[ \ket{\psi} = \sum_\sigma \int \ket{\vb*{x},\sigma} \psi^\sigma(\vb*{x}) \dd{^3 x}. \]
• Transformation of basis under rotation: \[ U[R]\ket{\vb*{x},\sigma} = \ket{R\vb*{x},\lambda} D^{1/2}[R]{^\lambda}{_\sigma}. \]
• Transformation of wavefunction under rotation: \[ \psi'{^\lambda}(\vb*{x}) = D^{1/2}[R]^{\lambda}{_\sigma} \psi^\sigma(R^{-1} \vb*{x}). \]

Irreducible Wavefunctions, Irreducible Tensors, and Wigner-Eckart Theorem

• Irreducible wavefunctions and fields: a set of multi-component functions \[ \set{\phi^m(\vb*{x}) | m = -j,\cdots,j} \] that transform under rotation as \[ \phi \xrightarrow{R} \phi';\quad \phi'^m(\vb*{x}) = D^j[R]{^m}{_n} \phi^n(R^{-1} x). \]
• Irreducible tensors: a set of multi-component operators \[ \set{O^s_\lambda | \lambda = -s,\cdots,s} \] that transform under rotation as \[ U(R) O^s_\lambda U(R)^{-1} = \sum_{\lambda'} O^s_{\lambda'} D^s(R){^{\lambda'}}{_\lambda}. \]
  • Lie brackets of irreducible tensors: \[ \begin{align*} [J^2,O^s_\lambda] &= s(s+1) O^s_{\lambda}, \\ [J_z,O^s_\lambda] &= \lambda O^s_\lambda, \\ [J_\pm,O^s_\lambda] &= \sqrt{s(s+1) - \lambda(\lambda\pm 1)}O^s_{\lambda\pm 1}. \end{align*} \]
• Vector operators: multi-component operators that transforms under rotation as \[ U[R]X_i U[R]^{-1} = X_j R{^j}{_i}. \]
  • Lie brackets of vector operators: \[ [J_k, A_l] = i\epsilon^{klm} A_m. \]
  • \(n\)-th rank tensor: \[ [J_k, T_{l_1\cdots l_n}] = i\qty{\epsilon^{kl_1 m}T_{ml_2\cdots l_n} + \cdots + \epsilon^{kl_n m}T_{l_1\cdots l_{n-1}m}}. \]

Pauli Spinor Field

• \(1/2\)-representation: \[ U[R] \Psi^\sigma(\vb*{x}) U[R]^{-1} = \Psi^\lambda(R\vb*{x}) D^{1/2}(R^{-1}){^\sigma}{_\lambda}. \]

Topology in Physics (I)

Typical Lie Groups

SO(3)

• Homeomorphism between:
  • \(\mathrm{SO}(3)\);
  • Unit solid ball in \(\mathbb{R}^3\) with antipodal points identified.

Fundamental Groups

Fundamental Group of SO(3)

• Two kinds of loops:
  • Loops lies in the interior of the unit ball.
  • Loops connects two antipodal points.
• Fundamental group \(\mathbb{Z}^2\).

Fundamental Group of Torus

The following code demonstrates why the fundamental group of a torus is abelian.

torus[t_, p_] := {2*Cos[t] + Cos[t]*Cos[p], 2*Sin[t] + Sin[t]*Cos[p], 
   Sin[p]};
torusStroke[t_, p_] := {2*Cos[t] + 1.01 Cos[t]*Cos[p], 
   2*Sin[t] + 1.01 Sin[t]*Cos[p], 1.01 Sin[p]};

Manipulate[
 Show[ParametricPlot3D[
   torus[2 \[Pi] t, 2 \[Pi] p], {t, 0, 1}, {p, 0, 1}], 
  ParametricPlot3D[
   torusStroke[2 \[Pi] (t - 0.125), 
    2 \[Pi] (Piecewise[{{u/(1 - u) t, 
         t < 1 - u}, {(1 - u)/u (t - 1) + 1, t > 1 - u}}])], {t, 0, 
    1}, PlotStyle -> {Thickness -> 0.01}, PlotPoints -> 200, 
   ColorFunction -> (Hue[
       Piecewise[{{#4/(1 - u)*0.5, #4 < 
           1 - u}, {0.5 + (#4 - (1 - u))/u*0.5, #4 > 
           1 - u}}]] &)]], {u, 0.01, 0.99}]

Fibre Bundles

• Tangent bundles are not necessarily trivial: \(TS^2\) is not \(S^2 \times \mathbb{R}^2\), since vector fields on \(S^2\) should vanish somewhere and therefore \(S^2\times \vb{v}\) for \(\vb{v}\neq 0\) could not be mapped to a section.

Index Theorems

Elliptic Operators and Fredholm Operators

• Operators: maps to sections \[ D: \Gamma(M,E) \rightarrow \Gamma(M,F). \]

Elliptic Operators

• Differential operator: with \(E\) and \(F\) being complex vector bundles over \(M\), \[ D: \Gamma(M,E) \rightarrow \Gamma(M,F). \]
• Notation: multi-index, \[ \begin{align*} M &= (\mu_1, \cdots, \mu_m),\quad \text{where}\quad \mu_j\in \mathbb{Z},\quad \mu_j \ge 0, \\ \abs{M} &= \mu_1 + \cdots + \mu_m, \\ D_M &= \pdv{^{\abs{M}}}{x^M}. \end{align*} \]
• Operator of order \(N\): \[ [Ds(x)]^\alpha = \sum_{\substack{\abs{M}\le N \\ 1\le a\le k}} A^{M\alpha}{_a}(x) D_M s^a(x). \]
  • \(N=1\): e.g. the Dirac operator, \[ [D\psi(x)]^\alpha = i(\gamma^\mu){^\alpha}{_\beta} \partial_\mu \psi^\beta(x) + m\psi^\alpha(x). \]
  • \(N=2\): e.g. the Laplacian operator.
• Symbol of \(D\): a \(k\times k'\) matrix, given by \[ \sigma(D,\xi) = \sum_{\abs{M} = N} A^{M\alpha}{_a}(x) \xi_M. \]
  • Independently of coordinates: \[ \sigma(D,\xi) s = \frac{1}{N!} D(f^N \tilde{s})\vert_p. \]
    • Section \(\tilde{s}\) satisfies \[ \tilde{s}(p) = s, \]
    • \(f\) satisfies \(f(p) = 0\),
    • \[\dd{f(p)} = \xi.\]
• Elliptic operator: \(\sigma(D,\xi)\) invertible for each \(x\in M\) and \(\xi \neq 0\).
• Composites, powers and roots of elliptic operators are elliptic.
• Let \[ \sigma(D,\xi) = A_{11}\xi^1 \xi^1 + 2A_{12} \xi^1 \xi^2 + A_{22}\xi^2 \xi^2. \] Then \(D\) is elliptic if and only if \[ \sigma(D,\xi) = 1 \] is an ellipse in \(\xi\)-space.

• \(x^\mu \in \mathbb{R}^m\). \(E\) and \(F\) be real line bundles over \(\mathbb{R}^m\).
• Laplacian: \[ \bigtriangleup = \pdv{^2}{(x^1)^2} + \cdots + \pdv{^2}{(x^m)^2}. \]
• Symbol: \[ \sigma(\Delta,\xi) = \sum_\mu (\xi_\mu)^2. \]
• The d'Alembertian is not invertible.

Fredholm Operators

• Adjoint: with inner products defined on \(E\) and \(F\), \[ D^\dagger: \Gamma(M,F) \rightarrow \Gamma(M,E), \] defined by \[ \langle s', Ds \rangle_F = \langle D^\dagger s', s\rangle_E. \]
• Cokernel: \[ \operatorname{coker} D = \Gamma(M,F)/\operatorname{im} D. \]
• Fredholm operator: an elliptic operator \(D\) whose kernels and cokernels are finite dimensional.
• Analytical index: \[ \begin{align*} \operatorname{ind} D &= \dim \ker D - \dim \operatorname{coker} D \\ &= \dim \ker D - \dim \ker D^\dagger. \end{align*}{} \]

Elliptic Complexes

• Elliptic complex: a sequence of Fredholm operators \[ \cdots \rightarrow \Gamma(M,E_{i-1}) \xrightarrow{D_{i-1}} \Gamma(M,E_i) \xrightarrow{D_i} \Gamma(M,E_{i+1}) \xrightarrow{D_{i+1}} \cdots \] that are nilpotent, i.e. \[ D_i \circ D_{i-1} = 0. \]
  • Laplacian: \[ \bigtriangleup = D_{i-1} D^\dagger_{i-1} + D^\dagger_i D_i. \]
• Hodge decomposition: \[ s_i = D_{i-1} s_{i-1} + D^\dagger_i s_{i+1} + h_i, \] where \(h_i\) is in the kernel of \(\Delta_i\).
• De Rham cohomology groups: \[ H^i(E,D) = \ker D_i / \operatorname{im} D_{i-1}. \]
  • \[ \dim H^i(E,D) = \dim \mathrm{Harm}^i(E,D). \]
• Index of elliptic complex: \[ \operatorname{ind} D = \sum_{i=0}^m (-1)^i \dim H^i(E,D) = \sum_{i=0}^m (-1)^i \dim \ker \bigtriangleup_i. \]
  • Special case: for \[ 0 \overset{i}{\hookrightarrow} \Gamma(M,E) \xrightarrow{D} \Gamma(M,F) \xrightarrow{\varphi} 0, \] the index is given by \[ \dim D = \dim\ker D - \dim\operatorname{coker} D^\dagger. \]
• Even bundle and odd bundle: \[ E_+ = \oplus_r E_{2r};\quad E_- = \oplus_r E_{2r+1}. \]
  • Laplacians: \[ \begin{align*} \Delta_+ &= A^\dagger A = \oplus_r \bigtriangleup_{2r}, \\ \Delta_- &= A A^\dagger = \oplus_r \bigtriangleup_{2r+1}. \end{align*} \]
  • Index: \[ \operatorname{ind}(E_\pm, A) = \dim\ker \bigtriangleup_+ - \dim\ker\bigtriangleup_- = \sum_r (-1)^r \dim\ker\bigtriangleup_r = \operatorname{ind}(E,D). \]

• De Rham complex \(\Omega(M)\) given by \[ 0 \xrightarrow{i} \Omega^0(M) \xrightarrow{\dd{}} \Omega^1(M) \xrightarrow{\dd{}} \cdots \xrightarrow{\dd{}} \Omega^m(M) \xrightarrow{\dd{}} 0. \]
• Index identified with Euler characteristic: \[ \begin{align*} \mathrm{ind}(\Omega^*(M), \dd{}) &= \sum_{r=0}^m (-1)^r \dim H^r(M;\mathbb{R}) \\ &= \chi(M) = \int_M e(TM) \\ &= \sum_{r=0}^m (-1)^r \dim \ker \bigtriangleup_r. \end{align*}{} \]

Atiyah-Singer Index Theorem

• \((E,D)\) be an elliptic complex over an \(m\)-dimensional compact manifold \(M\) without a boundary. \[ \operatorname{ind}(E,D) = (-1)^{m(m+1)/2}\int_M \left.\operatorname{ch}\qty({\oplus_r (-1)^r E_r})\frac{\operatorname{Td}(TM^{\mathbb{C}})}{e(TM)}\right\vert_{\mathrm{vol}}. \]
  • The division by \(e(TM)\) is carries out at the formal level.
  • Only \(m\)-forms are picked up in the integration.
• Let \[ \Gamma(M,E) \xrightarrow{D} \Gamma(M,F) \] be a two-term elliptic complex. The index of \(D\) is given by \[ \begin{align*} \operatorname{ind} D &= \dim\ker D - \dim\ker D^\dagger \\ &= (-1)^{m(m+1)/2} \int_M \left.(\operatorname{E} - \operatorname{F})\frac{\operatorname{Td}(TM^{\mathbb{C}})}{e(TM)}\right\vert_{\mathrm{vol}}. \end{align*}{} \]

Representation of Finite Groups

ねぇ、あなたは何色になりたい？

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Group Representation

Definitions

• Representation of a group: a homomorphism from a hroup \(G\) to a group of operators \(U(G)\) on a linear vector space \(V\).
  • Dimension: \(\dim V\).
  • Faithful: \(U\) is a isomorphism.
  • Degenerate: not faithful.
• Equivalent representation: two representations related by a similarity transformation, i.e. \[ U'(G) = S U(G) S^{-1}. \]
• Characters of representation: \(\chi(g)\) defined by \[ \chi(g) = \Tr U(g) = \sum_i D(g){^i}{_i}. \]
  • \(D(g)\): the matrix of \(U(g)\).
• Invariant subapce: \(V_1 \subset V\) such that \[ U(g)V_1 \in V_1,\quad \forall g. \]
  • Minimal/Proper: \(V_1\) doesn't contain any non-trivial invariant subspace.
• Irreducible representation: \(U(G)\) that doesn't admit non-trivial invariant in \(V\).
  • Reducible: not irreducible.
    • In matrix form: \[ D(g) = \begin{pmatrix} D_1(g) & * \\ 0 & D_2(g) \end{pmatrix}. \]
  • Fully reducible/Decomposible: (if \(V\) is Hermitian) orthogonal complement of the invariant subspace also invariant.
• Unitary representation:
  • \(V\) is an Hermitian space, and
  • \(U(g)\) are unitary for all \(g\).
• Direct sum representation: \(V = V_1 \oplus V_2\) is a sum of two invariant subspaces.
• Notation:
  • \(\abs{G}\): order of \(G\).
  • \(\mu,\nu\): labels for inequivalent IR of \(G\).
  • \(d_\mu\): dimension of \(\mu\)-IR.
  • \(D^\mu(g)\): the matrix of \(g\) in the \(\mu\)-IR, with respect to an orthonormal basis.
  • \(\chi^\mu_i\): character of class \(i\) in the \(\mu\)-IR.
  • \(\abs{C_i}\): number of elements in the class \(\xi_i\).
  • \(n_C\): number of conjugacy classes.
  • \[ D{^i}{_j} = \bra{e^i}D\ket{e_j}. \]

Theorems

• Reducible unitary representations are fully reducible.
  • Let \(V\) be the invariant subspace, and \[ \Pi^W = U^\dagger \Pi U. \] be the projection onto \(U^{-1} V\). From \[ U\Pi^V = \Pi^V U \Pi^V. \] we find \[ \dim (V^\perp\cap W) = \tr ((1 - \Pi^V) \Pi^W) = 0. \]
• Every representation on a Hermitian space is equivalent to a unitary representation.
  • We redefine the inner product by \[ (x,y) = \sum_g \bra{D(g)x}\ket{D(g)y} \] and note that \(D(g)\) is unitary under the new orthonormal basis.
• Reducible \(\Rightarrow\) fully reducible.
• Schur's Lemma (I): if \(U(G)\) is an IR and \[ AU(g) = U(g)A,\quad \forall A, \] then \(A = \lambda \mathbb{1}\).
  • We prove only for Hermitian \(A\). It's easily seen that the eigenspace of \(A\) is an invariant subspace of \(U(g)\).
• Schur's Lemma (II): if \(U(G)\) and \(U'(G)\) are two IRs on \(V\) and \(V'\) respectively, and \[ A: V'\rightarrow V \] satisfies \[ AU'(g) = U(g)A,\quad \forall g\in G, \] then either
  • \(A=0\), or
  • \(V\) is isomorphic to \(V'\) and \(U(G)\) is equivalent to \(U'(G)\).
  • Proof:
    • \(\operatorname{ker} A\) is invariant under \(U'\) and therefore \(A\) is injective.
    • \(\operatorname{im} A\) is invariant under \(U\) and therefore \(A\) is surjective.
• IR of any abelian group is of dimension one.
• Orthonormality of IR: for unitary \(D_\mu\), \[ \frac{d_\mu}{\abs{G}}\sum_g D^\dagger_\mu(g){^k}{_i}D^\nu(g){^j}{_l} = \delta^\nu_\mu \delta^j_i \delta^k_l. \]
  • \[D^\dagger_\mu(g){^k}{_i} = \qty[D^\mu(g){^i}{_k}]^*.\]
  • Proof:
    • Let \[ X_i^j = \sum_g D_\mu^\dagger(g) \ket{e_i}\bra{e^j} D^\nu(g). \]
    • We have \[ X_i^j D^\nu(g) = D_\mu^\dagger(g) X_i^j. \] Therefore, either \(X^j_i\) is \(0\) or \(D_\mu\) is equivalent to \(D_\nu\).
    • If \(D_\mu\) is equivalent to \(D_\nu\), we have \[X_i^j = \lambda \mathbb{1}.\]
    • Taking trace on both sides we get the desired result.
• The vectors \[ \ket{\mu,i,j} = \begin{pmatrix} D_\mu(g_1){^i}{_j} & \cdots & D_\mu(g_{\abs{G}}){^i}{_j} \end{pmatrix}{} \] are orthogonal. Each IR contribute \(d_\mu^2\) such vectors.
• The vectors are complete: \[ \sum_\mu d_\mu^2 = \abs{G}. \]
  • The number of IRs equal to the number of conjugacy classes.
• Completeness of IR: \[ \sum_{\mu,l,k} \frac{d_\mu}{\abs{G}} D^\mu(g){^l}{_k} D^\dagger_\mu(g'){^k}{_l} = \delta_{gg'}. \]
• Orthonormality of group characters: \[ \sum_i \frac{\abs{C_i}}{\abs{G}} \chi^{\dagger i}_\mu \chi^\nu_i = \delta^\nu_\mu. \]
  • \[ \chi^{\dagger i}_{\mu} = (\chi^\mu_i)^*. \]
• Completeness of group characters: \[ \sum_\mu \frac{\abs{C_i}}{\abs{G}} \chi^\mu_i \chi^{\dagger j}_\mu = \delta^j_i. \]
  • Combining the completeness relation of IR with the following lemma:
  • Let \(U^\mu(G)\) be an IR. \[ \sum_{h\in C_i} U^\mu(h) = \frac{\abs{C_i}}{d_\mu} \chi^\mu_i \mathbb{1}. \]
    • LHS commutes with any \(U^h(g)\).
• In the decomposition \[ U(G) = \bigoplus_{\mu} a_\mu U^\mu(G), \] we have \[ a_\nu = \sum_i \frac{\abs{C_i}}{\abs{G}} \chi^{\dagger i}_\nu \chi_i. \]
• \[ \mathrm{IR} \quad \Leftrightarrow \quad \sum_i \abs{C_i} \abs{\chi_i}^2 = \abs{G}. \]

Real, Complex, and Quaternionic Representations

• Real representation: \(\rho\) equivalent to a real representation.
  • In this case, we always have \(\rho \cong \overline{\rho}\).
• Quaternionic representation: \(\rho \cong \overline{\rho}\) but \(\rho\) is not a real representation.
• Complex representation: \(\rho\ncong \overline{\rho}\).
• Frobenius-Schur indicator: \[ \frac{1}{\abs{G}} \sum_{g\in G} \chi(g^2) = \begin{cases} 1, & \text{if } g \text{ is real}, \\ 0, & \text{if } g \text{ is complex}, \\ -1, & \text{if } g \text{ is quaternionic}. \end{cases}{} \]

For the quaternion group \[ Q_8 = \langle \overline{e},i,j,k \vert \overline{e}^2 = e, i^2=j^2=k^2=ijk=\overline{e} \rangle \] that consists of eight elements \[ \qty{e,i,j,k,\overline{e},\overline{i},\overline{j},\overline{k}}, \] the representation given by \[ \rho(i) = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix},\quad \rho(j) = w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \] is quternionic since \[ w\rho(g) w^{-1} = \overline{\rho}(g) \] for all \(g\) but \(\rho\) could not be sent to a real representation in that way.

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Typical Representations

The Regular Representation

• \(V\): the group ring.
  • Basis: \(\qty{g_j}\).
• \[ \mathrm{R}(g_i){^m}_{j} = \delta^{g_m}_{g_i g_j}. \]
  • \[ g_i g_j = g_m \Delta^m_{ij}. \]
  • \[ \chi^{\mathrm{R}}_i = \begin{cases} \abs{G}, & C_i = \qty{e}, \\ 0, & \text{otherwise}. \end{cases} \]
• \[ \mathrm{R}(G) = \bigoplus_\mu d_\mu U^\mu(G). \]

Direct Product Representations

• Direct product of vector spaces: \[ W = U \otimes V \] consists of \[ \set{\hat{\vb*{w}}_k | k = (i,j); i = 1,\cdots,d_U; j = 1,\cdots,d_V}. \]
  • \(\hat{\vb*{w}}_k\) is the formal product \[ \hat{\vb*{w}}_k = \hat{\vb*{u}}_i \cdot \hat{\vb*{v}}_j. \]
  • \[ \bra{w^{k'}}\ket{w_{k}} = \delta^{k'}_{k} = \delta^{i'}_i \delta^{j'}_j. \]
  • \[ W = \qty{x^k \ket{w_k}}. \]
  • \[ \bra{x}\ket{y} = x^\dagger_k y^k, \quad x^\dagger_k = (x^k)^*. \]
• Direct product of operator: \[ D = A \otimes B. \]
  • \[ D\ket{w_k} = \ket{w_{k'}} D^{k'}{_k}. \]
  • \[ D^{k'}{_k} = A^{i'}{_i} B^{j'}{_j}. \]
• Direct product of representation: \[ D^{\mu\times \nu}(g) = D^{\mu}(g)\otimes D^{\nu}(g). \]
• Character of direct product: \[ \chi^{\mu\times \nu} = \chi^\mu \chi^\nu. \]
• Decomposition \[ D^{\mu\times \nu}(G) = \bigoplus_{\lambda} a_\lambda D^\lambda. \]
• Invariant subspace: \[ W^\lambda_\alpha, \]
  • \(\lambda\): which representation acts on this invariant subspace?
  • \(\alpha\): if there are multiple invariant subspace, distinguish them with this index.
• Basis: new orthonormal basis, \[ \ket{w^\lambda_{\alpha l}} = \sum_{ij} \ket{w_{i,j}}\langle i,j(\mu\nu)\alpha,\lambda,l \rangle. \]
  • \((i,j)\) as row index.
  • \((\alpha,\lambda,l)\) as column index.
  • The new basis is demanded to be orthornormal (as the old basis is). We have large freedom to mix the basis in each invariant subspace. Therefore, the new basis (and hence the CG coefficients) are not uniquely determined.
  • The transformation is unitary.
• Action on old basis: \[ U(g)\ket{w_{i,j}} = \ket{w_{i',j'}} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j}. \]
• Action on new basis: \[ U(g)\ket{w^\lambda_{\alpha l}} = \ket{w^\lambda_{\alpha l'}} D^\lambda(g)^{l'}{_l}. \]
• Clebsch-Gordan coefficients: \[ \langle i,j(\mu\nu)\alpha,\lambda,l \rangle. \]
• Orthonormality of CG coefficients: \[ \sum_{\alpha,\lambda,l} \langle i',j'(\mu,\nu)\alpha,\lambda,l\rangle \langle \alpha,\lambda,l (\mu,\nu) i,j\rangle = \delta^{i'}_i \delta^{j'}_j. \]
  • \[ \langle \alpha,\lambda,l(\mu\nu) i,j \rangle = \langle i,j(\mu,\nu)\alpha,\lambda,l\rangle^*. \]
• Completeness of CG coefficients: \[ \sum_{i,j} \langle \alpha',\lambda',l'(\mu,\nu) i,j\rangle \langle i,j(\mu,\nu)\alpha,\lambda,l\rangle = \delta^{\alpha'}_{\alpha}\delta^{\lambda'}_{\lambda} \]
• Reduction of product representation: \[ \begin{align*} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j} &= \bra{i'j'}\ket{\alpha,\lambda,l'} D^\lambda(g)^{l'}{_l}\bra{\alpha,\lambda,l}\ket{i,j}, \\ \delta^{\alpha'}_{\alpha}\delta^{\lambda}_{\lambda'} D^\lambda(g)^{l'}{_l} &= \bra{\alpha',\lambda',l'}\ket{i',j'} D^\mu(g)^{i'}{_i} D^\nu(g)^{j'}{_j} \bra{i,j}\ket{\alpha,\lambda,l}. \end{align*}{} \]

Subduced Representations

• If \(T\) is a representation of a group \(G\), and
• \(G_1\subset G\) is a subgroup, then
• \(T\) subduces the representation \(T_1\) of \(G_1\) according to \[ \color{orange} T_1(g) = T(g),\quad \forall g\in G_1. \]
• When this is done for all the IRs of \(G\), we have a subduction table.
  • Subduction of IRs may be reducible.
• Subduced representation denoted \[ \color{orange} \rho_{G_1} = \rho_G \downarrow G_1. \]

Induced Representations

Induced Representation

Let \(H\) be a subgroup of \(G\) and \(\rho: H \rightarrow \mathrm{GL}(V)\) be a representation of \(H\). The induced representation \(\mathrm{Ind}^G_H(\pi)\) acts on \[ W = \bigoplus_{i=1}^n g_i V, \] where \(g_i \in G\) are representative elements of the quotient \(G/H\), by \[ g\cdot g_i v_i = g_{j(i)} h_i v_i. \] The induced representation is denoted \[ \color{orange} \rho_G = \rho\uparrow G. \]

Let \[ G = D_{12} = \bra{\rho,\sigma}\ket{\rho^6 = \sigma^2 = 1, \rho\sigma = \sigma \rho^{-1}}, \] and \[ H = \qty{1,\sigma,\rho^3, \sigma\rho^3}. \] A representation of \(H\) is given by \[ \rho(\sigma) = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \rho(\rho^3) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \rho(\sigma\rho^3) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. \] Note that \([G:H] = 3\) and therefore the induced representation consists of \(3\times 3\) matrices of \(2\times 2\) blocks. \[ \rho_G(\rho) = \begin{pmatrix} 0 & 0 & \rho(\rho^3) \\ \mathbb{1} & 0 & 0 \\ 0 & \mathbb{1} & 0 \end{pmatrix},\quad \rho_G(\sigma) = \begin{pmatrix} \rho(\sigma) & 0 & 0 \\ 0 & 0 & \rho(\sigma\rho^3) \\ 0 & \rho(\sigma\rho^3) & 0 \end{pmatrix}. \] We pick representative elements \[ g_1 = e,\quad g_2 = \rho,\quad g_3 = \rho^2. \] Let's figure out how \(g\in G\) acts on \(\bigoplus g_i V_i\), i.e. \[ g g_i v_i = g_? h_? v_i. \] That is to seek a \(g_?\) such that \[ g_?^{-1} g_i v_i = h_? \in H. \] We take \(g = \rho\) for example, and find that \begin{align*} g_2^{-1} g g_1 &= 1 \in H, \\ g_3^{-1} g g_2 &= 1 \in H, \\ g_1^{-1} g g_3 &= \rho^3 \in H. \end{align*} Therefore, we are now ready to answer the following questions:

1. How does \(g = \rho\) act on \(g_1 V_1\)?
   • \[g_1\cdot v_1 \rightarrow g_2\cdot \mathbb{1} v_1.\]
2. How does \(g = \rho\) act on \(g_2 V_2\)?
   • \[g_2 \cdot v_2 \rightarrow g_3\cdot \mathbb{1} v_2.\]
3. How does \(g = \rho\) act on \(g_3 V_3\)?
   • \[g_3 \cdot v_3 \rightarrow g_1\cdot \rho(\rho^3) v_3.\]

Now we obtain the \(\rho_G\) above.

Band Representations and Topological Quantum Chemistry

This could be written as \[ [\rho_G(g)]_{\alpha i,\beta j} = [\tilde{\rho}(g_\alpha^{-1} g g_\beta)]_{ij}, \] where \[ [\tilde{\rho}(g)]_{ij} = \begin{cases} [\rho(g)]_{ij}, & \text{if } g\in H, \\ 0, & \text{else}. \end{cases} \]

• The induced representation is always in block form, i.e. \(\rho(g)\) is always a matrix of blocks.
  • Big indices, labelling \(g_i\in G/H\): \(g\) sends \(g_i V\) to which \(g_? V\)?
  • Small indices, labelling \(v\in V\): how \(h = g_i^{-1}g\) acts on \(V\)?

Tensor Representations: Symmetry Dictates Selection Rule

• \(U(G)\) be a representation not necessarily reducible.
  • Acting on \[ V = \bigoplus_{\mu,\alpha} V^\alpha_\mu. \]
• Irreducible tensors: a set of operators \[ \set{O^\mu_i | i = 1,\cdots,d_\mu} \] acting on \(V\) and transforming as \[ U(g) O^\mu_i U(g)^{-1} = O^\mu_j D^\mu(g){^j}{_i}. \]
• \(O^\mu_i \ket{e^\nu_j}\) transforms like direct product: \[ U(g) O^\mu_i \ket{e^\nu_j} = O^\mu_k \ket{e^\nu_j} D^\mu(g){^k}{_i} D^\nu(g){^l}{_j}. \]
• New basis \[ O^\mu_i \ket{e^\nu_j} = \sum_{\alpha,\lambda,l} \langle \alpha,\lambda,l(\mu,\nu)i,j \rangle. \]
• Wigner-Eckart theorem: \[ \bra{e^l_\lambda} O^\mu_i \ket{e^\nu_j} = \sum_\alpha \langle \alpha,\lambda,l(\mu,\nu)i,j \rangle \bra{\lambda} O^\mu \ket{\nu}_\alpha. \]
  • Reduced matrix element: \[ \bra{\lambda} O^\mu \ket{\nu}_\alpha = \frac{1}{\abs{d_\lambda}} \sum_k \bra{e^k_\lambda}\ket{w^\lambda_{\alpha k}}. \]

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Obtaining Irreducible Representations

Irreducible Vectors

• \(U(G)\) be a representation not necessarily reducible.
  • Acting on \[ V = \bigoplus_{\mu,\alpha} V^\alpha_\mu. \]
• Irreducible set: a set of vectors \[\set{\hat{\vb*{e}}^\mu_i | i = 1,\cdots,d_\mu}\] transforming according to \[ U(g)\ket{e^\mu_i} = \ket{e^\mu_j} D^\mu(g){^j}{_i}. \]
• If
  • \(\qty{\hat{\vb*{u}}^\mu_i}\) and \(\qty{\hat{\vb*{v}}^\nu_j}\) are two sets of irreducible basis, and
  • \(\mu\) and \(\nu\) labels different IR, then
  • the two invariant subspaces spanned by these bases are orthogonal to each other.
  • Proof: prove that \[ \bra{v^j_\nu}\ket{u^\mu_i} = \frac{1}{\abs{G}} \sum_g \bra{v^j_\nu} U^\dagger(g) U(g) \ket{u^\mu_i} \] vanished.

States of different \(l\) (living in different spaces of the IRs of \(\mathrm{SO}(3)\)) doesn't overlap.

• Rotates one; kills the rest. \[ P^j_{\mu i} = \frac{d_\mu}{\abs{G}} \sum_g D^{-1}_\mu(g){^j}{_i} U(g). \]
  • For any \(\ket{x} \in V\), the sets of vectors \[ \set{\ket{e^\mu_i} = P^j_{\mu i}\ket{x}, i = 1,\cdots,d_\mu}. \] transform irreducibly according to \(D^\mu\) (as long as they are not all null). \[ U(g)P^j_{\mu i}\ket{x} = P^j_{\mu k}\ket{x} D^\mu(g){^k}{_i}. \]
  • \(P^j_{\mu i}\) kills and rotates in the invariant subspace. \[ P^j_{\mu i} \ket{e^\nu_{\alpha k}} = \ket{e^\nu_{\alpha i}} \delta^\nu_\mu \delta^j_k. \]
  • Consecutive kill-and-rotate: \[ P^j_{\mu i} P^l_{\nu k} = \delta^\nu_\mu \delta^j_k P^l_{\mu i}. \]
  • Inversion \[ U(g) = \sum_{\mu,i,j} P^{j}_{\mu i} D^\mu(g){^i}{_j}. \]
  • \[ U(g) P^l_{\nu k} = \sum_i P^l_{\nu i} D^\nu(g){^i}{_k}. \]
• Projection operators:
  • Onto the basis vector \(\hat{\vb*{e}}^\mu_i\) \[ P_{\mu i} = P^{i}_{\mu i}. \]
  • Onto the irreducible invariant space \(V_\mu\) spanned by \(\qty{\hat{\vb*{e}}^\mu_i}\) \[ P_\mu = \sum P_{\mu i}. \]
    • Completeness: \[ \sum_\mu P_\mu = \mathbb{1}. \]

The IRs of one-dimensional discrete translation group are given by \(\qty{e^{ikna}}\). From a local (Wannier) state \(\phi\) we generate a one-dimensional invariant (Bloch) basis \[ \ket{\psi_k} = \sum_n T_n\ket{\phi} e^{ikna}. \]

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Representations of the Permutation Groups

One-Dimensional Representations

• Trivial representation: \(E\).
  • Generated by the symmetrizer \(s\), essentially idempotent and primitive.
• Parity representation: \[D(p) = (-1)^{\operatorname{sign} p}.\]
  • Generated by the anti-symmetrizer \(a\), essentially idempotent and primitive.
• These are the only one-dimensional representations.

Partitions and Young Diagrams

• Partition:
  • Partition \[ \lambda = \qty{\lambda_1,\cdots,\lambda_r} \] of \(n\):
    • \[ \lambda_i \ge \lambda_{i+1}. \]
    • \[ \sum \lambda_i = n. \]
  • Partition \(\lambda = \mu\) if \[ \lambda_i = \mu_i,\quad \forall i. \]
  • \(\lambda>\mu\) or \(\lambda<\mu\) by lexicographic order.
• Parition to Young diagram:
  • \[ 4 = 2 + 1 + 1 \] represented as follows.
  • \[ 4 = 3 + 1 \] as follows.
• Partitions of \(n\) \(\leftrightarrow\) conjugacy classes of \(S_n\) \(\leftrightarrow\) IRs of \(S_n\).
• Young tableau, normal tableau, and standard tableau:
  • A tableau is obtained by filling the squares of a Young diagram with \(1,\cdots,n\).
  • Normal Young tableau: \(1,\cdots,n\) appear in order from left to right and from the top row to the bottom row.
    • Denoted by \(\Theta_\lambda\).
    • Examples as follows.

    1	2	3
    4

    and

    1	2
    3	4

    • An arbitrary tableau obtained from \(\Theta_\lambda\) by permutation \[ \Theta^p_\lambda = p\Theta_\lambda. \]
  • Standard Young tableau: number of each row appear increasing to the right, and in each column appear to the bottom.

Symmetrizers and Anti-symmetrizers of Young Tableaux

Let \(\Theta_2\) be given as follows.

1	2
3

The irreducible symmetrizer is given by \[ e_2 = \frac{[e+(12)][e-(31)]}{4}. \] Let \[ r_2 = (23) e_2. \] Then \(\qty{e_2,r_2}\) span an invariant subspace of the group ring, giving rise to the two-dimensional IR of \(S_3\).

Symmetry Classes of Tensors

• Horizontal and vertical permutations: \(\Theta^p_\lambda\) given,
  • Horizontal permutations: \(\qty{h^p_\lambda}\) leave invariant the sets of numbers appearing in the same row of \(\Theta^p_\lambda\).
    • \(\qty{h_\lambda}\) form a subgroup of \(S_n\).
  • Vertical permutations: \(\qty{v^p_\lambda}\) leave invariant the sets of numbers appearing in the same column of \(\Theta^p_\lambda\).
    • \(\qty{v_\lambda}\) form a subgroup of \(S_n\).
• Symmetrizer, anti-symmetrizer, and irreducible/Young symmetrizer: associated with the Young tableau \(\Theta^p_\lambda\),
  • Symmetrizer: summation over horizontal permutations, \[ s_\lambda^p = \sum_{h} h^p_\lambda. \]
    • Idempotent.
  • Anti-symmetrizer: summation over vertical permutations, \[ a^p_\lambda = \sum_v (-1)^{v_\lambda} v_\lambda^p. \]
    • Idempotent.
  • Irreducible/Young symmetrizer: sum over horinzontal and vertical permutations: \[ e^p_\lambda = \sum_v (-1)^{v_\lambda} h^p_\lambda v^p_\lambda. \]
    • Primitive idempotent.
• Properties of symmetrizers: superscript \(p\) omitted.
  • For \(\lambda\) in the group ring \(\mathbb{C}S_n\), these exists \(\xi\in\mathbb{C}\) such that \[ s_\lambda r a_\lambda = \xi e_\lambda. \]
  • There exists \(\xi \in \mathbb{\eta}\) such that \[ e_\lambda^2 = \eta e_\lambda. \]
• \(e^p_\lambda\) associated with \(\Theta_\lambda^p\) is a primitive idempotent. It generates an IR of \(S_n\) on the group ring.
• The representations generated by \(e^p_\lambda\) and \(e_\lambda\) are equivalent.
• Two irreducible symmetrizers \(e_\lambda\) and \(e_\mu\) generate inequivalent IR if \(\lambda \neq \mu\).
• If \(\lambda \neq \mu\), then \[e^p_\mu e^q_\lambda=0\] for all \(p,q\in S_n\).
• The left ideals generated by irreducible symmetrizers \(\qty{e_\lambda}\) associated with normal Young tableaux \(\qty{\Theta_\lambda}\) are all IRs of \(S_n\).
• The direct sum (yes they are lineraly-independent) of the left ideals generated by all standard tableaux is the group ring \(\mathbb{C}S_n\).

Irreducible Representations of the Permutation Groups

• \(\qty{i}\) denotes a collection of indices. We work on the space \(V^n_m\), i.e. \(n\) copies of a \(m\)-dimensional space \(V\).

• Direct product of operator acting on direct product of spaces: \[ g\ket{\qty{i}} = \ket{\qty{j}} D(g)^{\qty{j}}{_{\qty{i}}}. \]

  • \[ (gx)^{\qty{j}} = D(g)^{\qty{j}}{_{\qty{i}}} x^{\qty{i}}. \]
  • \[ (Dg)^{\qty{j}}{_{\qty{i}}} = g^{{j_1}}{_{i_1}} \cdots g^{{j_n}}{_{i_n}}. \]

• Action of permutation: \[ p\ket{\qty{i}} = \ket{p^{-1}\qty{i}}. \]

  • \[ (px)^{\qty{i}} = x^{p\qty{i}}. \]
  • \[ D(p)^{\qty{j}}{_{\qty{i}}} = \delta^{p\qty{j}}{_{\qty{i}}}. \]

• The two set of matrices \(\qty{D(p)}\) of where \(p\in S_n\) and \(\qty{D(g)}\) where \(g\in \mathrm{GL}(m)\) commute with each other.

• Tensors of symmetry class \(\lambda\): \[ T_\lambda(\alpha) = \set{re_\lambda \ket{\alpha} | r\in \mathbb{C}S_n}, \]

  • \(\lambda\) the Young diagram given.
  • \(T_\lambda\) is an irreducible invariant subspace with respect to \(S_n\).
  • If \(T_\lambda(\alpha)\) is not empty, then the realization of \(S_n\) on \(T_\lambda(\alpha)\) coincides with the IR generated by \(e_\lambda\).
  • \[ T'_\lambda = \set{e_\lambda\ket{\alpha} | \ket{\alpha} \in V^n_m}. \]
  • With Young symmetrizers, tensors may be completely decomposed into irreducible tensors \[ \ket{\lambda, \alpha, a}, \] where
    • \(\lambda\): the symmetry class (Young diagram),
    • \(\alpha\): distinct set of tensors \(T_\lambda(\alpha)\) invariant under \(S_n\),
    • \(a\): basis element whtin each \(T_\lambda(\alpha)\).
  • Indices of basis chosen in a way such that \[ p\ket{\lambda,\alpha,a} = \ket{\lambda,\alpha,b} D_\lambda(p){^{b}}{_{a}}. \]

• Tensors of symmetry \(\Theta^p_\lambda\): \[ \set{e^p_\lambda\ket{\alpha}}. \]

  • \(\Theta^p_\lambda\) the Young tableau given.

• Two tensor subspaces irreducibly invariant with respect to \(S_n\) and belonging to the same symmetry class either overlap completely or are disjoint.

• Two irreducible invariant tensor subspaces corresponding to two distinct symmetry classes are necessarily disjoint.

• If \(g\in \mathrm{GL}_m\) and \(\qty{\ket{\lambda,\alpha,a}}\) is a set of basis tensors, then the subspaces \(T'\lambda(a)\) spanned by \(\qty{\ket{\lambda,\alpha,a}}\) with fixed \(\lambda\) and \(a\) are invariant under \(\mathrm{GL}_m\). The representations of \(\mathrm{GL}_m\) on \(T'_\lambda(a)\) are independent of \(a\). \[ g\ket{\lambda,\alpha,a} = \ket{\lambda,\beta,a} D_\lambda(g){^\beta}{_\alpha}. \]

• The representations of \(\mathrm{GL}_m\) on the subspace \(T'_\lambda(a)\) of \(V^n_m\) are irreducible.

Let \(\Theta_{\lambda=s}\) be the Young tableau associated with the total symmetrizer, then \(T_\lambda(\alpha)\) is a totally symmetric tensor, \(S_n\) acts on which irreducibly (and trivially).

There is one and only one independent totally anti-symmetric tensor of rank \(n\) in \(n\)-dimensional space.

For second-rank tensors on \(m\)-dimensional \(V\), we have \[ e_s\ket{ij} = \frac{\ket{ij} + \ket{ji}}{2}, \] while \[ e_a\ket{ij} = \frac{\ket{ij} - \ket{ji}}{2}. \]

Let \(\Theta_m\) be given as follows.

1	2
3

The irreducible symmetrizer is given by \[ e_m = \frac{[e+(12)][e-(31)]}{4}. \] With \[ \begin{align*} \ket{\alpha} &= \ket{++-}, \\ \ket{\beta} &= \ket{--+}. \end{align*} \] we obtain \[ \begin{align*} e_m\ket{\alpha} = \ket{m,\alpha,1} &= \frac{2\ket{++-} - \ket{-++} - \ket{+-+}}{4}, \\ (23) e_m\ket{\alpha} = \ket{m,\alpha,2} &= \frac{2\ket{+-+} - \ket{-++} - \ket{++-}}{4}; \\ e_m\ket{\beta} = \ket{m,\beta,1} &= \frac{2\ket{--+} - \ket{+--} - \ket{-+-}}{4}, \\ (23) e_m\ket{\beta} = \ket{m,\beta,2} &= \frac{2\ket{-+-} - \ket{+--} - \ket{--+}}{4}. \end{align*}{} \] The action of \(S_3\) on the subspace generated by these two vectors gives rise to its two-dimensional representation.

---

Application: Degeneracy in Central Potential

An Example of Symmetry Group Acting on Wavefunctions

• The states \[ \set{\ket{nl,m}|m = -l,\cdots,l} \] could be related by operators in \(\mathrm{O(3)}\).

Application: Crystal Field Theory

An Example of Compatibility Relations

• My free atom has five \(d\)-orbitals. Without an external potential, the atom enjoys the full \(\mathrm{O}(3)\) symmetry. Therefore, its five \(d\)-orbitals are degenerate since \(\mathrm{O}(3)\) connects them.
• Now my atom is placed in a crystal of \(\mathrm{T_d}\) symmetry. The symmetries that \(d\)-orbitals enjoy are reduced. Therefore, they are not guaranteed to be degenerate since two orbitals may not be be connected by an operation in \(\mathrm{T_d}\).
  • In the worst case, when there is no symmetry present, we no longer have any degeneracy.
• The case has not come to the worst. We still have a non-trivial symmetry group and we wanna know if there could be any degenracy. We note the following one-to-one correspondence: \[ \text{degeneracy} \Leftrightarrow \text{invariant subspace} \Leftrightarrow \text{IR}. \]
• We take the following procedures to determine the possible degeneracies:
  1. Determine how \(\mathrm{T_d}\) acts on our \(d\)-orbitals. Find out the character. From this we get a representation of \(\mathrm{T_d}\).
  2. With the character of IRs of \(\mathrm{T_d}\), we could easily decompose the above representation into IRs and know the invariant subspaces, hence the degeneracy.
• We know how \(\mathrm{O}(3)\) acts on \(d\)-orbitals. The action of \(\mathrm{T_d}\) is obtained directly by subduction, and the characters are obtained just as if we are treating \(\mathrm{O}(3)\) operations.
  • The only question is, to each \(l\) we have two IRs of \(\mathrm{O}(3)\) of dimension \(2l+1\). Which one should we use?
  • Since \(d\)-orbitals are invariant under parity, we should therefore use the IR where parity has positive character.
• Character table of \(\mathrm{T_d}\): where \(6S_4\) denotes the rotoreflections by \(90^\circ\) and \(6\sigma_{\mathrm{d}}\) the reflections.

• We determine the characters of \(\mathrm{O}(3)\) on the above conjugacy classes with (\(j=2\)) \[ \chi = \frac{\sin (j+1/2)\psi}{\sin (\psi/2)}. \]
• Character table of \(\mathrm{O}(3)\).

• The decomposition is given by \[ D_2^+ = E\oplus T_1. \]
• The five \(d\)-orbitals splits into a two-fold level and a three-fold level.

Wrap Up: Compatibility Relations

• Originally, the atom has a higher symmetry \(\mathrm{O}(3)\).
• As we move the atom into a crystal, the symmetry is reduced and therefore the originally irreducible representation (\(d\)-orbitals) \(D_2^+\) becomes reducible.
• The IR \(D_2^+\) is decomposed into a direct sum. The possible decompositions are those decompositions into IRs of subgroups of \(\mathrm{O}_3\). The decomposition will be different if the reduced symmetry is given by \(\mathrm{O}_h\). These possible decompositions are compatibility relations.
• Compatibility relations \(\Leftrightarrow\) patterns of crystal field splitting.

Application: Band Representations

Point Group Action on Bloch Functions

• The actions that carry a point \(\vb*{k}\) in the reciprocal space to its equivalent \[ \vb*{k}' = \vb*{k} + \vb*{K} \] form the little group at point \(\vb*{k}\).
• If \(g\) is an element of the point group, then \[ g \psi_{\vb*{k}}(\vb*{r}) = \psi_{\vb*{k}}(g^{-1}\vb*{r}) = \psi_{g\vb*{k}}(\vb*{r}). \]
• If, in particular, \(h\) is an element of the little group of \(\vb*{k}\), then the set of wavefunctions \[ \set{h \psi_{\vb*{k}}(\vb*{r})} \] are degenerate, just like what we have in a central potential.

Compatibility Relations: Moving Away From Points of High Symmetry

• The symmetry group of points on the line \(XY\) joining two points of high symmetry \(X\) and \(Y\) is a subgroup of the symmetry group thereof. Therefore, the representations decomposes along the line into direct sums. The decomposition should satisfy the compatibility relations, just like what we have in crystal field splitting.

Application: Stark Effect

• We ignore the spin-orbit interaction.
• When \(E=0\), the \(l\)-orbitals (\(2l+1\)-fold degenerate) transforms according to \(D^l\).
• For \(E=E\hat{\vb*{z}}\), the \(l\)-orbitals split, each of which transforms according to the \(m\)-th representation of \(\mathrm{SO}(2)\). Therefore, the eigenkets are of the form \(\ket{m}\).
• That's why we could directly apply the perturbation theory to \(\ket{lm}\).

Fibre Bundles

ゲージのばの庭

Fibre Bundles

Fibre Bundles

Example: Tangent Bundles

• Tangent Bundle \(TM\): \[ TM = \bigcup_{p\in M} T_p M. \]
• \(M\) is called the base space.
• \(TU_i\) is a \(2m\)-dimensional differentiable manifold.
• Projection: \[ \pi: (p, V) \in TU_i \rightarrow u\in U_i. \]
• Fibre at \(p\): \(T_p M\).
• Structure group of \(TM\): \[ \mathrm{GL}(n,\mathbb{R}) \ni (G^\nu_\mu) = \pdv{y^\nu}{x^\mu}. \]
• Section: \[ s: M\rightarrow TM \] such that \[ \pi \circ s = \mathrm{id}_M. \]
• Local section: \[ s_i: U_i \rightarrow TU_i. \]

Definition of Fibre Bundle

• Differentiable fibre bundle \((E,\pi,M,F,G)\) consists of
  • Total space: a differentiable manifold \(E\),
  • Base space: a differentiable manifold \(M\),
  • (Typical) fibre: a differentiable manifold \(F\),
  • Projection: \[ \pi: E\rightarrow M. \]
    • The inverse image \[ \pi^{-1}(p) = F_p \cong F \] is called the fibre at \(p\).
  • Structure group: a Lie group \(G\) which acts \(F\) on the left.
  • A set of open covering \(\qty{U_i}\) of \(M\) with a diffeomorphism \[ \phi_i: U_i \times F \rightarrow \pi^{-1}(U_i). \]
    • \(\phi_i\) is called the local trivialization.
  • Let \[ \phi_{i,p}(f) = \phi_i(p,f), \] the map \[ \phi_{i,p}: F \rightarrow F_p \] is a diffeomorphism. On \(U_i\cap U_j \neq \emptyset\), we require \[ t_{ij}(p) = \phi^{-1}_{i,p} \circ \phi_{j,p}: F \rightarrow F \] be an element of \(G\). \(t_{ij}\) are called the transition functions. \[ \phi_j(p,f) = \phi_i(p,t_{ij}(p)f). \]
• The transition functions should satisfy (assuming intersection of \(U_i\), \(U_j\), \(U_k\)):
  • \[ t_{ii}(p) = \mathrm{id}. \]
  • \[ t_{ij}(p) = t_{ji}(p)^{-1}. \]
  • \[ t_{ij}(p) \cdot t_{jk}(p) = t_{ik}(p). \]
• If all the transition functions can be take to be \(\mathrm{id}\), then the fibre bundle is called a trivial bundle, i.e. \(M\times F\).
• Section: a smooth map \[ s: M\rightarrow E \] which satisfies \[ \pi \circ s = \mathrm{id}_M. \]
  • \(\Gamma(M,F)\): the set of sections on \(M\).
  • Local section: defined on \(U\).
  • \(\Gamma(U,F)\): the set of local sections on \(U\).
  • Not all fibre bundles admit global sections.

Let \[ E \xrightarrow{\pi} S^1, \] and the typical fibre \[ F = [-1,1]. \] Let \[ U_1 = (0,2\pi), \quad B = (-\pi,\pi), \] and \(A = (0,\pi)\) and \(B = (\pi,2\pi)\) be the intersection of \(U_1\cap U_2\).

• Local trivialization:
  • On \(A\): \[ \phi_1^{-1}(u) = (\theta, t),\quad \phi_2^{-1}(u) = (\theta, t). \]
    • Transition function: \[ t_{12}(\theta) = \mathrm{id}. \]
  • On \(B\): two choices,
    • \[ \phi_1^{-1}(u) = (\theta,t),\quad \phi_2^{-1}(u) = (\theta,t), \]
      • Transition function: \[t_{12}(\theta) = \mathrm{id}.\]
      • \(G=\qty{e}\), cylinder, trivial bundle \(S^1\times F\).
    • \[ \phi_1^{-1}(u) = (\theta,t),\quad \phi_2^{-1}(u) = (\theta,-t). \]
      • Transition function: \[t_{12}(\theta): t\mapsto -t.\]
      • \(G = \mathbb{Z}_2\), Möbius strip.

Reconstruction of Fibre Bundles

For given \(M\), \(\qty{U_i}\), \(t_{ij}(p)\), \(F\) and \(G\), we can reconstruct the fibre bundle \((E,\pi,M,F,G)\).

• \[ \color{red} E = X/\sim. \]
  • \[ X = \bigcup_i U_i \times F. \]
  • Equivalence relation \(\sim\): \[ (p,f)\in U_i \times F \sim (q,f') \in U_j \times F \] if and only if \[ p = q\quad \text{and} \quad f' = t_{ji}(p) f. \]
  • Projection \[ \pi: [(p,f)] \mapsto p. \]
  • Local trivialization \[ \phi_i: (p,f) \mapsto [(p,f)]. \]

Bundle Maps

• Bundle map: a smooth map \[\overline{f}: E'\rightarrow E\] that
  • maps each fibre \(F'_p\) of \(E'\) onto \(F_q\) on \(E\).
  • \(\overline{f}\) induces a smooth map \[f: M' \rightarrow M\] such that \(f(p) = q\).
  • The diagram \[ \begin{CD} E' @>\overline{f}>> E \\ @VV\pi' V @VV\pi V \\ M' @>f>> M \end{CD} \] commutes.

Equivalent Bundles

• Two bundles are equivalent if
  • there exists a bundle map \[\overline{f}: E'\rightarrow E\]
  • such that \(f: M\rightarrow M\) is the identity map and
  • \(\overline{f}\) is a diffeomorphism,
  • i.e. the diagram \[ \begin{CD} E' @>\overline{f}>> E \\ @VV\pi' V @VV\pi V \\ M @>\mathrm{id}_M>> M \end{CD} \] commutes.

Pullback Bundles

• Pullback of \(E\) by \[f: N\rightarrow M,\]
  • \[ f^*E = \qty{(p,u)\in N\times E | f(p) = \pi(u)}. \]
• The diagram \[ \begin{CD} f^*E @>\pi_2>> E \\ @VV\pi_1V @VV\pi V\\ N @>f>> M \end{CD} \] commutes.
• \[ t^*_{ij}(p) = t_{ij}(f(p)). \]

Let \(M\) and \(N\) be differentiable manifolds with \[\dim M = \dim N = m.\] Let \(f: N\rightarrow M\) be a smooth map. \(f\) induces \(\pi_2\) such the the diagram \[ \begin{CD} TN @>\pi_2>> TM \\ @VV\pi_1V @VV\pi V\\ N @>f>> M. \end{CD} \] If \(TN\) is a pullback bundle \(f^*(TM)\), then \(\pi_2\) maps \(T_p N\) to \(T_{f(p)}M\) diffeomorphically.

Homotopy Axiom

• \(f,g: M'\rightarrow M\) are homotopic if
  • there exists a smooth map \[ F: M'\times [0,1] \rightarrow M \] such that \(F(p,0) = f(p)\) and \(F(p,1) = g(p)\) for any \(p\in M'\).
• Let \(E\xrightarrow{\pi}M\) be a fibre bundle and \(f,g\) be homotopic maps from \(N\) to \(M\). Then \(f^* E\) and \(g^* E\) are equivalent bundles over \(N\).
• Let \(E\xrightarrow{\pi}M\) be a fibre bundle. \(E\) is trivial if \(M\) is contractible to a point.

Vector Bundles

Definition and Examples of Fibre Bundles

• Vector bundle: \(E\xrightarrow{\pi} M\),
  • a fibre bundle,
  • whose fibre is a vector space, i.e. \(F = \mathbb{R}^k\).
  • \(k\) is called the fibre dimension, denoted by \[ k = \dim E. \]
  • Transition functions belong to \(\mathbb{GK}(k)\).
• Line bundle: a vector bundle whose fibre is one-dimensional.
• Canonical line bundle \(L\): \[ L = \set{(p,v)\in \mathbb{C}P^n \times \mathbb{C}^{n+1} | v=ap,a\in \mathbb{C}}. \]

Let \(U_{\mathrm{N}}\) be \(S^2\) without south pole and \(U_{\mathrm{S}}\) be \(S^2\) without north pole. Let \((X,Y)\) and \((U,V)\) be the respective stereographic coordinates. Putting \(X = r\cos\theta\) and \(Y = r\sin\theta\) we obtain the transition \[ t_{\mathrm{SN}} = \pdv{(U,V)}{(X,Y)} = \frac{1}{r^2} \begin{pmatrix} -\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}. \]

• Let \(M\) be an \(m\)-dimensional manifold embedded in \(\mathbb{R}^{m+k}\).
• Let \(N_p M\) be the vector space normal to \(T_p M\) in \(\mathbb{R}^{m+k}\).
• Normal bundle: \[ NM = \bigcup_{p\in M} N_p M. \]
  • Fibre in \(\mathbb{R}^k\).

For example, the normal bundle of \(S^2\) embedded in \(\mathbb{R}^3\) is a trivial bundle \(S^2 \times \mathbb{R}\).

A wavefunction is a section of the trivial complex line bundle \(L = \mathbb{R}^3 \times \mathbb{C}\).

Frames

• Frame: linearly independent sections \[ \qty{e_1(p),\cdots,e_k(p)} \] defined on each \(U_i\).
• Local trivialization \[ \phi_i^{-1}(V) = (p,\qty{V^\alpha(p)}). \]
• Change of frame: \[ \begin{align*} \tilde{e}_\beta(p) &= e_\alpha(p) G(p){^\alpha}{_\beta}, \\ \tilde{V}^\beta &= G^{-1}(p){^\beta}{_\alpha} V^\alpha. \end{align*} \]

Cotangent Bundles and Dual Bundles

• Contangent bundle: \[ T^* M = \bigcup_{p\in M} T^*_p M. \]
• Change of frame: \[ \begin{align*} \dd{y^\mu} &= \dd{x^\nu} \qty(\pdv{y^\mu}{x^\nu})_p, \\ \tilde{\omega}_\mu ^= G{_\mu}{^\nu}(p) \omega_\nu. \end{align*} \]
• Dual bundle \(E^* \xrightarrow{\pi} M\): fibre \(F^*\) of which is the set of linear maps from \(F\) to \(\mathbb{R}\) or \(\mathbb{C}\).
  • Dual basis on \(F^*_p\) by \[ \langle \theta^\alpha(p), e_\beta(p)\rangle = \delta{^\alpha}{_\beta}. \]

Sections of Vector Bundles

• \[ (s+s')(p) = s(p) + s'(p) \].
• For \(f\in \mathcal{F}(M)\), \[ (fs)(p) = f(p) s(p). \]
• Null section: \(s_0 \in \Gamma(M,E)\) such that \[ \phi_i^{-1}(s_0(p)) = (p,0). \]

Let \(L\) be the canonical line bundle \(L\) over \(\mathbb{C}P^n\). Let The local section \(s_\mu\) over \(U_\mu\) is defined by \[ s_\mu = \qty{\xi{^0}{_{(\mu)}}, \cdots, 1, \cdots, \xi{^n}{_{(\mu)}}} \in \mathbb{C}^{n+1}. \] The transition is given by \[ s_\nu = \frac{z^\mu}{z^\nu} s_\mu. \] The dual section is defined by \[ s^*_\mu(s_\mu) = 1. \] The transition is given by \[ s^*_\nu = \frac{z^\nu}{z^\mu} s_\mu^*. \] A fibre metric \(h_{\mu\nu}(p)\) is defined by \[ (s,s')_p = h_{\mu\nu}(p) \overline{s^\mu(p)} s'^\nu(p). \]

The Product Bundle and Whitney Sum Bundle

• Product bundle: \[ E\times E' \xrightarrow{\pi\times \pi'} M\times M'. \]
  • Curvature on product bundle: \[ \mathcal{F}_{E\otimes F} = \mathcal{F}_E \otimes \mathbb{1} + \mathbb{1} \otimes \mathcal{F}_F. \]
• Typical fibre \(F \oplus F'\).
• Whitney sum bundle \(E\oplus E'\): pullback of \(E\times E'\) by \[ f: p\in M\rightarrow (p,p) \in M\times M, \]
  • denoted by \(E\oplus E'\).
  • \(E\xrightarrow{\pi} M\) and \(E' \xrightarrow{\pi'} M\) are vector bundles.
  • Commutative diagram: \[ \begin{CD} E \oplus E' @>\pi_2>> E\times E' \\ @VV\pi' V @VV\pi \times \pi'V \\ M @>f>> M \times M \end{CD} \]
  • Curvature on Whitney sum bundle: \[ \mathcal{F}_{E\oplus F} = \mathcal{F}_E \oplus \mathcal{F}_F. \]
• Transition \[ T_{ij}(p) = \begin{pmatrix} t^E_{ij}(p) & 0 \\ 0 & t^{E'}_{ij}(0) \end{pmatrix}. \]

Let \(E = TS^2\) and \(E' = NS^2\) defined in \(\mathbb{R}^3\). The trivialization \(\Phi_{i,j}\) is given by \[ \Phi^{-1}_{i,j}(u,v) = (p,q; V,W) = (p; V, W), \] where \[ \phi_i^{-1} = (p,V),\quad \psi_i^{-1} = (q,W). \] The Whitney sum \[ TS^2 \oplus NS^2 \] is therefore a trivial bundle \(S^2 \times \mathbb{R}^3\).

Tensor Product Bundles

• Tensor product bundle: denoted by \(E\otimes E'\).
  • \(E\xrightarrow{\pi} M\) and \(E\xrightarrow{\pi'} M\).
• Fibres are given by \[ F_p \otimes F'_p. \]
• \(\bigwedge^r E\) is spanned by \[ \qty{e_{\alpha_1} \wedge \cdots \wedge e_{\alpha_r}}. \]
• \[ \Omega^r(M) = \Gamma(M, \bigwedge^r(T^* M)). \]

Principal Bundles

Definition and Examples of Principal Bundles

• Principal bundle:
  • Fibre \(F\) is identical to the structure group \(G\).
  • Denoted by \(P(M,G)\).
  • Often called \(G\)-bundle over \(M\).
• Action on the right: for \(u\in P\) and \(a\in G\), \[ ua = \phi_i(p, g_i a), \]
  • \(\phi_i^{-1}(u) = (p, g_i)\).
  • The action is transitive.
  • The action is free, i.e. if \(ua=u\) for some \(u\) then \(a = e\).
• Canonical local trivialization:
  • Section \(s_i(p)\) defined over \(U_i\).
  • For \(u\in \pi^{-1}(p)\), there is a unique \(g_u\) such that \[ u = s_i(p) g_u. \]
  • Define \[ \phi_i^{-1}(u) = (p, g_u). \]
  • In this local trivialization, the section \(s_i(p)\) is expressed as \[ s_i(p) = \phi_i(p,e). \]
  • \[ \phi_i(p,g) = s_i(p)g. \]
  • Two sections on \(U_i\cap U_j\) are related by \[ s_i(p) = s_j(p) t_{ji}(p). \]

Let \(P\) be a principal bundle with fibre \(\mathrm{U}(1) = S^1\) and base space \(S^2\). Let \[ \begin{align*} U_{\mathrm{N}} &= \set{(\theta,\phi)| 0 \le \theta \le \pi/2+\epsilon}, \\ U_{\mathrm{S}} &= \set{(\theta,\phi)| \pi/2-\epsilon \le \theta\le \pi}. \end{align*}{} \] \(U_{\mathrm{N}} \cap U_{\mathrm{S}}\) is the equator. Let \(\phi_{\mathrm{N}}\) and \(\phi_{\mathrm{S}}\) be the local trivializations such that \[ \phi_{\mathrm{N}}^{-1}(u) = (p, e^{i\alpha_{\mathrm{N}}}),\quad \phi_{\mathrm{S}}^{-1}(u) = (p, e^{i\alpha_{\mathrm{S}}}). \] Fibre coordinates related on the equator as \[ e^{i\alpha_{\mathrm{N}}} = e^{in\phi} e^{i\alpha_{\mathrm{S}}}, \] characterized by \(\pi_1(\mathrm{U}(1)) = \mathbb{Z}\). Trivial bundle for \(n=0\), and twisted otherwise. Right action \(g = e^{i\Lambda}\) is a \(\mathrm{U}(1)\)-gauge transformation.

With \[S^4 = \mathbb{R}^4 \cup \qty{\infty},\] and the open covering \[ \begin{align*} U_{\mathrm{N}} &= \set{(x,y,z,t) | x^2 + y^2 + z^2 + t^2 \le R^2 + \epsilon}, \\ U_{\mathrm{S}} &= \set{(x,y,z,t) | x^2 + y^2 + z^2 + t^2 \ge R^2 - \epsilon}. \end{align*}{} \] We have \[U_{\mathrm{N}}\cap U_{\mathrm{S}} \cong S^3.\] For an \(\mathrm{SU}(2)\) bundle \(P\) over \(S^4\), the transition function \[t_{\mathrm{NS}}: S^3 \rightarrow \mathrm{SU}(2)\] is characterized by \(\pi_3(\mathrm{SU}(2)) = \mathbb{Z}\). The integer is called the instanton number. Nontrivial bundles are given by \[ t_{\mathrm{NS}}(p) = \frac{1}{\sqrt{x^2+y^2+z^2+t^2}^n}\qty(t\mathbb{1} + i\sum_i x^i \sigma_i)^n. \]

\(S^3\) as a \(\mathrm{U}(1)\) bundle over \(S^2\).

• Hopf map \(\pi:S^3\rightarrow S^2\) \[ \begin{align*} \xi^1 &= 2(x^1 x^3 + x^2 x^4), \\ \xi^2 &= 2(x^2 x^3 - x^1 x^4), \\ \xi^3 &= (x^1)^2 + (x^2)^2 - (x^3)^2 - (x^4)^2. \end{align*}{} \]
• Denote \[ \begin{align*} z^0 &= x^0 + ix^1, \\ z^1 &= x^2 + ix^3. \end{align*}{} \]
• Stereographic projection of \(S^2\) given by \[ Z = \frac{z^0}{z^1},\quad W = \frac{z^1}{z^0}, \] are both invariant under \(\mathrm{U}(1)\) transformation \[ (z^0,z^1) \mapsto (\lambda z^0, \lambda z^1). \]
• Local trivializations: \[ \phi_{\mathrm{S}}^{-1}: \pi^{-1}(U_{\mathrm{S}}) \rightarrow U_{\mathrm{S}} \times \mathrm{U}(1) \] by \[ (z^0,z^1) \mapsto (\frac{z^0}{z^1}, \frac{z^1}{\abs{z^1}}), \] and \[ \phi_{\mathrm{N}}^{-1}: \pi^{-1}(U_{\mathrm{N}}) \rightarrow U_{\mathrm{N}} \times \mathrm{U}(1) \] by \[ (z^0,z^1) \mapsto (\frac{z^1}{z^0}, \frac{z^0}{\abs{z^0}}). \]
  • On the equator \[ t_{\mathrm{NS}}(\xi) = \xi^1 + i\xi^2 \in \mathrm{U}(1). \]
• \(t_{\mathrm{NS}}(\xi)\) traverses the unit circle once as we circumnavigate the equator. The \(\mathrm{U}(1)\) bundle \[ S^3 \xrightarrow{\pi} S^2 \] is therefore characterized by the homotopy class \(1\) of \[ \pi_1(\mathrm{U}(1)) = \mathbb{Z}. \]
• This Hopf map describes a magnetic monopole of unit strength.

• Unit quaternion represents \(S^3\).
• Let \[ S^1_{\mathbb{H}} = \set{(q^0,q^1)\in \mathbb{H}^2 | \abs{q^0}^2 + \abs{q^1}^2 = 1}, \] which represents \(S^7\).
• Let \[ \mathbb{H}P^1 = \qty{\eta(q^0,q^1)\in \mathbb{H}^2 | \eta \in \mathbb{H} - \qty{0}}, \] which represents \(S^4\).
• We have the Hopf map \(\pi: S^7 \rightarrow S^4\). The transition function is defined by the class \(1\) of \[ \pi_3(\mathrm{SU}(2)) \cong \mathbb{Z}. \]
• This Hopf map describes a instanton of unit strength.

• Let \(H\) be a closed Lie subgroup of a Lie group \(G\). Then \(G\) is a principal bundle with fibre \(H\) and base space \(M = G/H\).
• Local trivializations defined by \[ \phi_i^{-1}(g) = ([g], f_i(g)). \]
  • \[f_i(g) = s([g])^{-1}g.\]
  • \(s\) is a section over \(U_i\).
• Examples: \[ \begin{align*} \mathrm{O}(n) / \mathrm{O}(n-1) &= \mathrm{SO}(n) / \mathrm{SO} (n-1) = S^{n-1}, \\ \mathrm{U}(n) / \mathrm{U}(n-1) &= \mathrm{SU}(n) / \mathrm{SU}(n-1) = S^{2n-1}. \end{align*}{} \]

Associated Bundles

• Principal bundle \(P(M,G)\) given.
• Let \(F\) be a new fibre.
• Action of \(g\in G\) on \(P\times F\) given by \[ (u,f) \rightarrow (ug, g^{-1}f). \]
• Associated fibre bundle: \[ (E,\pi,M,G,F,P). \]
  • An equivalence class \((P\times F)/G\) in which two point \[ (u,f) \sim (ug, g^{-1}f). \]
• Associated vector bundle:
  • \[ (u,v) \sim (ug, \rho(g)^{-1} v). \]
  • \(\rho\) is a \(\dim V\)-dimensional representation of \(G\).
• A vector bundle naturally induces a principal bundle associated with it, by \[ P(E) = P(M,G). \]

Frame bundle \[ LM = \bigcup_{p\in M} L_p M, \] where \(L_p M\) is the set of frames at \(p\). A frame \[ u = \qty{X_1,\cdots,X_m} \] is expressed as \[ X_\alpha = X{^\mu}{_\alpha} \partial_\mu. \] The local trivialization may be given by \[ \phi_i^{-1}(u) = (p, (X{^\mu}{_\alpha})) \in U_i \times \mathrm{GL}(m, \mathbb{R}). \]

• Action of \(\mathrm{GL}(m,\mathbb{R})\) given by \[ Y_\beta = X_\alpha a{^\alpha}{_\beta}. \]
• Transition function given by \[ {X}{^\mu}{_\alpha} = \pdv{x^\mu}{y^\nu} \tilde{X}{^\nu}{_\alpha}. \]

The universal covering group of \(\mathrm{O}^+_{\uparrow}(3,1)\) is \(\mathrm{SL}(2,\mathbb{C})\). The kernel of the homomorphism \[ \varphi: \mathrm{SL}(2,\mathbb{C}) \rightarrow \mathrm{O}^+_{\uparrow}(3,1) \] is \(\qty{I_2,-I_2}\).

• Weyl spinor: a section of \[ (W,\pi,M,\mathbb{C}^2,\mathrm{SL}(2,\mathbb{C})). \]
• Dirac spinor: a section of \[ (D,\pi,M,\mathbb{C}^4,\mathrm{SL}(2,\mathbb{C})\oplus \overline{\mathrm{SL}(2,\mathbb{C})}). \]

Triviality of Bundles

• A principal bundle is trivial if and only if it admits a global section.
• A vector bundle \(E\) is trivial if and only if its associated principal bundle \(P(E)\) admits a global section.

Connections on Fibre Bundles

Connections on Principal Bundles

Definitions of Connection

• Vertical subspace \(V_u P\):
  • a subspace of \(T_u P\),
  • tangent to \(G_p\) at \(u\),
  • where \(P(M,G)\) is a principal bundle given, \(u\) an element thereof, and \(G_p\) a fibre at \(p=\pi(u)\).
• Right action defined by \[ R_{\exp(tA)} u = u\exp(tA). \]
  • \(A\in\frak{g}\).
• Fundamental vector field \[ A^{\#} f(u) = \dv{}{t} f(u\exp(tA))\vert_{t=0}. \]
  • \(f:P\rightarrow\mathbb{R}\).
  • \(A^{\#}\) is tangent to \(G_p\) at \(u\).
• Isomorphism \[ \#: A\in{\frak{g}} \rightarrow A^{\#}\in V_u P. \]
• Horizontal subspace \(H_u P\): complement of \(V_u P\) in \(T_u P\).
• Ehresmann Connection on \(P(M,G)\):
  • A unique separation of the tangent space \(T_u P\) such that \[ T_u P = H_u P \oplus V_u P, \]
  • a smooth vector field \(X\) on \(P\) is separated into smooth vector fields \[ X = X^H + X^V, \]
  • for arbitrary \(u\in P\) and \(g\in G\), \[ H_{ug} P = R_{g*} H_u P. \]

(image from Supplementary chapters for "Geometric Control of Mechanical Systems")

The Connection One-Form

• Ehresmann connection one-form \(\omega\):
  • A projection of \(T_u P\) onto the vertical component \[ V_u P \cong {\frak{g}}. \]
  • \[ \omega \in {\frak{g}} \otimes T^* P. \]
  • The projection should satisfy
    • \[ \omega(A^{\#}) = A,\quad \text{for}\quad A\in{\frak{g}}. \]
    • \[ R^*_g \omega = \mathrm{Ad}_{g^{-1}}\omega, \] i.e. for \(X\in T_u P\), \[ R^*_g \omega_{ug}(X) = \omega_{ug}(R_{g*} X) = g^{-1} \omega_u (X) g. \]
• Ehresmann connection: \[ \color{orange} H_u P = \qty{X\in T_u P | \omega(X) = 0}. \]
  • In harmony with the previous definition.

The Local Connection Form and Gauge Potential

• Let \[ \mathcal{A}_i = \sigma_i^* \omega \in {\frak{g}}\otimes \Omega^1(U_i). \]
  • \(A_i\) is a Lie-algebra-valued one-form on \(U_i\).
  • \(\sigma_i\) is a given section of \(P\) on \(U_i\).
  • With \(\mathcal{A}_i\) given we could also recontruct \(\sigma_i\).
• Given a \(\frak{g}\)-valued one-form \(\mathcal{A}_i\) on \(U_i\) and a local section \(\sigma_i\), there exists a connection one-form \(\omega\) such that \(\mathcal{A}_i = \sigma_i^* \omega\).
  • \(\omega\) may be given by \[ \omega_i = g_i^{-1} \pi^* \mathcal{A}_i g_i + g_i^{-1} \dd{_P} g_i. \]
    • \(\dd{_P}\) is the exterior derivative on \(P\).
    • \(g_i\) is the canonical local trivialization defined by \[ \phi_i^{-1}(u) = (p, g_i), \] for \[ u = \sigma_i(p) g_i. \]
  • For \(\omega\) to be uniquely (consistently) defined, we demand the compatibility relation \[ \color{red} \mathcal{A}_j = t_{ij}^{-1} \mathcal{A}_i t_{ij} + t_{ij}^{-1} \dd{t_{ij}}. \]
    • Since \(t_{ij}\in G\), the first term maps elements in \(\frak{g}\) to another element in \(\frak{g}\).
    • In the second term, the \(\dd{}\) takes with respect to \(M\). Therefore, the second term yields an element in \(\frak{g}\) by acting on \(X\in T_p M\).
    • In particular, on the chart \(U\) we could perform the gauge transform by using a new trivialization \(\sigma_2(p) = \sigma_1(p) g(p)\). \[ \mathcal{A}_{2\mu} = g^{-1}(p) \mathcal{A}_{1\mu} g(p) + g^{-1}(p) \partial_\mu g(p). \]
  • \(\mathcal{A}_i\) is identified with the gauge potential or Yang-Mills potential.
  • The physical field is encoded in \(\omega\).
    • By picking a specific trivialization \(\sigma\) (coordinate system of the bundle), we slice \(\omega\) and pull it back to \(M\), yielding the potential.
    • By the equation for \(\omega\) above, its value on \(P\) is determined by its value on a single slice (thanks to the projection properties or the invariance property of Ehresmann connection).
    • Due to the presence of \(g_i\) in the equation for \(\omega\), it's seemingly that \(\omega\) depends on the trivialization we choose. We don't want that to happen. We demand \(\omega\) stays the same even if we switch to another trivialization.
    • Therefore, we simultaneously (1) go from one trivialization to another, and (2) transform \(\mathcal{A}_i\) to \(\mathcal{A}_j\), i.e. a gauge transformation.
    • The transformation may be as well understood as slicing \(\omega\) in a different way.

For a \(\mathrm{U}(1)\)-bumdle, we define a new trivialization by \[ t_{ij}(p) = \exp[i\Lambda(p)]. \] The potentials are therefore related as \[ \mathcal{A}_j(p) = \mathcal{A}_i(p) + i\partial_\mu \Lambda. \] The connection \(\mathcal{A}_\mu\) differs from the standard vector potential by the Lie algebra factor, i.e. \[ \mathcal{A}_\mu = iA_\mu. \]

Horizontal Lift and Parallel Transport

• Horizontal lift of \(\gamma:[0,1] \rightarrow M\):
  • a curve \(\tilde{\gamma}: [0,1] \rightarrow P\),
    • \(P(M,G)\),
  • \(\pi\circ\tilde{\gamma} = \gamma\),
  • the tangent vector to \(\tilde{\gamma}(t)\) belongs to \(H_{\tilde{\gamma}(t)}P\).
• For \(\gamma:[0,1]\rightarrow M\) and \(u_0\in \pi^{-1}(\gamma(0))\), there exists a unique horizontal lift \(\tilde{\gamma}(t)\) such that \(\tilde{\gamma}(0)\) = u_0.
• Let \(\tilde{\gamma}'\) be another horizontal lift of \(\gamma\), such that \(\tilde{\gamma}'(0) = \gamma(0)g\). Then \(\tilde{\gamma}'(t) = \tilde{\gamma}(t)g\).
• Explicitly, for \(\omega\) and \(\sigma_i\) given, \[ g_i(\gamma(t)) = \mathcal{P} \exp\qty({-\int_{\gamma(0)}^{\gamma(t)} \mathcal{A}_{i\mu}(\gamma(t))\dd{x^\mu}}). \]
  • \(\mathcal{P}\) denotes path ordering. \[ \mathcal{P}[A(\gamma(t))B(\gamma(s))] = \begin{cases} A(\gamma(t))B(\gamma(s)), & \text{if } t>s, \\ B(\gamma(s))A(\gamma(t)), & \text{if } s>t. \end{cases} \]
  • As a differential equation: \[ \dv{g_i(t)}{t} = -\mathcal{A}_i(X) g_i(t). \]
• Parallel transport:
  • \(\gamma: [0,1] \rightarrow M\).
  • \(u_0 \in \pi^{-1}(\gamma(0))\).
  • There is a unique horizontal lift \(\tilde{\gamma(t)}\) of \(\gamma(t)\) through \(u_0\),
  • and hence a unique point \[u_1 = \tilde{\gamma}(1) \in \pi^{-1}(\gamma(1)).\]
  • \(u_1\) is called the parallel transport of \(u_0\) along \(\gamma\).
  • This defines a map \[ \Gamma(\tilde{\gamma}): u_0 \in \pi^{-1}(\gamma(0)) \rightarrow u_1\in \pi^{-1}(\gamma(1)). \]
  • \(\Gamma\) commutes with \(R_g\), i.e. \[ R_g \Gamma(\tilde{\gamma}) = \Gamma(\tilde{\gamma}) R_g. \]
• Explicitly, \[ u_1 = \sigma_i(1) \mathcal{P} \exp\qty(-\int_0^1 \mathcal{A}_{i\mu} \dv{x^\mu(\gamma(t))}{t} \dd{t}). \]

• \(P(M,\mathbb{R}) \cong M\times \mathbb{R}\) where \(M = \mathbb{R}^2 - \qty{0}\).
• The local trivialization be given by \[ \phi: ((x,y),f) \mapsto u. \]
• \[ \omega = \frac{y\dd{x} - x\dd{y}}{x^2 + y^2} + \dd{f}. \]
• \[ \gamma: t\in [0,1] \rightarrow (\cos 2\pi t, \sin 2\pi t)\in M. \]
• \[ X = \dv{}{t} = \dv{x}{t} \pdv{}{x} + \dd{y}{t} \pdv{}{y} + \dv{f}{t} \dd{}{f}. \]
• \[ 0 = \omega(X) = \dv{x}{t}\frac{y}{r^2} - \dv{y}{t}\frac{x}{r^2} + \dv{f}{t} = -2\pi + \dv{f}{t}. \]
• Horizontal lift \[ \tilde{\gamma}(t) = ((\cos 2\pi t, \sin 2\pi t), 2\pi t). \]

Holonomy

Definition of Holonomy

• A loop \(\gamma: [0,1] \rightarrow M\) gives rise to a transformation \[ \tau_\gamma: \pi^{-1}(p) \rightarrow \pi^{-1}(p) \] by parallel transport.
• \(\tau_\gamma\) is compatible with the right action of the group, \[ \tau_\gamma(ug) = \tau_\gamma(u)g. \]
• Holonomy group \(\Phi_u \subset G\): \[ \Phi_u = \set{g\in G | \tau_\gamma(u) = ug,\gamma\in C_p(M)}. \]
  • \(p = \pi(u)\).
  • \[ C_p(M) = \set{\gamma:[0,1] \rightarrow M | \gamma(0) = \gamma(1) = p}. \]
• Restricted holonomy group: \[ \Phi_u^0 = \set{g\in G | \tau_\gamma(u) = ug, \gamma\in C_p^0(M)}. \]
  • \(C_p^0\) denotes the set of loops at \(p\) homotopic to the constant loop at \(p\).

In the previous example, \[ \tau_\gamma(g) = g + 2\pi. \]

Curvature

Covariant Derivatives in Principal Bundles

• Covariant derivative of \(\phi\): \[ \color{orange} \mathrm{D}\phi(X_1,\cdots,X_{r+1}) = \dd{_P \phi}(X_1^H,\cdots,X_{r+1}^H). \]
  • \[\phi\in \Omega^r(P)\otimes V.\]
  • \[X_1,\cdots,X_{r+1}\in T_u P.\]
  • \[\dd{_P \phi} = \dd{_P \phi^\alpha} \otimes e_\alpha.\]

Curvature

• Curvature two-form \(\Omega\): \[ \color{orange} \Omega = \mathrm{D}\omega \in \Omega^2(P)\otimes {\frak{g}}. \]
  • \[ R^*_a \Omega = a^{-1}\Omega a. \]
• For \(\xi = \xi^\alpha\otimes T_\alpha\), \(\eta = \eta^\alpha \otimes T_\alpha\) where \[\zeta^\alpha \in \Omega^p(P),\quad \eta^\alpha\in \Omega^q(P)\] and \(\qty{T_\alpha}\) a basis of \(\frak{g}\), \[ \begin{align*} \color{orange} [\zeta,\eta] &\color{orange}= \zeta\wedge \eta - (-1)^{pq} \eta\wedge \zeta \\ &\color{orange}= [T_\alpha,T_\beta] \otimes \zeta^\alpha \wedge \eta^\beta. \end{align*} \]
  • \[ [\zeta,\zeta] = 2\zeta\wedge \zeta. \]
  • If \(X\in H_u p\) and \(Y\in V_u P\) then \[ [X,Y] \in H_uP. \]
• Cartan's structure equation: for \(X,Y\in T_u P\), \[ \color{red} \Omega(X,Y) = \dd{_P \omega}(X,Y) + [\omega(X), \omega(Y)]. \]
  • Equivalently, \[ \color{red} \Omega = \dd{_P \omega} + \omega \wedge \omega. \]

Geometrical Meaning of the Curvature

• Ambrose-Singer theorem. Let \(P(M,G)\) be a \(G\) bundle over a connected manifold \(M\). The Lie algebra of the holonomy group \(\Phi_{u_0}\) of a point \(u_0\in P\) agrees with the subalgebra of \(\frak{g}\) spanned by the elements of the form \[ \Omega_u(X,Y),\quad X,Y\in H_u P, \] where \(u\in P\) is a point on the same horizontal lift as \(u_0\).

Local Form of the Curvature

• Yang-Mills field strength (local form) \[ \color{orange} \mathcal{F} = \sigma^* \Omega. \]
  • \(\sigma\) is a local section defined on a chart \(U\subset M\).
• In terms of gauge field \[ \color{red} \mathcal{F} = \dd{\mathcal{A}} + \mathcal{A}\wedge \mathcal{A}. \]
  • \(\mathcal{A} = \sigma^* \omega\).
  • \(\dd{}\) is the exterior derivative on \(M\).
  • \[ \color{red} \mathcal{F}(X,Y) = \dd{\mathcal{A}}(X,Y) + [\mathcal{A}(X), \mathcal{A}(Y)]. \]
• Component form \[ \color{red} \mathcal{F}_{\mu\nu} = \partial_\mu \mathcal{A}_\nu - \partial_\nu \mathcal{A}_\mu + [\mathcal{A}_\mu, \mathcal{A}_\nu]. \]
  • \[ \mathcal{F} = \frac{1}{2}\mathcal{F}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}. \]
  • Let \(\qty{T_\alpha}\) be a basis of the Lie algebra, \[ \color{red} F_{\mu\nu}{^\alpha} = \partial_\mu A{_\nu}{^\alpha} - \partial_\nu A{_\mu}{^\alpha} + f_{\beta\gamma}{^\alpha} A{_\mu}{^\beta} A{_\nu}{^\gamma}. \]
    • \[[T_\beta, T_\gamma] = f_{\beta\gamma}{^\alpha} T_\alpha.\]
• Compatibility condition: \[ \color{red} \mathcal{F}_j = t_{ij}^{-1} \mathcal{F}_i t_{ij}. \]

The Bianchi Identity

• Bianchi identity: \[ \color{red} \mathrm{D}\Omega = 0. \]
• \[ \color{red} \mathcal{D}\mathcal{F} = \dd{\mathcal{F}} + [\mathcal{A}, \mathcal{F}] = \dd{\mathcal{F}} + \mathcal{A} \wedge \mathcal{F} - \mathcal{F} \wedge \mathcal{A} = 0. \]
  • \[ \color{orange} \mathcal{D}\eta = \dd{\eta} + [\mathcal{A},\eta]. \]

The Covariant Derivative on Associated Vector Bundles

The Covariant Derivative on Associated Bundles

• \(P(M,G)\) be a \(G\) bundle.
• \(E = P\times_\rho V\) be a associated bundle thereof.
• Representatives as \[ s(p) = [(\sigma_i(p), \xi(p))]. \]
• \(\gamma\) a curve in \(M\). \(\tilde{\gamma}\) the horizontal lift thereof.
• Parallel transported section: \[ \color{orange} s(\gamma(t)) = [(\tilde{\gamma}(t), \eta(\gamma(t)))] \] where \(\eta\) is constant along \(\gamma(t)\).
• Covariant derivative \[ \color{orange} \grad_X s = \qty[\qty({\tilde{\gamma}(0), \left.\dv{}{t}\eta(\gamma(t))\right\vert_{t=0} })]. \]
  • \(X\) the tangent vector of \(\gamma(t)\) at \(t=0\).
  • \[ \grad_X: \Gamma(M,E) \rightarrow \Gamma(M,E). \]
  \[ \grad: \Gamma(M,E) \rightarrow \Gamma(M,E)\otimes \Omega^1(M) \] defined by \[ \grad s(X) = \grad_X s. \]
  • Properties of covariant derivatives are satisfied.
    • Linearity in \(s\): \[ \grad(a_1 s_1 + a_2 s_2) = a_1 \grad s_1 + a_2 \grad s_2. \]
    • Linearity in \(X\): \[ \grad_{f_1 X_1 + f_2 X_2} s = f_1 \grad_{X_1} s + f_2 \grad_{X_2} s. \]
    • Leibniz rule: \[ \grad(fs) = (\dd{f}) s + f\grad s. \]
  • Independent of
    • horizontal lift, and
    • local section (trivialization).
    • \(\mathcal{A}_i\) transforms under the change of local trivialization in a way such that \(\grad_X s\) is well-defined.

Local Expression for Covariant Derivative

• \(\tilde{\gamma}\), the horizontal shift of \(\gamma\), is written as \[ \tilde{\gamma}(t) = \sigma_i(t) g_i(t). \]
• \[ \qty(\mathcal{A}_{i\mu}){^\beta}{_\alpha} = \mathcal{A}_{i\mu}{^\gamma}(T_\gamma){^\beta}{_\alpha}. \]
• \(e{_\alpha}{^0}\) denotes the \(\alpha\)th basis vector of \(V\).
• \[ e_\alpha(t) = [(\tilde{\gamma}(t) g_i(t)^{-1}, e{_\alpha}{^0})] = [(\tilde{\gamma}(t), g_i(t)^{-1} e{_\alpha}{^0})]. \]
• Covariant derivative of basis: \[ \grad_X e_\alpha = [(\tilde{\gamma}(0) g_i(0)^{-1}, \mathcal{A}_i(X) e{_\alpha}{^0})]. \]
• Local expression: \[ \begin{align*} \color{red} \grad e_\alpha &\color{red}= \mathcal{A}{_i}{^\beta}{_\alpha} e_\beta, \\ \color{red} \grad_{\partial/\partial x^\mu} e_\alpha &\color{red}= \mathcal{A}_{i\mu}{^\beta}{_\alpha} e_\beta. \end{align*}{} \]
  • For a general vector field, \[ \color{red} \grad_X s = X^\mu \qty{\pdv{\xi{_i}{^\alpha}}{x^\mu} + \mathcal{A}_{i\mu}{^\alpha}{_\beta}\xi{_i}{^\beta}} e_\alpha. \]

• \(FM\) be frame bundles over \(M\).
  • \[FM = P(M,\mathrm{GL}(m,\mathbb{R})).\]
• \(TM\) be its associated bundle.
  • \[TM = FM \times_\rho \mathbb{R}^m.\]
  • \(m = \dim M\).
• \[\mathcal{A}_i = \Gamma{^\alpha}_{\mu\beta} \dd{x^\mu}.\]
  • \[ \grad_{\partial/\partial x^\mu} e_\alpha = [(\sigma_i(0), \Gamma_\mu e{_\alpha}{^0})] = \Gamma{^\beta}_{\mu\alpha} e_\beta. \]
  • \[ \grad_{\partial/\partial x^\mu} s = \qty(\frac{\partial}{\partial x^\mu} X{_i}{^\alpha} + \Gamma{^\alpha}_{\mu\beta} X^\beta) e_\alpha. \]
• \(\mu\) is the \(\Omega^1(M)\) index while \(\alpha\) and \(\beta\) are \(\frak{gl}(m,\mathbb{R})\) indices.

• \(P(M,\mathrm{U}(1))\).
• \(E = P\times_\rho \mathbb{C}\).
• \(\sigma_i\) be a local section. \(\tilde{\gamma}\) be a horizontal lift defined by \[ \tilde{\gamma}(t) = \sigma_i(t) e^{i\varphi(t)}. \]
• Basis section \[ e = [(\sigma_i(p), 1)]. \]
• My field (section) \[ \phi(p) = [(\sigma_i(p), \Phi(p))] = \Phi(p)e = \Phi(t)[(\tilde{\gamma}(t), e^{-i\varphi(t)})]. \]
• Covariant derivative \[ \grad_X \phi = X^\mu \qty(\pdv{\Phi}{x^\mu} + \mathcal{A}_{i\mu} \Phi)e. \]

• \(P(M,\mathrm{SU}(2))\).
• \(E = P \times_\rho \mathbb{C}^2\).
• Gauge potential
• \[ \mathcal{A}_i = \mathcal{A}_{i\mu} \dd{x^\mu} = A_{i\mu}{^\alpha}\qty(\frac{\sigma_\alpha}{2i})\dd{x^\mu}. \]
• \(e{_\alpha}{^0}\) where \(\alpha=1,2\) be basis vectors of \(\mathbb{C}^2\).
• Basis section \[ e_\alpha(p) = [(\sigma_i(p), e{_\alpha}{^0})]. \]
• My field (section) \[ \phi(t) = [(\tilde{\gamma}(t), U(t)^{-1} \Phi^\alpha(t) e{_\alpha}{^0})]. \]
  • \[\tilde{\gamma}(t) = \sigma_i(p) U(t),\quad U(t) \in \mathrm{SU}(2).\]
• Covariant derivative \[ \grad_X \phi = X^\mu \qty(\pdv{\Phi^\alpha}{x^\mu} + \mathcal{A}_{i\mu}{^\alpha}{_\beta}) e_\alpha. \]

Curvature Rederived

• \(\grad\) on vector-valued \(p\)-form: \[ \color{orange} \grad(s\otimes \eta) = (\grad s) \wedge \eta + s\otimes \dd{\eta}. \]
  • \(\eta \in \Omega^p(M)\).
• \[ \color{red} \grad \grad e_\alpha = e_\beta \otimes \mathcal{F}{_i}{^\beta}{_\alpha}. \]

A Connection Which Preserves the Inner Product

• Riemann structure: a positive-definite symmetric inner product \[ g_p: \pi^{-1}(p) \otimes \pi^{-1}(p) \rightarrow \mathbb{R}. \]
  • \(E\xrightarrow{\pi} M\) a vector bundle.
• Metric connection: \(\grad\) such that \[ \color{orange} \dd{[g(s,s')]} = g(\grad s, s') + g(s, \grad s'). \]
  • \[ \dd{g_{\alpha\beta}} = \mathcal{A}{_i}{^\gamma}{_\alpha} g_{\gamma\beta} + \mathcal{A}{_i}{^\gamma}{_\beta} g_{\alpha\gamma}. \]
  • Torsion-free: \[ \Gamma{^\gamma}_{\rho\sigma} \dd{x^\rho} = \mathcal{A}{^\gamma}{_\alpha}. \]
• \(\grad\) associated with the orthonormal frame is a metric connection.
  • Orthonormal frame: \(\qty{\hat{e}_\alpha}\) such that \[ g(\hat{e}_\alpha, \hat{e}_\beta) = \delta_{\alpha\beta}. \]
  • Structure group \(O(k)\). Lie algebra \({\frak{o}}(k)\) spanned by skew symmetric matrices. \[ \omega{^\alpha}{_\beta} = -\omega{^\beta}{_\alpha}. \]

Holomorphic Vector Bundles and Hermitian Inner Products

• Holomorphic vector bundle \(E\):
  • \(\pi: E\rightarrow M\) is a holomorphic surjection, where \(E\) and \(M\) are complex manifolds.
  • The typical fibre is \(\mathbb{C}^k\) and the structure group is \(\mathrm{GL}(k,\mathbb{C})\).
  • The local trivialization \[ \phi_i: U_i \times \mathbb{C}^k \rightarrow \pi^{-1}(U_i) \] is a biholomorphism.
  • The transition function \[ t_{ij}: U_i \cap U_j \rightarrow G = \mathrm{GL}(k,\mathbb{C}) \] is a holomorphic map.
• Hermitian structure: \[ h_p: \pi^{-1}(p) \times \pi^{-1}(p) \rightarrow \mathbb{C}. \]
  • \[h_p(u,av + bw) = ah_p(u,v) + bh_p(u,w).\]
  • \[ h_p(u,v) = \overline{h_p(v,w)}. \]
  • \[ h_p(u,u) \ge 0. \]
    • Equality holds only if \[u = \phi_i(p,0).\]
  • \[ h(s_1, s_2) \in \mathcal{F}(M)^\mathcal{C}. \]
• Unitary frame: \[ h(\hat{e}_i, \hat{e}_j) = \delta_{ij}. \]
  • The unitary frame bundle is not a holomorphic vector bundle, because \(\mathrm{U}(m)\) is not a complex manifold.
• Hermitian connection: \[ \grad: \Gamma(M,E) \rightarrow \Gamma(M,E\otimes T^* M^{\mathbb{C}}). \]
  • \[\grad(fs) = (\dd{f})s + f\grad s.\]
  • \[ \dd{h(s_1,s_2)} = h(\grad s_1, s_2) + h(s_1, \grad s_2). \]
  • \[ \grad s = \mathrm{D}s + \overline{\mathrm{D}}s. \]
    • \(\mathrm{D}\) and \(\overline{\mathrm{D}}\) being a \((1,0)\)-form and \((0,1)\)-form valued section, respectively.
      • We demand \[ \overline{\mathrm{D}} = \overline{\partial}. \]
• For a given \(E\) and Hermitian metric \(h\), there exists a unique Hermitian connection \(\grad\).
• \(\mathcal{A}\) and \(\mathcal{F}\) under a unitary frame is skew Hermitian: \[ \mathcal{A}{^\alpha}{_\beta} = -\overline{\mathcal{A}}{^\beta}{_\alpha},\quad \mathcal{F}{^\alpha}{_\beta} = -\overline{\mathcal{F}}{^\beta}{_\alpha}. \]
• \(\mathcal{F}\) is a \((1,1)\)-form (bidegree).

The holomorphic tangent bundle \(TM^+\) is a holomorphic vector bundle. The typical fibre is \(\mathbb{C}^m\) and the local basis is \(\qty{\partial/\partial z^\mu}\).

Gauge Theories

U(1) Gauge Symmetry

• \(P = \mathbb{R}^4 \times \mathrm{U}(1)\).
• Gauge potential \[ \mathcal{A} = \mathcal{A}_\mu \dd{x^\mu}. \]
  • Field strength \[ \mathcal{F} = \dd{\mathcal{A}}. \]
    • Component form \[ \mathcal{F}_{\mu\nu} = \pdv{\mathcal{A}_\nu}{x^\mu} - \pdv{\mathcal{A}_\mu}{x^\nu}. \]
  • Bianchi identity \[ \dd{\mathcal{F}} = \mathcal{F}\wedge \mathcal{A} - \mathcal{A} \wedge \mathcal{F} = 0. \]
    • Component form \[ \partial_\lambda \mathcal{F}_{\mu\nu} + \partial_\nu \mathcal{F}_{\lambda\mu} + \partial_\mu \mathcal{F}_{\nu\lambda} = 0. \]
    • The two homogeneous Maxwell equations.
• Maxwell action: \[ S_{\mathrm{M}}[\mathcal{A}] = \frac{1}{4}\int \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} \dd{^4 x} = -\frac{1}{4} \int F_{\mu\nu} F^{\mu\nu} \dd{^4 x}. \]

Dirac Monopole

• On \(U_{\mathrm{N}}\) and \(U_{\mathrm{S}}\), \[ \begin{align*} \mathcal{A}_{\mathrm{N}} &= \sigma_{\mathrm{N}}^* \omega = ig(1-\cos\theta)\dd{\phi}, \\ \mathcal{A}_{\mathrm{S}} &= \sigma_{\mathrm{S}}^* \omega = -ig(1+\cos\theta)\dd{\phi}. \end{align*}{} \]
• \[t_{\mathrm{NS}}\in\mathrm{U}(1)\] satisfies \[ \mathcal{A}_{\mathrm{N}} = t^{-1}_{\mathrm{NS}} \mathcal{A}_{|mathrm{S}} t^{\mathrm{NS}} + t^{-1}_{\mathrm{NS}}\dd{t_{\mathrm{NS}}} = \mathcal{A}_{\mathrm{S}} + i\dd{\varphi}. \]
  • \[ t_{NS}(\phi) = e^{i\varphi(\phi)}. \]
• Quantization condition: \[ 2g\in \mathbb{Z}. \]
• \(2g\) represents the homotopy class to which the bundle belongs.

The Aharonov-Bohm Effect

• \(M = \mathbb{R}^2 - \qty{0}\).
• \(P(M,\mathrm{U}(1))\).
• \(E = P\times_\rho \mathbb{C}\).
• On \(S^1\), \[ \vb*{A}(\vb*{r}) = \qty(-\frac{y\Phi}{2\pi r^2}, \frac{x\Phi}{2\pi r^2}),\quad \mathcal{A} = i\frac{\Phi}{2\pi}\dd{\theta}. \]
• \(\psi(\theta)\) be parallel transported: \[ \mathcal{D}\psi(\theta) = \qty{\dd{} + i\frac{\Phi}{2\pi} \dd{\theta}} \psi(\theta) = 0. \]
• Holonomy \[ \Gamma: \psi(0) \mapsto e^{-i\Phi}\psi(0). \]

Yang-Mills Theory

• \(P(\mathbb{R}^4, \mathrm{SU}(2))\).
• Connection one-form \[ \mathcal{A} = A{_\mu}{^\alpha} T_\alpha \dd{x^\mu}. \]
  • \[ T_\alpha = \frac{\sigma_\alpha}{2i}. \]
• Field strength \[ \mathcal{F} = \dd{\mathcal{A}} + \mathcal{A} \wedge \mathcal{A} = \frac{1}{2} \mathcal{F}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}. \]
  • \[ \mathcal{F}_{\mu\nu} = F_{\mu\nu}{^\alpha} T_\alpha. \]
  • \[ F_{\mu\nu}{^\alpha} = \partial_\mu A_{\nu\alpha} - \partial_\nu A_{\mu\alpha} + \epsilon_{\alpha\beta\gamma} A_{\mu\beta} A_{\nu\gamma}. \]
• Bianchi identity \[ \mathcal{D}\mathcal{F} = \dd{\mathcal{F}} + [\mathcal{A}, \mathcal{F}] = 0. \]
• Yang-Mills action \[ \mathcal{S}_{\mathrm{YM}}[\mathcal{A}] = -\frac{1}{4}\int \tr{\mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu}} = \frac{1}{2} \int_M \tr(\mathcal{F} \wedge * \mathcal{F}). \]
• Equations of motion: \[ \mathcal{D}_\mu \mathcal{F}^{\mu\nu} = \mathcal{D}*\mathcal{F} = 0. \]

Instantons

• \(P(\mathbb{R}^4, \mathrm{SU}(2))\), Euclidean metric.
• \(\mathbb{R}^4\) compatified to \(S^4\).
• Euclidean action \[ S^{\mathrm{E}}_{\mathrm{YM}} = \frac{1}{4}\int \tr{\mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu}} = -\frac{1}{2} \int_M \tr(\mathcal{F} \wedge * \mathcal{F}) = \mp \frac{1}{2} \int_M \tr(\mathcal{F} \wedge \mathcal{F}). \]
• Self-dual condition: \[ \mathcal{F}_{\mu\nu} = \pm * \mathcal{F}_{\mu\nu}. \]
• Asymptotic behavior: for a large \(L\), \[ \mathcal{A}_\mu(x) \rightarrow g(x)^{-1} \partial_\mu g(x),\quad \text{as}\quad \abs{x}\rightarrow L. \]
• \[ U_{\mathrm{N}} = \qty{x\in \mathbb{R}^4 | \abs{x} \le L+\epsilon},\quad U_{\mathrm{S}} = \qty{x\in \mathbb{R}^4 | \abs{x} \ge L - \epsilon}. \]
• No twist on \(U_{\mathrm{S}}\): \[ \mathcal{A}_S(x) = 0. \]
• Transition \[ \mathcal{A}_{\mathrm{N}} = t_{\mathrm{NS}}^{-1} \dd{t_{\mathrm{NS}}}. \]
• Transition \(S^3 \rightarrow \mathrm{SU}(2)\) classified by \[ \pi_3(\mathrm{SU}(2)) \cong \mathbb{Z}. \]
  • Class \(0\): constant map \[ g_0: x\mapsto e. \]
  • Class \(1\): identity map \[ g_1: x_1\mapsto \frac{1}{r}\qty[x^4 \mathbb{1} + x^i \sigma_i]. \]
  • Class \(n\): \[ g_n: x \mapsto \frac{1}{r^n}\qty[x^4 \mathbb{1} + x^i \sigma_i]^n. \]
• \[\mathcal{F}^2 = \mathcal{F} \wedge \mathcal{F} = \dd{K}.\]
  • \[ K = \tr\qty[\mathcal{A} \dd{\mathcal{A}} + \frac{2}{3} \dd{\mathcal{A}}^3]. \]
  • \[ \int_{S^4} \tr \mathcal{F}^2 = -\frac{1}{3} \int_{S^3} \tr{\mathcal{A}^3}. \]
• Degree of mapping \[ n = \frac{1}{24\pi^2} \int_{S^3} \tr(g^{-1} \dd{g})^3 = \frac{1}{2} \int_{S^4} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2. \]

Berry's Phase

Definition of Berry's Phase

• \(H(\vb*{R}(t))\) where \(\vb*{R}\) collects all parameters of \(H\).
• For \(\ket{\psi(0)} = \ket{n}\), \[ \ket{\psi(t)} = \exp\qty[i\eta(t) - i\int_0^t E_n(\vb*{R}(s))\dd{s}]\ket{n,\vb*{R}(t)}. \]
  • \[ \eta_n(T) = i\int_{\vb*{R}(0)}^{\vb*{R}(T)} \bra{n,\vb*{R}} \grad_{\vb*{R}} \ket{n,\vb*{R}} \dd{\vb*{R}}. \]

Berry's Phase, Berry's Connection, and Berry's Curvature

• \(M\) a manifold describing the parameter space.
• \(\vb*{R}\) the local coordinate.
• \(P(M,\mathrm{U}(1))\).
• Kets are in equivalence classes \[ [\ket{\vb*{R}}] = \set{g\ket{\vb*{R}} | g\in \mathrm{U}(1)}. \]
  • Index \(n\) omitted.
• Projection \[ \pi(g\ket{\vb*{R}}) = \ket{\vb*{R}}. \]
• Fixing the phase \(\Leftrightarrow\) choosing a section.
• Local section \[ \sigma(\vb*{R}) = \ket{\vb*{R}}. \]
• Local trivialization \[ \phi^{-1}(\ket{\vb*{R}}) = (\vb*{R},e). \]
• Right action \[ \phi^{-1}(\ket{\vb*{R}}\cdot g) = (\vb*{R}, g). \]
• Berry's connection: \[ \mathcal{A} = \mathcal{A}_\mu \dd{R}^\mu = \bra{\vb*{R}}(\dd{\ket{\vb*{R}}}) = -(\dd{\bra{\vb*{R}}})\ket{\vb*{R}}. \]
  • \(\mathcal{A}\) is anti-Hermitian.
• For \[ \ket{\vb*{R}}_j = \ket{\vb*{R}}_i t_{ij}(\vb*{R}), \] we have the transition \[ \mathcal{A}_j(\vb*{R}) = {_j}\bra{\vb*{R}}\dd{\ket{\vb*{R}}}_j = \mathcal{A}_i(\vb*{R}) + t_{ij}(\vb*{R})^{-1} \dd{t_{ij}(\vb*{R})}. \]
• Berry's curvature: \[ \mathcal{F} = \dd{\mathcal{A}} = \qty(\pdv{\bra{\vb*{R}}}{R^\mu})\qty(\pdv{\ket{\vb*{R}}}{R^\nu}) \dd{R^\mu} \wedge \dd{R^\nu}. \]
• Biased Hamiltonian \[ \mathcal{H}(\vb*{R})\ket{\vb*{R}(t),t} = i \dv{}{t}\ket{\vb*{R}(t),t}. \]
  • \[ \ket{\vb*{R}(t),t} = e^{i\eta(t)}\ket{\vb*{R}(t)}. \]
• Berry's phase as holonomy: \[ \tilde{\vb*{R}}(1) = \exp\qty(-\oint \bra{\vb*{R}}\dd{\ket{\vb*{R}}}) \cdot \ket{\vb*{R}(0)}. \]

• Electron in a magnetic field: \[ H(\vb*{B}) = \vb*{B}\cdot \vb*{\sigma}. \]
• Stick with the positive energy state \(\ket{\vb*{R}}_{\mathrm{N}}\).
• \[ \mathcal{A}_{\mathrm{N}} = {_\mathrm{N}}\bra{\vb*{R}} \dd{\ket{\vb*{R}}_\mathrm{N}}. \]
• Field strength \[ \mathcal{F} = \dd{\mathcal{A}} = \frac{i}{2} \frac{R_1 \dd{R_2} \wedge R_3 + R_2 \dd{R_3} \wedge \dd{R_1} + R_1 \dd{R_1}\wedge \dd{R_2}}{R^3}. \]
• With \(\vb*{B}\) given in spherical coordinates we have \[ \mathcal{A}_{\mathrm{S}} = \mathcal{A}_{\mathrm{N}} + e^{i\phi} \dd{e^{-i\phi}}. \]
• Similarity to the Dirac monopole due to the local \(\mathrm{U}(1)\) invariance here.

Characteristic Classes

Invariant Polynomials and Chern-Weil Homomorphism

Invariant Polynomials

• \(S^r(M(k,\mathbb{C}))\) denote the vector space of symmetric \(r\)-linear \(\mathbb{C}\)-valued functions on \(M(k,\mathbb{C})\).
• \[ S^*(M(k,\mathbb{C})) = \bigoplus_{r=0}^\infty S^r(M(k,\mathbb{C})). \]
• Product in \(S^*\): \[ \tilde{P}\tilde{Q}(X_1,\cdots,X_{p+q}) = \frac{1}{(p+q)!}\sum_{\sigma} \tilde{P}(X_{\sigma(1)}, \cdots,X_{\sigma(p)})\tilde{Q}(X_{\sigma(p+1)},\cdots,X_{\sigma(p+q)}). \]
• \(\tilde{P}\) invariant if \[ \tilde{P}(gA_1g^{-1},\cdots,gA_r g^{-1}) = \tilde{P}(A_1,\cdots,A_r). \]
  • Denoted \(\tilde{P}\in I^r\).
• \[ I^*({\frak g}) = \bigoplus_{r\ge 0} I^r({\frak g}). \]
  • An algebra under the induced multiplication \[ I^p({\frak g}) \oplus I^q({\frak g}) \rightarrow I^{p+q}({\frak g}). \]
• Invariant polynomial: \[ P(A) = \tilde{P}(A,\cdots,A). \]
  • \(P\in I^r({\frak{g}})\).
  • \(A\in{\frak{g}}\).
  • Any invariant polynomial may be written as a sum of \(\tr(A^r)\).
  • Acting on \(\frak{g}\)-valued \(p\)-forms by extension: \[ \tilde{P}(A_1 \eta_1,\cdots,A_r \eta_r) = \eta_1 \wedge \cdots \wedge \eta_r \tilde{P}(A_1,\cdots,A_r). \]
• Polarization: from any invariant polynomial \(P\) be define \(\tilde{P}\in I^r\) by expanding \[ P(t_1A_1 + \cdots + t_r A_r) \] and extract the coefficient of \[ t_1 t_2 \cdots t_r, \] and multiply it by \(1/r!\).
• Chern-Weil theorem: \(P\) an invariant polynomial, then
  • \[ \dd{P(\mathcal{F})} = 0. \]
  • Let \(\mathcal{F}\) and \(\mathcal{F}'\) be curvature two-forms corresponding to different connections \(\mathcal{A}\) and \(\mathcal{A}'\). Then \[ P(\mathcal{F}') - P(\mathcal{F}) \] is exact.
  • Notes:
    • We should clarity how \(\dd{}\) is acting: if \[ \mathcal{F} = {\frak{g}}_i \eta^{(i)}_{\mu\nu}(x) \dd{x^\mu} \wedge \dd{x^\nu}, \] then \(\dd{}\) acts on \[ \eta^{(i)}_{\mu\nu}(x) \dd{x^\mu} \wedge \dd{x^\nu} \] and \({\frak{g}}_i\) are deemed as constants.
    • A useful identity is (for \(\Omega_i\) being \({\frak{g}}_i\)-valued forms) \[ \dd{\tilde{P}_r}(\Omega_1,\cdots,\Omega_r) = \sum_{i=1}^r (-1)^{p_1 + \cdots + p_{i-1}}\tilde{P}_r(\Omega_1, \cdots, \dd{\Omega}_i,\cdots,\Omega_r). \]
    • With \[ \mathcal{A}_t = \mathcal{A} + t\theta;\quad \theta = (\mathcal{A}' - \mathcal{A}), \] and \[ \mathcal{F}_t = \dd{\mathcal{A}}_t + \mathcal{A}_t \wedge \mathcal{A}_t = \mathcal{F} + t\mathcal{D}\theta + t^2 \theta^2, \] we have \[ P_r(\mathcal{F}') - P_r(\mathcal{F}) = \dd{\qty[r\int_0^1 \tilde{P}_r(\mathcal{A}' - \mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t) \dd{t}]}. \]
  • Transgression: \[ TP_r(\mathcal{A}',\mathcal{A}) = r\int_0^1 \dd{t} \tilde{P}_r(\mathcal{A}' - \mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t). \]
• Invariant polynomial is closed and (in the sense of cohomology) independent of the gauge potential chosen.
  • \(P\) stays in the same cohomology class after changing the gauge potential \(\mathcal{A}\).
  • The cohomology class is called the characteristic class, denoted by \[ \chi_E(P), \] where \(E\) is the fibre bundle over which connections and curvatures are defined.

The symmetrized trace \[ \tilde{P}(A_1,\cdots,A_r) = \operatorname{str}(A_1,\cdots,A_r) = \frac{1}{r!} \sum_\sigma \tr(A_{\sigma(1)} \cdots A_{\sigma(r)}) \] is symmetric, \(r\)-linear and invariant. \[ P(A) = \tr(A^r) \] is an invariant polynomial. Acting on \(p\)-forms \(P\) becomes \[ \operatorname{str}(A_1 \eta_1,\cdots,A_r \eta_r) = \eta_1 \wedge \cdots \wedge \eta_r \operatorname{str}(A_1,\cdots,A_r). \] The diagonal combination becomes \[ P(A\eta) = \eta\wedge \cdots \wedge \eta P(A). \]

From \[P(A) = \tr(A^3)\] we define the polarization \[ \tilde{P}(A_1,A_2,A_3) = \frac{1}{6!}\cdot 3\tr(A_1A_2A_3 + A_2A_1A_3) = \operatorname{str}(A_1,A_2,A_3). \]

• Theorem. Let \(P\) be an invariant polynomial in \(I^*(G)\) and \(E\) a fibre bundle over \(M\) with structure group \(G\).
  • The map \[ \chi_E: P\in I^*(G) \rightarrow \chi_E(P)\in H^*(M) \] is a homomorphism (the Weil homomorphism), i.e. \[ (P_r P_s)(\mathcal{F}) = P_r(\mathcal{F}) \wedge P_s(\mathcal{F}) \] is in the cohomology class defined by \[ \chi_E(P_r) \wedge \chi_E(P_s). \]
  • If \(f:N\rightarrow M\) is a differentiable map, and \(f^* E\) is the pullback bundle, then we have the naturality \[ \chi_{f^* E} = f^* \chi_E, \] i.e. \[ f^* \chi_E(P) = \chi_{f^* E}(P). \]
    • It could be verified that \(f^*\mathcal{A}\) is a connection in \(f^* E\). Without this it could not be guaranteed that \(\chi_{f^* E}(P)\) makes sense.
• Characteristic classes of trivial bundles are trivial.

Chern Class

Definition of Chern Classes

• Total Chern class: for a complex vector bundle whose fibre is \(\mathbb{C}^k\), \[ \begin{align*} c(\mathcal{F}) &= \det\qty(\mathbb{1} + \frac{i\mathcal{F}}{2\pi}) \\ &= 1 + c_1(\mathcal{F}) + c_2(\mathcal{F}) + \cdots. \end{align*}{} \]
  • \(j\)th Chern class: \[ c_j(\mathcal{F}) = \Omega^{2j}(M). \]
• Diagonalization: at each point the field is diagonalized as \[ g^{-1}\qty(\frac{i\mathcal{F}}{2\pi}) g = \operatorname{diag}(x_1,\cdots,x_k). \]
  • \(x_k\) are two-forms (scalar-valued).
  • Total Chern class \[ \det\qty(\mathbb{1} + \frac{i\mathcal{F}}{2\pi}) = S_0(x) + S_1(x) + \cdots + S_k(x). \]
  • \(S_i\) are elementary symmetric functions: \[ \begin{align*} c_0 = S_0(x) &= 1, \\ c_1 = S_1(x) &= \sum x_j = \frac{i}{2\pi} \tr \mathcal{F}, \\ c_2 = S_2(x) &= \sum_{i<j} x_i \wedge x_j = \frac{1}{2}\qty(\frac{i}{2\pi})^2 \qty[\tr \mathcal{F} \wedge \tr \mathcal{F} - \tr(\mathcal{F} \wedge \mathcal{F})], \\ &\vdots \\ c_k = S_k(x) &= x_1 \wedge \cdots \wedge x_k = \qty(\frac{i}{2\pi})^k \det \mathcal{F}. \end{align*}{} \]

• \(F\) a complex vector bundle \(\mathbb{C}^2\) over \(M\).
• \(G = \mathrm{SU}(2)\).
• \(\dim M = 4\).
• Field \[ \mathcal{F} = \mathcal{F}^\alpha \frac{\sigma_\alpha}{2i}, \] where \[ \mathcal{F}^\alpha = \frac{1}{2} \mathcal{F}{^\alpha}_{\mu\nu} \dd{x^\mu} \wedge \dd{x^\nu}. \]
• Total Chern class \[ \begin{align*} c(\mathcal{F}) &= \det\qty(\mathbb{1} + \frac{i}{2\pi} \mathcal{F}^\alpha \frac{\sigma_\alpha}{2i}) \\ &= \det \begin{pmatrix} 1 + (i/2\pi)\mathcal{F}^3/(2i) & (i/2\pi) (\mathcal{F}^1 - i\mathcal{F}^2)/(2i) \\ (i/2\pi) (\mathcal{F}^1 + i\mathcal{F}^2)/(2i) & 1-(i/2\pi)(\mathcal{F}^3)/(2i) \end{pmatrix} \\ &= 1 + \frac{1}{4}\qty(\frac{i}{2\pi})^2\qty{\mathcal{F}^3 \wedge \mathcal{F}^3 + \mathcal{F}^1 \wedge \mathcal{F}^1 + \mathcal{F}^2 \wedge \mathcal{F}^2}. \end{align*}{} \]
• Individual Chern class: \[ \begin{align*} c_0(\mathcal{F}) &= 1, \\ c_1(\mathcal{F}) &= 0, \\ c_2(\mathcal{F}) &= \qty(\frac{i}{2\pi})^2 \sum_\alpha \frac{\mathcal{F}^\alpha \wedge \mathcal{F}^\alpha}{4}. \end{align*}{} \]

Properties of Chern Classes

• \(E\) a vector bundle with \(G = \mathrm{GL}(k,\mathbb{C})\) and \(F = \mathbb{C}^k\).
• Naturality of total Chern class: if \(f:N\rightarrow M\) is a smooth map, then \[ c(f^* E) = f^* c(E). \]
• Let \(E'\) be another vector bundle and \(E\oplus E'\) be their Whitney sum bundle. Then \[ c(E\oplus E') = c(E) \wedge c(E'). \]
  • Note that the curvature of a Whitney sum bundle is block diagonal: \[ \mathcal{F}_{E\oplus E'} = \operatorname{diag}(\mathcal{F}_E, \mathcal{F}_{E'}). \]

Splitting Principle

See Splitting Principle.

• Let \(E\rightarrow M\) be a vector bundle of rank \(n\) over a paracompact space \(M\). There exists a space \(N\) called the flag bundle associated to \(E\) and a map \(p:N\rightarrow M\) such that
  • the induced cohomology homomorphism \[ p^*: H^*(M) \rightarrow H^*(N) \] is injective, and
  • the pullback bundle \(p^* E\) breaks up as a direct sum of line bundles \[ p^*(E) = L_1 \oplus \cdots \oplus L_n. \]

Universal Bundles and Classifying Spaces

• Let \(M\) be a manifold with \(\dim M = m\) and let \(E\rightarrow M\) be a complex vector bundle with the fibre \(\mathbb{C}^k\). Then there exists a natural number \(N\) such that for all \(n>N\),
  • there exists a map \(f:M\rightarrow G_{k,n}(\mathbb{C})\) such that \[ E \cong f^* L_{k,n}(\mathbb{C}); \] where \(L_{k,n}(\mathbb{C})\) is the canonical \(k\)-plane byndle \(L_{k,n}\) over \(G_{k,n}(\mathbb{C})\) with fibre \(\mathbb{C}^k\);
  • \(f,g: M\rightarrow G_{k,n}(\mathbb{C})\) are homotopic if and only if \[ f^* L_{k,n}(\mathbb{C}) \cong g^* L_{k,n}(\mathbb{C}). \]
• Classifying space \(G_k(\mathbb{C})\): \[ G_k(\mathbb{C}) = \bigcup_{n=k}^\infty G_{k,n}(\mathbb{C}). \]
• Natural inclusions: \[ G_{k,k}(\mathbb{C}) \hookrightarrow G_{k,k+1}(\mathbb{C}) \hookrightarrow \cdots \hookrightarrow G_{k}(\mathbb{C}). \]
• Universal bundle: \[ L_k \rightarrow G_k(\mathbb{C}) \] with fibre \(\mathbb{C}^k\).
  • For any fibre bundle \(E\) with fibre \(\mathbb{C}^k\) we have a map \(f\) such that \[ E = f^* L_k(\mathbb{C}). \]
• A character class is a map \[ \chi: E\rightarrow \chi(E) \in H^*(M) \] such that
  • Naturality: \[ \chi(f^* E) = f^* \chi(E). \]
  • If \(E\) is equivalent to \(E'\): \[ \chi(E) = \chi(E'). \]
• Chern class \(c(E)\) defined axiomatically by
  • \[ c(f^* E) = f^* c(E). \]
  • \[ c(E) = c_0(E) + c_1(E) + \cdots + c_k(E), \] where \[ c_i(E) \in H^{2i}(M) \] and \(c_i(E) = 0\) for \(i>k\).
  • \[ c(E\oplus F) = c(E) \wedge c(F). \]
  • Normalization \[ c(L) = 1 + x \] where \(L\) is the canonical line bundle over \(\mathbb{C}P^n\).

Chern Characters

Definition of Chern Characters

• Total Chern character: \[ \operatorname{ch}(\mathcal{F}) = \tr \exp \qty(\frac{i\mathcal{F}}{2\pi}) = \sum_{j=0} \frac{1}{j!} \tr\qty(\frac{i\mathcal{F}}{2\pi})^j. \]
• \(j\)th Chern character: \[ \mathrm{ch}_j(\mathcal{F}) = \frac{1}{j!} \mathrm{tr}\qty(\frac{i\mathcal{F}}{2\pi})^j. \]
• Diagonalized: \[ \operatorname{ch}(\mathcal{F}) = \sum_{j=1}^k \exp(x_j). \]
• In terms of Chern classes: \[ \begin{align*} \mathrm{ch}_0(\mathcal{F}) &= k, \\ \mathrm{ch}_1(\mathcal{F}) &= c_1(\mathcal{F}), \\ \mathrm{ch}_2(\mathcal{F}) &= \frac{1}{2}\qty[c_1(\mathcal{F})^2 - 2c_2(\mathcal{F})]. \end{align*}{} \]

• \(P\) a \(\mathrm{U}(1)\) bundle over \(S^2\).
• \(\mathcal{A}_{\mathrm{N}}\) and \(\mathcal{A}_{\mathrm{S}}\) are local connections.
• \(\mathcal{F}^n = 0\) for \(n \ge 2\).
• Chern character \[ \mathrm{ch}(\mathcal{F}) = \mathbb{1} + \frac{i\mathcal{F}}{2\pi}. \]
• Magnetic charge \[ 2g = N = \frac{i}{2\pi} \int_{S^2} \mathcal{F} = \int_{S^2} \mathrm{ch}_1(\mathcal{F}). \]

• \(P\) a \(\mathrm{SU}(2)\) bundle over \(S^4\).
• Total Chern character \[ \mathrm{ch}(\mathcal{F}) = 2 + \cancelto{0}{\tr\qty(\frac{i \mathcal{F}}{2\pi})} + \frac{1}{2} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2. \]
• Instanton number \[ \frac{1}{2} \int_{S^4} \tr\qty(\frac{i\mathcal{F}}{2\pi})^2 = \int_{S^4} \mathrm{ch}_2(\mathcal{F}). \]

• \(P\) a \(\mathrm{U}(1)\) bundle over \(M\) a \(2m\)-dimensional manifold.
• The \(m\)th Chern character \[ \frac{1}{m!} \tr\qty(\frac{i\mathcal{F}}{2\pi})^m = \qty(\frac{i}{4\pi})^m \epsilon^{\mu_1\nu_1 \cdots \mu_m \nu_m} F_{\mu_1\nu_1 \cdots \mu_m \nu_m} \dd{x^1} \wedge \cdots \wedge \dd{x^{2m}}. \]

• \(L\) a complex line bundle.
• Curvature under the Fubini-Study metric \[ \mathcal{F} = -\partial \overline{\partial} \ln(1+\abs{z}^2) = -\frac{\dd{z}\wedge \dd{\overline{z}}}{(1+z\overline{z})^2} = 2i \frac{r\dd{r}\wedge \dd{\theta}}{(1+r^2)^2}. \]
• The first Chern character \[ \mathrm{ch}_1(\mathcal{F}) = -\frac{1}{\pi} \frac{r\dd{r}\wedge \dd{\theta}}{(1+r^2)^2}. \]
• Integral of the first Chern character \[ \mathrm{Ch}_1(L) = -\frac{1}{\pi} \int \frac{r\dd{r} \dd{\theta}}{(1+r^2)^2} = -1. \]

Properties of the Chern Characters

• Naturality: \[ \mathrm{ch}(f^* E) = f^* \mathrm{ch}(E). \]
• Let \(E\) and \(F\) be vector bundles over \(M\). The Chern classes of their direct product and Whitney sum are given by, respectively, \[ \begin{align*} \mathrm{ch}(E\otimes F) &= \mathrm{ch}(E) \wedge \mathrm{ch}(F), \\ \mathrm{ch}(E\oplus F) &= \mathrm{ch}(E) + \mathrm{ch}(F). \end{align*}{} \]

Todd Classes

• Todd class: \[ \begin{align*} \mathrm{Td}(\mathcal{F}) &= \prod_j \frac{x_j}{1-e^{-x_j}} \\ &= \prod_j \qty({1 + \frac{1}{2} x_j + \sum_{k\ge 1} (-1)^{k-1} \frac{B_{k}}{(2k)!} x^{2k}_j}) \\ &= 1 + \frac{1}{2} c_1(\mathcal{F}) + \frac{1}{12}\qty[c_1(\mathcal{F})^2 + c_2(\mathcal{F})] + \cdots. \end{align*}{} \]

Pontrjagin and Euler Classes

Pontrjagin Classes

• \(E\) a real vector bundle.
  • Orthonormal frames assigned.
• Structure group \(\mathrm{O}(k)\).
  • Lie algebra: not diagonalizable, but reducible to \[ \begin{pmatrix} 0 & \lambda_1 & & & \\ -\lambda_1 & 0 & & & \\ & & 0 & \lambda_2 & \\ & & -\lambda_2 & 0 & \\ & & & & \ddots \end{pmatrix}. \]
• Total Pontrjagin class: \[ p(\mathcal{F}) = \det\qty(\mathbb{1} + \frac{\mathcal{F}}{2\pi}). \]
  • For skew-symmetric \(\mathcal{F}^T = -\mathcal{F}\), \[ p(\mathcal{F}) = p(\mathcal{F}^T). \]
• Expansion \[ p(\mathcal{F}) = 1 + p_1(\mathcal{F}) + p_2(\mathcal{F}) + \cdots. \]
  • \(p_j\) is a polynomial of order \(2j\) and is an element of \(H^{4j}(M;\mathbb{R})\).
• Diagonalized \[ p(\mathcal{F}) = \prod_{i=1}^{\lfloor k/2 \rfloor} (1+x_i^2), \] where \[ \frac{\mathcal{F}}{2\pi} \rightarrow \begin{pmatrix} -ix_1 & & & & \\ & ix_1 & & & \\ & & -ix_2 & & \\ & & & ix_2 & \\ & & & & \ddots \end{pmatrix}. \]
• Pontrjagin class \[ p_j(\mathcal{F}) = \sum_{i_1<\cdots<i_j} x_{i_1}^2 \cdots x_{i_j}^2. \]
  • In terms of \(\mathcal{F}\), \[ \begin{align*} p_1(\mathcal{F}) &= -\frac{1}{2}\qty(\frac{1}{2\pi})^2 \tr \mathcal{F}^2, \\ p_2(\mathcal{F}) &= \frac{1}{8}\qty(\frac{1}{2\pi})^4 \qty[(\tr \mathcal{F}^2)^2 - 2\tr \mathcal{F}^4], \\ & \vdots \\ p_{\lfloor k/2 \rfloor}(\mathcal{F}) &= \qty(\frac{1}{2\pi})^k \det \mathcal{F},\quad \text{if } k \text{ is even}. \end{align*}{} \]
• Pontrjagin class of Whitney sum: \[ p(E\oplus F) = p(E) \wedge p(F). \]
• Relation to Chern classes: \[ p_j(E) = (-1)^j c_{2j}(E^{\mathbb{C}}). \]

• \(M\) a four-dimensional Riemannian manifold.
• Orthonormal frame employed.
• Structure group \(\mathrm{O}(4)\).
• Curvature two-form \[ \mathcal{R} = \frac{1}{2} \mathcal{R}_{\alpha\beta} \theta^\alpha \wedge \theta^\beta. \]
• Pontrjagin class: \[ \det\qty(\mathbb{1} + \frac{\mathcal{R}}{2\pi}) = \underbrace{1}_{p_0} \underbrace{\mathbin{-}\frac{1}{8\pi}^2 \tr \mathcal{R}^2}_{p_1} + \underbrace{\frac{1}{128\pi^4}\qty[(\tr \mathcal{R}^2)^2 - 2\tr \mathcal{R}^4]}_{p_2}. \]

Euler Classes

• \(M\) a \(2l\)-dimensional orientable Riemannian manifold.
• Euler class: square root of \(p_l\), defined by \[ e(A)e(A) = p_l(A). \]
  • Both sides understood as functions of a \(2l\times 2l\) matrix \(A\).
  • We solve for \(e(A)\) first and then substitute \(\mathcal{R}\) for \(A\).
  • If \(M\) is odd-dimensional we define \[ e(M) = 0. \]
• Pfaffian: for \(P\) a skew-symmetric \(2l\times 2l\)-matrix only, \[\operatorname{Pf}(A) = \frac{(-1)^l}{2^l l!} \sum_\sigma \operatorname{sign}(\sigma) A_{\sigma(1)\sigma(2)} \cdots A_{\sigma(2l-1) \sigma(2l)}.\]
  • \[ \det A = (\operatorname{Pf}(A))^2. \]
  • \[ \operatorname{Pf}(X^t A X) = \operatorname{Pf}(A) \det A. \]
  • Since the determinant of an odd-dimensional skew-symmetric matrix vanishes, we put \(e(M) = 0\) for an odd-dimensional manifold.
• Euler class in terms of \(\mathcal{R}\): \[ e(M) = \operatorname{Pf}\qty(\frac{\mathcal{R}}{2\pi}) = \frac{(-1)^l}{(4\pi)^l l!} \sum_\sigma \operatorname{sign}(\sigma) \mathcal{R}_{\sigma(1)\sigma(2)} \cdots \mathcal{R}_{\sigma(2l-1)\sigma(2l)}. \]
  • Generating function \[ e(x) = x_1 \cdots x_l, \] where \[ x_j = -\frac{\lambda_j}{2\pi}. \]

• \(M = S^2\). Consider \(TS^2\).
• Curvature two-form \[ \mathcal{R}_{\theta\phi} = \sin\theta \dd{\theta} \wedge \dd{\phi}. \]
• \(p_1(S^2) = 0\) as a differential form: \[ \begin{align*} p_1(S^2) = -\frac{1}{8\pi^2} \tr \mathcal{R}^2 = \qty(\frac{1}{2\pi} \sin\theta \dd{\theta} \wedge \dd{\phi})^2. \end{align*} \]
• Reading off the Euler form: \[ e(S^2) = \frac{1}{2\pi} \sin\theta \dd{\theta} \wedge \dd{\phi}. \]
• Euler characteristic: \[ \int_{S^2} e(S^2) = 2. \]

• \(M\) a four-dimensional orientable manifold.
• The structure group of \(TM\) is \(\mathrm{O}(4)\).
• Euler class \[ e(M) = \frac{1}{2(4\pi)^2} \epsilon^{ijkl} \mathcal{R}_{ij} \wedge \mathcal{R}_{kl}. \]
• Pontrjagin class: \[ p_2(M) = \frac{1}{128\pi^4}\qty[(\tr \mathcal{R}^2)^2 - 2\tr \mathcal{R}^4] = x_1^2 x_2^2. \]

Hirzebruch L-Polynomial and Â-Genus

• Hirzebruch \(L\)-Polynomial: \[ \begin{align*} L(x) &= \prod_{j=1}^k \frac{x_j}{\tanh x_j} \\ &= \prod_{j=1}^k \qty(1 + \sum_{n\ge 1} (-1)^{n-1} \frac{2^{2n}}{(2n)!}B_n x^{2n}_j). \end{align*}{} \]
• In terms of Pontrjagin classes: \[ L(\mathcal{F}) = 1 + \frac{1}{3} p_1 + \frac{1}{45}(-p_1^2 + 7p_2) + \cdots. \]
• \[ L(E\oplus F) = L(E) \wedge L(F). \]
• \(\hat{A}\) genus, Dirac genus: \[ \begin{align*} \hat{A}(\mathcal{F}) &= \prod_{j=1}^k \frac{x_j/2}{\sinh(x_j / 2)} \\ &= \prod_{j=1}^k \qty({1 + \sum_{n\ge 1} (-1)^n \frac{(2^{2n} - 2)}{(2n)!} B_n x^{2n}_j}). \end{align*}{} \]
• \[ \hat{A}(E\oplus F) = \hat{A}(E) \wedge \hat{A}(F). \]
• In terms of Pontrjagin classes: \[ \hat{A}(\mathcal{F}) = 1 - \frac{1}{24}p_1 + \frac{1}{5760}(7p_1^2 - 4p_2) + \cdots. \]

• \(M\) a compact connected and orientable four-dimensional manifold.
• Define \[ \sigma: H^2(M,\mathbb{R}) \times H^2(M,\mathbb{R}) \rightarrow \mathbb{R} \] by \[ \sigma([\alpha], [\beta]) = \int_M \alpha \wedge \beta. \]
• \(\sigma\) is a \(b^2 \times b^2\) symmetric matrix, where \[b^2 = \dim H^2(M;\mathbb{R}).\]
• Hirzebruch signature: \[ \tau(M) = p - q. \] - \(p\) and \(q\) are the number of positive and negative eigenvalues of \(\sigma\), respectively.
• Hirzebruch signature theorem: \[ \tau(M) = \int_{M} L_1(M) = \frac{1}{3} \int_M p_1(M). \]

Chern-Simons Forms

Definition of Chern-Simons Forms

• Chern-Simons form of \(P_j(\mathcal{F})\), where \(P_j(\mathcal{F})\) is an arbitrary \(2j\)-form characteristic class: \[ Q_{2j-1}(\mathcal{A},\mathcal{F}) = TP_j(\mathcal{A},0) = j\int_0^1 \tilde{P}_j(\mathcal{A}, \mathcal{F}_t,\cdots,\mathcal{F}_t) \dd{t}. \]
  • We set \(\tilde{A}' = 0\) and \(\tilde{F}' = 0\).
  • $Q$ exists only locally.【.text-danger】
  • \(Q_{2j-1}\) is an \((2j-1)\)-form.
  • \(Q_{2j-1}\) defines the topology of the boundary.

The Chern-Simons Form of the Chern Character

• \[ \mathcal{A}_t = t\mathcal{A}. \]
• \[ \mathcal{F}_t = t\mathcal{F} + (t^2 - t)\mathcal{A}^2. \]
• \[ Q_{2j-1}(\mathcal{A},\mathcal{F}) = \frac{1}{(j-1)!}\qty(\frac{i}{2\pi})^j \int_0^1 \dd{t} \operatorname{str}(\mathcal{A}, \mathcal{F}_t^{j-1}). \]
  • Examples: \[ \begin{align*} Q_1(\mathcal{A},\mathcal{F}) &= \frac{i}{2\pi} \tr \mathcal{A}, \\ Q_3(\mathcal{A},\mathcal{F}) &= \frac{1}{2} \qty(\frac{i}{2\pi})^2 \tr \qty(\mathcal{A} \dd{\mathcal{A}} + \frac{2}{3}\mathcal{A}^3), \\ Q_5(\mathcal{A},\mathcal{F}) &= \frac{1}{6} \qty(\frac{i}{2\pi})^3 \tr \qty[\mathcal{A}(\dd{\mathcal{A}})^2 + \frac{3}{2}\mathcal{A}^3 \dd{\mathcal{A}} + \frac{3}{5}\mathcal{A}^5]. \end{align*}{} \]

Cartan's Homotopy Operator and Applications

• \(S(\mathcal{A},\mathcal{F})\) a polynomial.
• Homotopy operator: \[ k_{01} S(\mathcal{A}_t,\mathcal{F}_t) = \int_0^1 \delta t l_t S(\mathcal{A}_t, \mathcal{F}_t). \]
• Cartan's homotopy formula: \[ S(\mathcal{A}_1,\mathcal{F}_1) - S(\mathcal{A}_0,\mathcal{F}_0) = (\dd{k_{01}} + k_{01}\dd{})S(\mathcal{A_t},\mathcal{F}_t). \]
• \(l_t\): an anti-derivative operator, \[ \begin{align*} l_t \mathcal{A}_t &= 0, \\ l_t \mathcal{F}_t &= \delta t(\mathcal{A}_1 - \mathcal{A}_0), \\ l_t(\eta_p \omega_q) &= (l_t \eta_p)\omega_q + (-1)^p \eta_p l_t(\omega_q). \end{align*} \]

With \[ \mathrm{ch}_{j+1}(\mathcal{F}) = \dd{\qty[k_{01} \mathcal{ch}_{j+1}(\mathcal{F}_t)]} = \dd{Q_{2j+1}}, \] we find \[ Q_{2j+1} = k_{01} \mathrm{ch}_{j+1}(\mathcal{F}_t) = \frac{1}{j!}\qty(\frac{i}{2\pi})^{j+1} \int_0^1 \delta t \mathrm{str}(\mathcal{A}, \mathcal{F}^j_t). \]

• The Chern-Simons form is not necessarily gauge invariant: \[ Q_{2j+1}(\mathcal{A}^g,\mathcal{F}^g) - Q_{2j+1}(\mathcal{A},\mathcal{F}) = Q_{2j+1}(g^{-1} \dd{g}, 0) + \dd{\alpha_{2j}}, \]
• where \[ \begin{align*} \mathcal{A}^g &= g^{-1}(\mathcal{A} + \dd{}) g, \\ \mathcal{F}^g &= g^{-1} \mathcal{F} g. \end{align*}{} \]
• The first term on the RHS \[Q_{2j+1}(g^{-1}\dd{g},0)\] is closed.

• Lagrangian: \(\mathrm{U}(1)\) bundle in three-dimensional spacetime, with \(\mathcal{A} = iA\) and \(\mathcal{F} = iF\), \[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{1}{4} m \epsilon^{\lambda\mu\nu} F_{\lambda\mu} A_\nu. \]
• The second term is the Chern-Simons forms of the second Chern character \(F^2\).
• Field equation \[ \partial_\mu F^{\mu\nu} + m {*} F^\nu = 0. \]
• Under \[ A_\mu \rightarrow A_\mu + \partial_\mu \theta, \] the Lagrangian changes by \[ \mathcal{L} \rightarrow \mathcal{L} + \frac{1}{4} m \epsilon^{\lambda\mu\nu} F_{\lambda\mu}(A_\nu + \partial_\nu \theta) = \mathcal{L} + \frac{1}{2} m \partial_\nu({*} F^\nu \theta). \]

Stiefel-Whitney Classes

Spin Bundles

• \(LM\) the frame bundle related to \(TM\), where \(M\) a \(m\)-dimensional manifold.
• Spin structure: \(\tilde{t}_{ij}\in \mathrm{Spin}(m)\) such that \[ \varphi(t_{ij}) = t_{ij} \] where \(\varphi\) is the projection from \(\mathrm{Spin}(m)\) to \(\mathrm{SO}(m)\).
• Not all manifolds admit spin structures.

&circC;ech Cohomology Groups

• &circC;ech \(r\)-cochain: totally symmetric function to \(\mathbb{Z}_2\), i.e. for any permutation \(\sigma\), \[ f(i_{\sigma(0)}, \cdots, i_{\sigma(r)}) = f(i_0,\cdots,i_r). \]
• \(C^r(M;\mathbb{Z}_2)\) be the multiplicative group of &circC;ech \(r\)-cochains.
• Coboundary operator \[\delta: C^r(M;\mathbb{Z}_2) \rightarrow C^{r+1}(M;\mathbb{Z}_2)\] defined by \[ (\delta f)(i_0,\cdots,i_{r+1}) = \prod_{j=1}^{r+1}f(i_0,\cdots,\cancel{i}_j,\cdots,i_{r+1}). \]
• \(\delta\) is nilpotent: \[ \delta^2 f = 1. \]
• Cocycle group \(Z^r(M;\mathbb{Z}_2)\), coboundary group \(B^r(M;\mathbb{Z}_2)\), and &circC;ech cohomology group \(H^r(M;\mathbb{Z}_2)\): \[ \begin{align*} Z^r(M;\mathbb{Z}_2) &= \set{f\in C^r(M;\mathbb{Z}_2) | \delta f = 1}, \\ B^r(M;\mathbb{Z}_2) &= \set{f\in C^r(M;\mathbb{Z}_2) | f = \delta f'}, \\ H^r &= \operatorname{ker} \delta_r / \operatorname{im} \delta_{r-1} = Z^r(M,\mathbb{Z}_2)/B^r(M,\mathbb{Z}_2). \end{align*} \]

Definition of Stiefel-Whitney Classes

• Stiefel-Whitney class \(w_r\): a characteristic class which takes values in \(H^r(M;\mathbb{Z}_2)\).
• \(\qty{U_i}\) a simple open covering of \(M\).
  • The intersection of any number of charts is either empty of contractible.
• \(\qty{e_{i\alpha}}\) be a local orthonormal frame of \(TM\) over \(U_i\).
• &circC;ech \(1\)-cochain defined by \[ f(i,j) = \det(t_{ij}) = \pm 1. \]
  • \(\delta f = 1\), i.e. \[ f\in Z^1(M;\mathbb{Z}_2). \]
• First Stiefel-Whitney class: \[ w_1(M) = \qty[f] \in H^1(M;\mathbb{Z}_2). \]
  • \(f\) on a new frame \[\tilde{e}_{i\alpha} = h_i e_{i\alpha}\] still defines the same cohomology class.
• \(M\) is orientable if and only if \(w_1(M)\) is trivial.
• Second Stiefel-Whitney class: \[ w_2(M) \in H^2(M;\mathbb{Z}_2) \] defined by \([f]\) where \[ \tilde{t}_{ij}\tilde{t}_{jk}\tilde{t}_{ki} = f(i,j,k) \mathbb{1}. \]
  • \(f\) on a new lift \(-\tilde{t}_{ij}\) still defines the same cohomology class.
• If \(M\) is orientable, then a spin bundle over \(M\) exists if and only if \(w_2(M)\) is trivial.
• Examples:
  • \(x\) denotes the generator of \(H^2(\mathbb{C}P^m;\mathbb{Z}_2)\), \[ w_1(\mathbb{C}P^m) = 1,\quad w_2(\mathbb{C}^m) = \begin{cases} 1, & m \text{ odd}, \\ x, & m \text{ even}. \end{cases} \]
  • \[ w_1(S^m) = w_2(S^m) = 1. \]
  • \(\Sigma_g\) denotes the Riemann surface of genus \(g\), \[ w_1(\Sigma_g) = w_2(\Sigma_g) = 1. \]

Japanese Phonology and Phonetics

Hiraganas

ん

• [m] when the following sound is [m], [p], [b].
  • 鉛【えん】筆【ぴつ】
  • 南【なん】部【ぶ】 (Nambu)
• [ŋ] when the following sound is [k], [g].
  • 天【てん】気【き】
• [N] when at the end of utterances. Touch the root of your tongue with your uvula.
  • 本【ほん】
• [n] elsewhere. Touch the roof of your mouth with your tongue just a little farther back than your upper gums.

Phonology

大和言葉

See 大和言葉.

• There should be no 濁音/半濁音 at the beginning of a word. However, exceptions do exist in modern Japanese.
  • 抱【だ】く (c. 999) from 抱【いだく】 (c. 850), where the beginning い was dropped, the latter from 抱【うだく】 (c. 810), ultimately from 抱【むだ】く (c. ) from old Japanese.
    • 抱【むだ】く (c. 759) from 身【む】 + 綰【た】く, where 身【む】 was shifted from 身【み】, itself cognate with 実【み】, while 綰【た】く was probably derived from 手【た】.
  • 薔薇【ばら】 from 茨【いばら】, ultimately from 茨【うばら】 which shifted also to 茨【むばら】.
  • 場【ば】 from 場【にぱ】, cognate with 庭【にわ】.
  • This rule does not apply to オノマトペ (e.g. ピカピカ), 動植物名 (e.g. ブリ, ブナ), alternation from voiceless consonants (e.g. ばば, じじ), and words that emphasizes negative meaning (e.g. 蟹【がに】股【また】, 狡【ずる】い).
• There should be no ら行音 at the beginning of a word, a property that shared with Altaic languages.
• Upon forming compounds, 連濁 may occur, and the consonants in the first word may alter, e.g. 木【こ】立【だ】ち from 木【き】 + 立【た】ち, and 酒【さか】樽【だる】 from 酒【さけ】 + 樽【たる】.

On the etymology of 大和【やまと】: 諸説ある

• 山【やま】の麓【ふもと】
• 山【やま】門【かど】
  • 門【かど】 itself from か + 戸【と】, the latter of which is to be distinguished from 所【と】.
    • か was probably prepended the same way as in 籠【かご】, where ご from こ appears in 込【こ】む/込【こ】める.
  • Why not 山【やま】戸【と】?
• 山【やま】跡【あと】
  • 跡【あと】 from 足【あ】 + 所【と】.
  • Why not 山【やま】所【と】?
• See also 邪馬台国.

Apophony

• い acts as an unbound nominalizing suffix (emphatic nominal particle), e.g.
  • 木【き】 (standalone form) from 木【こ】 (bounded form, only in compounds) + い.
  • 火【ひ】 from 火【ほ】 + い.
  • 丈【たけ】 from 高【たか】 + い.
  • 手【て】 from 手【た】 + い.
  • 目【め】 from 目【ま】 + い.
  • 白【しろ】 from 白【しら】 + い.
  • 口【くち】 from 口【くつ】 + い.
  • 酒【さけ】 from 酒【さか】 + い.
  • 雨【あめ】 from 雨【あま】 + い.
  • 天【あめ】 from 天【あま】 + い.

Rendaku

• Lyman's Law states that there can be no more than one voiced obstruent (a consonant sound formed by obstructing airflow) within a morpheme.
  • 山【やま】門【かど】
  • 一【ひと】人【り】旅【たび】
  • 山【やま】火【か】事【じ】
• For some lexical items, rendaku does not manifest if there is a voiced obstruent near the morphemic boundary, including preceding the boundary.
• Rendaku usually only applies to native Japanese words, but it also occurs infrequently in Sino-Japanese words (Japanese words of Chinese origin) especially where the element undergoing rendaku is well integrated ("vulgarized").
• Rendaku also tends not to manifest in compounds which have the semantic value of "X and Y" (so-called dvandva or copulative compounds).
• Rendaku is also blocked by what is called a "branching constraint". In a right-branching compound, the process is blocked in the left-branching elements.
• See here for more rules.

Vocabulary

Verbs

• 言う as いう or ゆう?
  • 終止形/連体形 pronounced as ゆう.
  • Otherwise pronounced as いって/いえば.
  • Always written as 言【い】う.
• 萌【も】える is possibly cognate with 燃【も】える.

Nouns

• Why is China ちゅうごく while Korea is かんこく? Why the difference between ご and こ?.
• Why is ホワイト spelled as it is?

Dates

• Coordinate terms.

Looking up Etymology

• 語源由来辞典
• 日本語の語源

Names

Surnames

• 小鳥遊【たかなし】 from 鷹【たか】 + 無【な】し, possibly from 高【たか】梨【なし】

Nancy was additionally a member of Countrywide’s Model Validation Committee and one of many senior managers in cost of Countrywide Commercial Real Estate Finance.

Japanese Adjectives and Verbs

Word Formation

i-adjectives

• しい adjectives:
  • Mostly words for feeling.
  • From old Japanese しく.
  • 悲しい, 嬉しい, etc.
  • May be appended after 未然形, e.g. 勇ましい, 忙しい, etc.
• ない adjectives:
  • Having that quality, having that state, very much that quality or state.
  • From ancient copula or stative verb ぬ.
  • Or from なし.
  • ない is not for 無い here, cf. inflammable is flammable.
  • 切【せつ】ない
  • 危【あぶ】ない

na-adjectives

• やか adjectives:
  • Subjective.
  • や for softness and か for visible.
  • 鮮やか, 穏やか, 爽やか, etc.
• らか adjectives:
  • Similar to やか.
  • 明らか, 柔らか, etc.

たる-adjectives

• Archaic.
  • 連体形-たる, 連用形-と/として.
  • 堂々たる, 茫然たる, etc.

なる-adjectives

• 単なる, 聖なる, etc.

Verb Suffixes

• めく: to show signs of, to have the appearance of, to behave like
  • ときめく: とき as in どきどき
  • 時【とき】めく
  • 呻【うめ】く
  • 粧【めか】す: causative, to dress up oneself, to make like

Uncommon Verbs

• 灯【とも】る/灯【とも】す: to burn, to be lit / to light, to turn on
  • from 留【とど】まる or 留【とど】める
  • 取【と】 + 燃【も】
  • 火を灯す.
• 撓【おお】る/撓【しお】る, 撓【しわ】る, 撓【しな】る: to bend / to bend, to bully
• 温【ぬく】もる/温【ぬく】める: to become warm / to warm, to heat, to nurse, to renew
  • from 夏【なつ】 + 来【く】
• 凪【な】ぐ: to calm down, to die down

Note on 敬語

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Density Functional Theory (II)

Kohn-Sham Equations

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Foundations

The Hohenberg-Kohn Theorems

• Hamiltonian: \[ H = -\frac{\hbar^2}{2m} \sum_i \grad_i^2 + \sum_i V_{\mathrm{ext}}(\vb*{r}_i) + \frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vb*{r}_i - \vb*{r}_j}}. \]
• Theorem. For any system of interacting particles in an external potential \(V_{\mathrm{ext}}(\vb*{r})\), the potential \(V_{\mathrm{ext}}(\vb*{r})\) is determined uniquely, up to a constant, by the ground state particle density \(n_0(\vb*{r})\).
  • Therefore, teh wavefunctions of all states are determined, given only \(n_0(\vb*{r})\).
• Theorem. A universal functional \(E[n]\) in terms of the density \(n(\vb*{r})\) can be defined, valid for any external potential \(V_{\mathrm{ext}}(\vb*{r})\). For any particular \(V_{\mathrm{ext}}(\vb*{r})\), the exact ground state energy of the system is the global minimum of the functional.
  • The functional \(E[n]\) alone is sufficient to determined the exact ground state energy.
• Spin included: \[ E = E_{\mathrm{HK}}[n_\uparrow, n_\downarrow]. \]

The Kohn-Sham Equations

Framework

• Assumption. The exact ground state can be represented by the ground state density of an axuiliary system of non-interacting particles.
• The auxiliary Hamiltonian is chosen to have the usual kinetic operator and an effective local potential \(V^\sigma_{\mathrm{eff}}(\vb*{r})\).
• Energy functional: \[ E_{\mathrm{KS}} = T_s[n] + \int \dd{\vb*{r}} V_{\mathrm{ext}}(\vb*{r}) n(\vb*{r}) + E_{\mathrm{Hatree}}[n] + E_{II} + E_{\mathrm{xc}}[n]. \]
  • Density \[ n(\vb*{r}) = \sum_\sigma \sum_{i=1}^{N^\sigma} \abs{\psi_i^\sigma(\vb*{r})}^2. \]
  • Kinetic energy \[ T_s = \frac{1}{2}\sum_\sigma \sum_{i=1}^{N^\sigma} \int \dd{^3 \vb*{r}} \abs{\grad \psi_i^\sigma (\vb*{r})}^2. \]
  • Hatree energy \[ E_{\mathrm{Hatree}}[n] = \frac{1}{2} \iint \dd{^3 \vb*{r}} \dd{^3 \vb*{r}'} \frac{n(\vb*{r}) n(\vb*{r}')}{\abs{\vb*{r} - \vb*{r}'}}. \]
  • \(V_{\mathrm{ext}}\) includes the potential due to the nuclei and any other external fields.
  • \(E_{II}\) is the interaction between the nuclei.
  • \(E_{\mathrm{xc}}\) is the exchange-correlation energy.
• Kohn-Sham equations: \[ \color{red} (H^\sigma_{\mathrm{KS}} - \epsilon^\sigma_i) \psi^\sigma_i = 0. \]
  • \[ H^\sigma_{\mathrm{KS}} = -\frac{1}{2}\grad^2 + V^\sigma_{\mathrm{KS}}(\vb*{r}). \]
  • \[ V^\sigma_{\mathrm{KS}}(\vb*{r}) = V_{\mathrm{ext}}(\vb*{r}) + V_{\mathrm{Hatree}}(\vb*{r}) + V^\sigma_{\mathrm{xc}}(\vb*{r}). \]
  • \[ V_{\mathrm{Hatree}}(\vb*{r}) = \frac{\delta E_{\mathrm{Hatree}}}{\delta n(\vb*{r},\sigma)}. \]
  • \[ V^\sigma_{\mathrm{xc}}(\vb*{r}) = \frac{\delta E_{\mathrm{xc}}}{\delta n(\vb*{r},\sigma)}. \]

Exchange-Correlation

• Exchange-correlation functional \[ E_{\mathrm{xc}}[n] = \int \dd{\vb*{r}} n(\vb*{r}) \epsilon_{\mathrm{xc}}([n], \vb*{r}). \]
  • \(n(\vb*{r})\) is the total density.
  • \(\epsilon_{xc}\) may depend on \(n_\uparrow\) and \(n_\downarrow\).
• Exchange-correlation potential \[ V^\sigma_{\mathrm{xc}}(\vb*{r}) = \epsilon_{\mathrm{xc}}([n],\vb*{r}) + n(\vb*{r}) \frac{\delta \epsilon_{\mathrm{xc}}([n],\vb*{r})}{\delta n(\vb*{r},\sigma)}. \]

Meaning of the Eigenvalues

• The highest eigenvalue in a finite system is \(-I\), where \(I\) is the ionization energy,
• Slater-Janak theorem: \[ \epsilon_i = \dv{E_{\mathrm{total}}}{n_i}. \]
  • \(n_i\) is the occupation of the state \(i\).

Time-Dependent Density Functional Theory

• \[ i\hbar \dv{\psi_i(t)}{t} = H(t)\psi_i(t). \]
• \(V_{\mathrm{xc}}\) depends on \(n(\vb*{r}, t')\) for all \(t' \le t\).
• Adiabatic LDA: approximation that \(V_{\mathrm{xc}}\) depends on \(n\) at the same \(t\) only.

Complex Manifolds

---

Definition and Examples

Definitions

• Holomorphic \(f: \mathbb{C}^m \rightarrow \mathbb{C}{}\):
  • Cauchy-Riemann relations: \[ \pdv{f_1}{x^\mu} = \pdv{f_2}{y^\mu},\quad \pdv{f_2}{x^\mu} = -\pdv{f_1}{y^\mu}. \]
  • \(f: \mathbb{C}^m \rightarrow \mathbb{C}{}\) is holomorphic if each \(f^\lambda\) is holomorphic.
• Complex manifold \(M\):
  • \(M\) is a topological space,
  • \(M\) is provided with an atlas \(\qty{(U_i,\varphi_i)}\),
  • each \(\varphi_i\) maps the open set \(U_i\) to an open subset \(U\subset \mathbb{C}^m\), and
  • for each pair of \(U_i\cap U_j \neq \emptyset\), \[ \psi_{ji} = \varphi_j\circ \varphi_i^{-1} \] is holomorphic.
• \(\dim_{\mathbb{C}} M = m\), while
• \(\dim M = 2m\).
• If the union of two atlases is again an atlas which satisfy the definition of a complex manifold, they are said to define the same complex structure.

Examples

Stereographic projection \(S^2 \rightarrow \mathbb{C}\) naturally promote \(S^2\) to a complex manifold.

Torus:

• Let \(L\) be a lattice \[L(\omega_1,\omega_2) = \set{\omega_1 m + \omega_2 n|m,n\in \mathbb{Z}}.\]
  • \(\omega_1/\omega_2 \notin \mathbb{R}\).
  • Take \(\Im(\omega_2/\omega_1)>0\).
• Homeomorphism \[ T^2 \cong \mathbb{C}/L(\omega_1,\omega_2). \]
• The pair \((\omega_1,\omega_2)\) defined a complex structure on \(T^2\).
• Two lattices \(L(\omega_1,\omega_2)\) coincides if and only if there exists a matrix \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{PSL}(2,\mathbb{Z}) = \mathrm{SL}(2,\mathbb{Z})/\mathbb{Z}_2 \] where \(\pm A\) are identified such that \[ \begin{pmatrix} \omega_1' \\ \omega_2' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}. \]
• Take \((1,\tau)\) to be the generators of a lattice.
  • \(\tau\) and \(\tau'\) define the same complex structure if \[ \tau = \frac{a\tau + b}{c\tau + d} \] where \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{PSL}(2,\mathbb{Z}). \]
• Modular transform: \(\tau \rightarrow \tau'\) generated by
  • \(\tau \rightarrow \tau+1\),
  • \(\tau \rightarrow -1/\tau\).

Complex projective space: denoted by \(\mathbb{C}P^n\).

• \(z\in \mathbb{C}^{n+1}\) identified with \(z'\) if \(z=az'\) for some complex number \(a\neq 0\).
• The \(n+1\) numbers \(z^0,\cdots,z^n\) are called homogeneous coordinates.
• For \[ U_\mu = \set{z\in \mathbb{C}\backslash\qty{0} | z^\mu \neq 0}, \] a chart could defined by \(\xi^\nu_{(\mu)} = z^\nu/z^\mu\).

Complex Grassman manifolds:

• \[ G_{k,n}(\mathbb{C}) = M_{k,n}(\mathbb{C})/\mathrm{GL}(k,\mathbb{C}). \]
  • \(M_{k,n}\) consists of all \(k\times n\) matrices of rank \(k\).
• In particular, \[ \mathbb{C}P^n = G_{1,n+1}(\mathbb{C}). \]
• A chart \(U_\alpha\) may be defined to be a subset of \(G_{k,n}\) such that \(\det A_\alpha \neq 0\).
  • The \(k(n-k)\) coordinates of \(U_\alpha\) are given by the non-trivial entries of the matrix \(A_\alpha^{-1} A\).

Algebraic variety:

• The common zeros of a set of homogeneous polynomials.
• A compact submanifold of \(\mathbb{C}P^n\).
• \(U_\mu\) defined as in a complex projective space.

Calculus on Complex Manifolds

Holomorphic Maps

• Holomorphic map: \(f:M\rightarrow N\) is holomorphic if \[\psi\circ f \circ \varphi^{-1}: \mathbb{C}^m \rightarrow \mathbb{C}^n\] is a holomorphic function.
  • Independent of the coordinates choosen.
• Biholomorpism: \(M\) is biholomorphic to \(N\) if there exists a diffeomorphism \(f:M\rightarrow N\) that is holomorphic.
  • The map \(f\) is called a biholomorpism.
• Holomorphic function: a holomorphic map \(f:M\rightarrow \mathbb{C}\).
  • Any holomorphic function on a compact complex manifold is constant.
  • The set of holomorphic functions on \(U\subset M\) is denoted by \(\mathcal{O}(U)\).

Complexifications

• Complexification \(\mathcal{F}(M)^{\mathbb{C}}\) of \(\mathcal{F}(M)\): the set of complex-valued smooth functions on \(M\).
  • Complexified functions do not satisfy the Cauchy-Riemann relation in general.
• Complexification of a vector space \(V\): consists of \(\qty{X+iY}\) where \(X,Y\in V\).
  • Vectors in \(V\) are said to be real.
  • Linear operator or linear function \(A\) on \(V\) extended as \[ A(X+iY) = A(X) + i A(Y). \]
• Complexification of tensors: \[ t = t_1 + i t_2. \]
  • A tensor \(t\) is real if \[ \overline{t(X+iY)} = t(X-iY). \]
• Complexified \(T_p M^{\mathbb{C}}\):
  • consists of \[ Z = X + iY, \] where \(X,Y\in T_p M\).
  • \(Z\) acts on \(\mathcal{F}(M)^{\mathbb{C}}\) as \[ Z[f] = X[f_1] - Y[f_2] + i\qty{X[f_2] + Y[f_1]}. \]
  • \(T_p^* M^{\mathbb{C}}\) consists of \[ \zeta = \omega + i\eta. \]
    • \[ (T^*_p M)^{\mathbb{C}} = (T_p M^{\mathbb{C}})^*. \]
• Complexified vector field \(\mathcal{X}(M)^\mathbb{C}\) consists of \[ Z = X+iY. \]
  • Lie bracket as \[ [X+iY,U+iV] = \qty{[X,U] - [Y,V]} + i\qty{[X,V] + [Y,U]}. \]
• Complexified one-form \(\Omega^1(M)^{\mathbb{C}}\) consists of \[ \xi = \omega + i\eta. \]

Almost Complex Structure

• Basis: \[ \begin{align*} \pdv{}{z^\mu} &= \frac{1}{2}\qty{\pdv{}{x^\mu} - i\pdv{}{y^\mu}}, \\ \pdv{}{\overline{z}^\mu} &= \frac{1}{2}\qty{\pdv{}{x^\mu} + i\pdv{}{y^\mu}}. \end{align*}{} \]
• Dual basis: \[ \begin{align*} \dd{z^\mu} &= \dd{x^\mu} + i \dd{y^\mu}, \\ \dd{\overline{z}^\mu} &= \dd{x^\mu} - i\dd{y^\mu}. \end{align*}{} \]
• Inner products: \[ \begin{align*} \langle \dd{z^\mu}, \pdv{\overline{z}^\nu} \rangle &= \langle \dd{\overline{z}^\mu}, \pdv{}{z^\nu} \rangle = 0, \\ \langle \dd{z^\mu}, \pdv{{z}^\nu} \rangle &= \langle \dd{\overline{z}^\mu}, \pdv{}{\overline{z}^\nu} \rangle = \delta{^\mu}{_\nu}. \end{align*}{} \]
• \(J_p: T_p M \rightarrow T_p M\) defined by \[ J_p\qty{\pdv{}{x^\mu}} = \pdv{}{y^\mu},\quad J_p\qty(\pdv{}{y^\mu}) = -\pdv{}{x^\mu}. \]
  • Real tensor.
  • \[ J_p^2 = -\operatorname{id}_{T_p M}. \]
  • Corresponding to the multiplication by \(\pm i\).
  • Independent of charts: \[ J_p\qty{\pdv{}{x^\mu}} = \pdv{}{y^\mu} \Rightarrow J_p\qty{\pdv{}{u^\mu}} = \pdv{}{v^\mu}. \]
  • In matrix form: \[ J_p = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}. \]
• Almost complex structure: a smooth tensor field \(J\) whose component at \(p\) is \(J_p\).
  • \(J\) may be patched across charts only on a complex manifold.
  • \(J\) completely specifies the complex structure.
  • \[ J_p \pdv{}{z^\mu} = i\pdv{}{z^\mu},\quad J_p \pdv{\overline{z}^\mu} = -i \pdv{\overline{z}^\mu}. \]
  • \[ \color{red} J_p = i\dd{z^\mu} \otimes \pdv{}{z^\mu} - i \dd{\overline{z}^\mu} \otimes \pdv{\overline{z}^\mu} \Leftrightarrow \begin{pmatrix} i I_m & 0 \\ 0 & -i I_m \end{pmatrix}. \]
    • For \(\displaystyle Z = Z^\mu \pdv{}{z^\mu}\), \[ J_p Z = iZ. \]
    • For \(\displaystyle Z = Z^\mu \pdv{}{\overline{z}^\mu}\), \[ J_p Z = -iZ. \]
    • \(T_p M^{\mathbb{C}}\) is separated into \[ T_p M^{\mathbb{C}} = T_p M^+ \oplus T_p M^{-}. \]
      • \[ \color{orange} T_p M^\pm = \set{Z\in T_p M^{\mathbb{C}} | J_p Z = \pm iZ}. \]
      • Projection operator \[ \mathcal{P}^\pm = \frac{1}{2}(I_{2m} \mp iJ_p). \]
      • \(Z\) may be decomposed into \[ Z = Z^+ + Z^-. \]
      • \(Z \in T_p M^+\) is called an holomorphic vector.
        • Spanned by \(\qty{\partial/\partial z^\mu}\).
      • \(Z \in T_p M^-\) is called an anti-holomorphic vector.
        • Spanned by \(\qty{\partial/\partial\overline{z}^\mu}\).
      • Likewise, a vector field may be decomposed as \[ \mathcal{X}(M)^\mathbb{C} = \mathcal{X}(M)^+ \oplus \mathcal{X}(M)^-. \]
      • \(Z = Z^+ + Z^-\) is real if and only if \[ Z^+ = \overline{Z^-}. \]
      • If \(X,Y\in \mathcal{X}(M)^{\pm}\) then \[ [X,Y] \in \mathcal{X}(M)^{\pm}. \]

Complex Differential Forms

Complexification of Real Differetial Forms

• Complex \(q\)-form \(\zeta = \omega + i\eta\).
• Exterior product defined by \[ (\omega + i\eta)\wedge (\varphi + i\psi) = (\omega \wedge \varphi - \eta \wedge \psi) + i(\omega \wedge \psi + \eta \wedge \varphi). \]
• Exterior derivative \[ \dd{\xi} = \dd{\omega} + i \dd{\eta}. \]
  • \(\dd{}{}\) is a real operator, i.e. \[ \overline{\dd{\eta}} = \dd{\overline{\zeta}}. \]
  • \[ \omega \wedge \xi = (-1)^{qr} \xi \wedge \omega. \]
  • \[ \dd{\omega \wedge \xi} = \dd{\omega} \wedge \xi + (-1)^q \omega \wedge \dd{\xi}. \]

Differential Forms on Complex Manifolds

• \(\Omega^{r,s}_p(M)^{\mathbb{C}}{}\) is
  • a subset of \(\Omega^{r+s}_p(M)^{\mathbb{C}}\),
  • where \[ \omega(V_1,\cdots,V_q) \neq 0 \] only if \(r\) of the \(V_i\) are in \(T_p M^+\) and \(s\) of the \(V_i\) are in \(T_p M^-\).
  • \((r,s)\) is the bidegree of \(\omega\).
  • A \((r,s)\)-form may be written as \[ \omega = \frac{1}{r! s!} \omega_{\mu_1 \cdots \mu_r \nu_1\cdots \nu_s} = \dd{z^{\mu_1}} \wedge \cdots \wedge \dd{z^{\mu_r}} \wedge \dd{\overline{z}^{\nu_1}} \wedge \cdots \wedge \dd{\overline{z}^{\nu_s}}. \]
  • \[r+s \le 2\dim_{\mathbb{C}}M.\]
• Proposistion: Let \(\dim_{\mathbb{C}} M = m\).
  • If \(\omega \in \Omega^{q,r}(M)\) then \[ \overline{\omega} \in \Omega^{r,q}(M). \]
  • If \(\omega \in \Omega^{q,r}(M)\) and \(\xi\in\Omega^{q',r'}(M)\), then \[ \omega \wedge \xi = \Omega^{q+q',r+r'}(M). \]
  • A complex \(q\)-form is uniquely written as \[ \omega = \sum_{r+s = q} \omega^{(r,s)}. \]
    • This gives rise to the decomposition \[ \Omega^q(M)^{\mathbb{C}} = \bigoplus_{r+s=q}\Omega^{r,s}(M). \]

Dolbeault Operators

• \[ \dd = \partial + \overline{\partial}. \]
  • \[ \partial: \Omega^{r,s}(M) \rightarrow \Omega^{r+1,s}(M). \]
  • \[ \overline{\partial}: \Omega^{r,s}(M) \rightarrow \Omega^{r,s+1}(M). \]
  • \(\partial\) and \(\overline{\partial}\) are called Dolbeault operators.

Let \[ \omega = \omega_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu}. \] Then \[ \partial \omega = \pdv{\omega_{\mu\bar{\nu}}}{z^\lambda} \dd{z^\lambda} \wedge \dd{z^\mu} \wedge \dd{\overline{z}^\nu}, \] and \[ \overline{\partial} \omega = \pdv{\omega_{\mu\bar{\nu}}}{\overline{z}^\lambda} \dd{\overline{z}^\lambda} \wedge \dd{z^\mu} \wedge \dd{\overline{z}^\nu}. \]

Theorem. Let \(\omega \in \Omega^q(M)^{\mathbb{C}}\) and \(\xi\in \Omega^p(M)^{\mathbb{C}}\). Then

• \[ \partial \partial = (\partial\overline{\partial} + \overline{\partial}\partial) \omega = \overline{\partial}\overline{\partial} \omega = 0. \]

• \[ \partial\overline{\omega} = \overline{\overline{\partial}\omega},\quad \overline{\partial}\overline{\omega} = \overline{\partial\omega}. \]

• \[ \partial(\omega \wedge \xi) = \partial \omega \wedge \xi + (-1)^q \omega\wedge \partial \xi. \]

• \[ \overline{\partial} (\omega \wedge \xi) = \overline{\partial} \omega \wedge \xi + (-1)^q \omega \wedge \overline{\partial}\xi. \]

• Holomorphic \(r\)-form: \(\omega\in \Omega^{r,0}(M)\) that satisfies \[ \overline{\partial}\omega = 0. \]

  • A holomorphic \(0\)-form is a holomorphic function.

• Doubeault complex: the sequence of \(\mathbb{C}\)-linear maps \[ \Omega^{r,0}(M) \xrightarrow{\bar{\partial}} \Omega^{r,1}(M) \xrightarrow{\bar{\partial}}\cdots \xrightarrow{\bar{\partial}} \Omega^{r,m-1}(M) \xrightarrow{\bar{\partial}} \Omega^{r,m}(M) \] where \(m = \dim_{\mathbb{C}}M\).

  • The set of \(\overline{\partial}\)-closed \((r,s)\) forms is called the \((r,s)\)-cocycle, denoted by \(Z^{r,s}_{\overline{\partial}}(M)\).
  • The set of \(\overline{\partial}\)-exact \((r,s)\) forms is called the \((r,s)\)-coboundary, denoted by \(B^{r,s}_{\overline{\partial}}(M)\).
  • \((r,s)\)-th \(\overline{\partial}\)-cohomology group \[ H^{r,s}_{\bar{\partial}}(M) = Z^{r,s}_{\bar{\partial}}(M) / B^{r,s}_{\bar{\partial}}(M). \]

Hermitian Manifolds and Hermitian Differential Geometry

• \(g\) may be extended by \[ g_p(X+iY,U+iV) = g_p(X,U) - g_p(Y,V) + i[g_p(X,V) + g_p(Y,U)]. \]
  • \[ g_{\mu\nu}(p) = g_p(\partial/\partial z^\mu, \partial/\partial z^\nu). \]
  • \[ g_{\mu\bar{\nu}}(p) = g_p(\partial/\partial z^\mu, \partial/\partial \overline{z}^\nu). \]
  • \[ g_{\bar{\mu}\nu}(p) = g_p(\partial/\partial \overline{z}^\mu, \partial/\partial z^\nu). \]
  • \[ g_{\bar{\mu}\bar{\nu}}(p) = g_p(\partial/\partial \overline{z}^\mu, \partial/\partial \overline{z}^\nu). \]
• Hermitian metric: for any \(X,Y\in T_p M\), \[ g_p(J_p X, J_p Y) = g_p(X,Y). \]
• Hermitian manifold: \((M,g)\) where \(g\) is a Hermitian metric.
• \(J_p X\) is orthogonal to \(X\): \[ g_p(J_p X,X) = 0. \]
• A complex manifold always admits a Hermitian metric.
• The Hermitian \(g\) takes the form \[ g = g_{\mu\bar{\nu}} \dd{z^\mu} \otimes \dd{\overline{z}^\nu} + g_{\bar{\mu}\nu} \dd{\overline{z}^\mu} \otimes \dd{z^\nu}. \]
• Inverse metric: \[ g_{\mu\bar{\lambda}} g^{\bar{\lambda}\nu} = \delta{_\mu}{^\nu},\quad g{^{\bar{\nu}\lambda}}g_{\lambda\bar{\mu}} = \delta{^{\bar{\nu}}}{_{\bar{\mu}}}. \]

Kähler Form

• Kähler form: a two-form field defined by \[ \Omega_p(X,Y) = g_p(J_p X, Y), \] where \(X,Y\in T_p M\).
• Invariant under \(J\): \[ \Omega(JX, JY) = \Omega(X,Y). \]
• \[ \Omega = -i g_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu} = -J_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu}. \]
• \(\Omega\) is a real form.
• A complex manifold is orientable.

Covariant Derivatives

• \[ \grad_\mu \pdv{}{z^\nu} = \Gamma{^\lambda}_{\mu\nu} \pdv{}{z^\lambda}. \]
• \[ \grad_{\bar{\mu}} \pdv{\overline{z}^\nu} = \Gamma{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} \pdv{}{\overline{z}^\lambda} = \overline{\Gamma{^\lambda}_{\mu\nu}} \pdv{}{\overline{z}^\lambda}. \]
• \[ \grad_\mu \pdv{}{\overline{z}^\nu} = \grad_{\bar{\mu}} \pdv{}{z^\nu} = 0. \]
• \[ \grad_\mu \dd{z^\nu} = -\Gamma{^\nu}_{\mu\lambda} \dd{z^\lambda}. \]
• \[ \grad_{\bar{\mu}} \dd{\overline{z}^\nu} = -\Gamma{^{\bar{\nu}}}_{\bar{\mu}\bar{\lambda}}\overline{z}^\lambda. \]
• Covariant derivatives of vector: \[\begin{align*} \grad_\mu X^+ &= (\partial_\mu X^\lambda + X^\nu \Gamma{^\lambda}_{\mu\nu}) \pdv{}{z^\lambda}, \\ \grad_\mu X^- &= \partial_\mu X^{\bar{\lambda}} \pdv{}{\overline{z}^\lambda}, \\ \grad_{\bar{\mu}} X^+ &= \partial_{\bar{\mu}} X^\lambda \pdv{}{z^\lambda}, \\ \grad_{\bar{\mu}} X^- &= (\partial_{\bar{\mu}} X^{\bar{\lambda}} + X^{\bar{\nu}} \Gamma{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} \pdv{}{\overline{z}^\lambda}). \end{align*}{}\]
• Metric compatibility:
  • \[ \Lambda{^\lambda}_{\kappa\mu} = g^{\bar{\nu}\lambda}\partial_\kappa g_{\mu\bar{\nu}},\quad \Gamma{^{\bar{\lambda}}}_{\bar{\kappa}\bar{\nu}} = g^{\bar{\lambda}\mu}\partial_{\bar{\kappa}} g_{\mu\bar{\nu}}. \]
• Hermitian connection: a metric compatible connection for which \(\Gamma\)(mixed indices) vanish.
• The almost complex structure \(J\) is convariantly constant with respect to the Hermitian connection. \[ \color{red} (\grad_\kappa J){_\nu}{^\mu} = (\grad_{\bar{\kappa}} J){_\nu}{^\mu} = (\grad_\kappa J){_{\bar{\nu}}}{^{\bar{\mu}}} = (\grad_{\bar{\kappa}} J){_{\bar{\nu}}}{^{\bar{\mu}}} = 0. \]

Torsion and Curvature

• Nonzero components of \(T\):
  • \[ T{^\lambda}_{\mu\nu} = g^{\bar{\xi}\lambda}(\partial_\mu g_{\nu\bar{\xi}} - \partial_\nu g_{\mu\bar{\xi}}). \]
  • \[ T{^\lambda}_{\bar{\mu}\bar{\nu}} = g^{\bar{\lambda}\xi}(\partial_{\bar{\mu}}g_{\bar{\nu}\xi} - \partial_{\bar{\nu}} g_{\bar{\mu}\xi}). \]
• Nonzero components of \(R\):
  • \[ \begin{align*} R{^\kappa}_{\lambda\bar{\mu}\nu} &= -R{^\kappa}_{\lambda\nu\bar{\mu}} = \partial_{\bar{\mu}}(g^{\bar{\xi}\kappa}\partial_\nu g_{\lambda\bar{\xi}}),\\ R{^{\bar{\kappa}}}_{\bar{\lambda}\mu\bar{\nu}} &= -R{^{\bar{\kappa}}}_{\bar{\lambda}\bar{\nu}\mu} = \partial_\mu(g^{\bar{\kappa} \xi} \partial_{\bar{\nu}} g_{\xi\bar{\lambda}}). \end{align*}{} \]
• Ricci form: \[ \mathscr{R} = i\mathscr{R}_{\mu\bar{\nu}} \dd{z^\mu} \wedge \dd{\overline{z}^\nu} = -i\partial\overline{\partial}\log G. \]
  • \[ G = \det(g_{\mu\overline{\nu}}) = \sqrt{g}. \]
  • \[ \mathscr{R}_{\mu\bar{\nu}} = R{^\kappa}_{\kappa\mu\bar{\nu}} = -\partial_{\bar{\nu}}\partial_\mu \log G. \]
  • \(\mathscr{R}\) is closed, i.e. \(\dd{\mathscr{R}} = 0\).
  • \(\mathscr{R}\) is not exact.
    • \(G\) is not a scalar and \[(\partial - \overline{\partial})G\] is not defined globally.
  • \(\mathscr{R}\) defines a non-trivial element \[ c_1(M) = [\mathscr{R}/2\pi] \in H^2(M;\mathbb{R}). \]
    • \(c_1(M)\) is invariant under a smooth change of metric \[ g \rightarrow g+\delta g. \]

Kähler Manifolds and Kähler Differential Geometry

Examples of Kähler Manifolds

• Kähler manifold:
  • a Hermitian manifold \((M,g)\),
  • whose Kähler form \(\Omega\) is closed, i.e. \[ \dd{\Omega} = 0. \]
  • The metric \(g\) is called the Kähler metric of \(M\).
• A Hermitian manifold \((M,g)\) is a Kähler manifold if and only if the almost complex structure \(J\) satisfies \[ \color{red}\grad_\mu J = 0, \] where \(\grad_\mu\) is assocaited with \(g\).
• The Riemann structure is compatible with the Hermitian structure in a Kähler manifold.
• Kähler potential: denoted by \(\mathcal{K}\),
  • Locally \[ g_{\mu\bar{\nu}} = \partial_\mu \partial_{\bar{\nu}} \mathcal{K}, \]
    • \(\mathcal{K}\in \mathcal{F}(U_i)\) for a chart \(U_i\).
  • \[ \Omega = i\partial \overline{\partial} \mathcal{K}. \]
  • Between different charts: \[ \mathcal{K}_j(w,\overline{w}) = \mathcal{K}_i(z,\overline{z}) + \phi_{ij}(z) + \psi_{ij}(\overline{z}), \]
    • \(\phi\) is holomorphic and \(\psi\) is anti-holomorphic in \(z\).

Let \(M = \mathbb{C}^m\). Let the metric be given by \[ \begin{align*} \delta\qty(\pdv{}{x^\mu},\pdv{}{x^\nu}) &= \delta\qty(\pdv{}{y^\mu}, \pdv{}{y^\nu}) = \delta_{\mu\nu}, \\ \delta\qty(\pdv{}{x^\mu},\pdv{}{y^\nu}) &= 0. \end{align*}{} \] Since \[ J \pdv{}{x^\mu} = \pdv{}{y^\mu},\quad J\pdv{}{y^\mu} = -\pdv{}{x^\mu}, \] we find that \(\delta\) is Hermitian. In complex coordinates, we have \[ \begin{align*} \delta\qty(\pdv{}{z^\mu}, \pdv{}{z^\nu}) &= \delta\qty(\pdv{}{\overline{z}^\mu}, \pdv{}{\overline{z}^\nu}) = 0, \\ \delta\qty(\pdv{}{z^\mu}, \pdv{}{\overline{z}^\nu}) &= \delta\qty(\pdv{}{\overline{z}^\mu}, \pdv{}{z^\nu}) = \frac{1}{2}\delta_{\mu\nu}. \end{align*}{} \] The Kähler form is given by \[ \Omega = \frac{i}{2} \sum_{\mu=1}^m \dd{z^\mu} \wedge \dd{\overline{z}^\mu}. \] The Kählter potantial is given by \[ \mathcal{K} = \frac{1}{2} \sum z^\mu \overline{z}^\mu. \] The Kähler manifold \(\mathbb{C}\) is called the complex Euclid space.

Any orientable complex manifold \(M\) with \(\dim_{\mathbb{C}} M = 1\) is Kähler. One-dimensional compact orientable complex manifolds are Riemann surfaces.

\(\mathbb{C}P^m\) is a Kähler manifold. We define the Ka¨hler potential on \(U_\alpha\) by \[ \log \mathcal{K}_\alpha(p) = \log \sum_{\nu=1}^{m+1} \abs{\frac{z^\nu}{z^\alpha}}^2. \] The Kähler form may be defined by \[ \Omega = i\partial \overline{\partial} \log \mathcal{K}_\alpha. \] It can be shown that \[ g(X,Y) = \Omega(X,JY) \] is Hermitian. It's positive-definite since \[ g(X,X) = 2 \qty[\sum_\mu \abs{X^\mu}^2 \qty(\sum_\lambda \abs{\zeta^\lambda}^2 + 1) - \abs{\sum_\mu X^\mu \zeta^\mu}^2]\qty(\sum_\lambda \abs{\zeta^\lambda}^2 + 1)^{-2}, \] where \(\zeta^\lambda\) are the inhomogeneous coordinates. The metric is called the Fubini-Study metric.

• \(S^2\) is the only sphere which admits a complex structure. \(S^2 \cong \mathbb{C}P^1\) and therefore \(S^2\) is a K¨hler manifold (\(S^6\) unknown still).
• \(S^{2m+1}\times S^{2n+1}\) always admits a complex structure. Only \(S^1 \otimes S^1\) admits a Kähler metric.
• Any complex submanifold of a Kähler manifold is Kähler.

Kähler Geometry

• The Kähler metric is torsion free: \[ \begin{align*} T{^\lambda}_{\mu\nu} &= g^{\bar{\xi}\lambda}(\partial_\mu g_{\nu\bar{\xi}} - \partial_\nu g_{\mu\bar{\xi}}) = 0, \\ T{^{\bar{\lambda}}}_{\bar{\mu}\bar{\nu}} &= g^{\bar{\lambda}\xi}(\partial_{\bar{\mu}} g_{\bar{\nu}\xi} - \partial_{\bar{\nu}}g_{\bar{\mu}\xi}) = 0. \end{align*}{} \]
• The Riemann tensor has an extra symmetry: \[ R{^\kappa}_{\lambda\mu\bar{\nu}} = R{^\kappa}_{\mu\lambda\bar{\nu}}. \]
• Ricci form \[ \mathscr{R}_{\mu\bar{\nu}} = \mathrm{Ric}_{\mu\bar{\nu}}. \]
• If \(\mathrm{Ric} = \mathscr{R} = 0\), the Kähler metric is said to be Ricci flat.
• Let \((M,g)\) be a Kähler manifold. If \(M\) admits a Ricci flat metric \(h\), then its first Chern class must vanish.
• Calabi-Yau manifold: a compact Kähler manifold with vanishing first Chern class.
• Calabi conjecture (proved): If \(c_1(M) = 0\), then the Kähler manifold \(M\) admits a Ricci-flat metric.

The Holonomy Group of Kähler Manifolds

• The holonomy group is contained in \(\mathrm{U}(m) \subset \mathrm{O}(2m)\).
• If \(g\) is the Ricci-flat metric of an \(m\)-dimensional Calabi-Yau manifold \(M\), the holonomy group is contained in \(\mathrm{SU}(m)\).

Harmonic Forms and ∂̅-Cohomology Groups

Adjoint Operators

• Inner product: \(\alpha,\beta \in \Omega^{r,s}(M)\), \[ \color{orange}(\alpha,\beta) = \int_M \alpha \wedge\overline{*}\beta. \]
• Hodge *: \[ \overline{*}\beta = \overline{*\beta} = *\overline{\beta}. \]
• Adjoint operators: \[ \color{red} \partial^\dagger = -*\overline{\partial} *,\quad \overline{\partial}^\dagger = -*\partial *. \]
  • \[ \color{red} (\alpha,\partial\beta) = (\partial^\dagger \alpha,\beta),\quad (\alpha,\overline{\partial}\beta) = (\overline{\partial}^\dagger \alpha,\beta). \]
  • \[ (\partial^\dagger)^2 = (\overline{\partial}^\dagger)^2 = 0. \]

Hodge Theory on Complex Manifolds

• \((r,s)\)th \(\overline{\partial}\)-cohomology group: \[ H^{r,s}_{\overline{\partial}}(M) = Z^{r,s}_{\bar{\partial}}(M) / B^{r,s}_{\bar{\partial}}(M). \]
• Laplacians: \[ \begin{align*} \bigtriangleup_\partial &= (\partial + \partial^\dagger)^2 = \partial\partial^\dagger + \partial^\dagger \partial, \\ \bigtriangleup_{\bar{\partial}} &= (\overline{\partial} + \overline{\partial}^\dagger)^2 = \overline{\partial}\overline{\partial}^\dagger + \overline{\partial}^\dagger \overline{\partial}. \end{align*}{} \]
• Harmonic forms:
  • \(\partial\)-harmonic: \[ \bigtriangleup_\partial \omega = 0. \]
    • Equivalently, \[ \partial\omega = \partial^\dagger \omega = 0. \]
  • \(\overline{\partial}\)-harmonic: \[ \bigtriangleup_{\bar{\partial}} \omega = 0. \]
    • Equivalently, \[ \overline{\partial}\omega = \overline{\partial}^\dagger \omega = 0. \]
• \(\overline{\partial}\)-harmonic forms: \[ \mathrm{Harm}^{r,s}_{\bar{\partial}}(M) = \set{\omega\in \Omega^{r,s}(M) | \bigtriangleup_{\bar{\partial}}\omega=0}. \]
• Hodge's theorem: \[ \Omega^{r,s}(M) = \overline{\partial} \Omega^{r,s-1}(M) \oplus \overline{\partial} \Omega^{r,s+1}(M) \oplus \mathrm{Harm}^{r,s}_{\bar{\partial}}(M). \]
  • Each \((r,s)\)-form \(\omega\) is uniquely expressed as \[ \omega = \overline{\partial}\alpha + \overline{\partial}^\dagger \beta + \gamma. \]
• If \(M\) is a Kähler manifold, then \[ \bigtriangleup = 2\bigtriangleup_{\partial} = 2\bigtriangleup_{\bar{\partial}}. \]
  • Any holomorphic form is harmonic with respect to the Kähler metric.
  • Every harmonic form of bidegree \((p,0)\) is holomorphic.

The Hodge Numbers of Kähler Manifolds

Let \(M\) be a Kähler manifold.

• Hodge number: \[ b^{r,s} = \dim H^{r,s}_{\bar{\partial}}(M). \]
• Hodge diamond: \[ \begin{pmatrix} & & & b^{m,m} & & & \\ & & b^{m,m-1} & & b^{m-1,m} & & \\ b^{m,0} & b^{m-1,1} & & \cdots & & b^{1,m-1} & b^{0,m} \\ & & b^{1,0} & & b^{0,1} & & \\ & & & b^{0,0} & & & \end{pmatrix}. \]
• Properties of hodge numbers:
  • \[ b^{r,s} = b^{s,r}. \]
  • \[ b^{r,s} = b^{m-r,m-s}. \]
  • Number of independent Hodge numbers:
    • \(m = \dim_{\mathbb{C}}M\) even: \[ (\frac{1}{2}m+1)^2. \]
    • \(m = \dim_{\mathbb{C}}M\) odd: \[ \frac{1}{4}(m+1)(m+2). \]
  • \[ b^p = \sum_{r+s=p} b^{r,s}. \]
  • \[ b^{2p-1} \text{ is even}. \]
  • \[ b^{2p}\ge 1. \]

Almost Complex Manifolds

• Almost complex manifold: the pair \((M,J)\),
  • \(J\) is a tensor field of type \((1,1)\),
  • \(J^2_p = -\operatorname{id}_{T_p M}\).
  • \(J\) is called the almost complex structure.
• Integrable almost complex structure \(J\):
  • For any \(X,Y\in\mathcal{X}^+(M)\), \[ [X,Y] \in \mathcal{X}^+(M). \]
• Nijenhuis tensor field: \[ \color{orange} N(X,Y) = [X,Y] + J[JX,Y] + J[X,JY] - [JX,JY]. \]
  • \[ N(X,Y) = X^\kappa Y^\nu [-J{_\lambda}{^\mu}(\partial_\nu J{_\kappa}{^\lambda}) + J{_\lambda}{^\mu}(\partial_\kappa J{_\nu}{^\lambda}) - J{_\kappa}{^\lambda}(\partial_\lambda) J{_\nu}{^\mu} + J{_\nu}{^\lambda} (\partial_\lambda J{_\kappa}{^\mu})] \pdv{}{x^\mu}. \]
• An almost complex structure \(J\) is integrable if and only if \(N(A,B) = 0\) for any \(A,B\in \mathcal{X}(M)\).

Orbifolds

Definition and Examples of Orbifolds

• Orbifold: \[ \Gamma = M/G. \]
  • \(M\) a manifold.
  • \(G\) a discrete group acting on \(M\).
  • \(\Gamma\) may not be a manifold.

\(\Gamma = \mathbb{C}/C_3\) is a third of the complex plane. The holonomy group is \(C_3\). \(\Gamma\) is a manifold.

Let \[ M = T^2 = \mathbb{C}/\qty{z \sim z+m+ne^{i\pi/3}}, \] and \(C_3\) acts on \(T^2\) as \[ z \mapsto e^{2\pi i/3 z}. \] Then \(\Gamma = T^2/C_3\) is homeomorphic to \(S^2\). The holonomy around each fixed point is \(C_3\).

\(T = \mathbb{C}^3/L\) is a three-dimensional complex torus where \(L\) is a lattice in \(\mathbb{C}^3\). \[ T = T_1 \times T_2 \times T_3. \] Let \(C_3\) act on \(T\) by \[ z_i \mapsto e^{2\pi i/3}z_i. \] There are \(27\) fixed points in the orbifold, each of which is a conical singularity. The orbifold is not a manifold.

band.ipynb

Jupyter

A Certain Tight-Binding Index

強相関電子系とモンテカルロが交差するとき、位相幾何は始まる

Topological Quantum Chemistry

How do we discover topological materials from band representations?

---

Filling-Enforced Quantum Band Insulator

Filling-enforced quantum band insulators in spin-orbit coupled crystals.

Atomic Insulator

• (TR symmetry) total spin per site is integer.
• Electron wavefunctions localized at Wyckoff positions.
• Minimum filling \(\nu^{\mathrm{AI}}_{\mathrm{min}}\).

Band Insulator

• Minimum filling \(\nu^{\mathrm{Band}}_{\mathrm{min}}\).
• (Kramers degeneracy) \(\nu^{\mathrm{Band}}_{\mathrm{min}}\) is even.
• If SOC is nonnegligible, a band insulator may not be an atomic insulator.
• Such nonatomic filling may not respect all crystal symmetries.
• Filling-enforced quantum band insulator: a band insulator realized at such nonatomic filling.
• Possibly realizable in SG No.199, 214, 220, 230.
• Physical manifestations of an FEQBI is still unknown.
  • For possible topological invariants, see Topological Invariants of a Filling-Enforced Quantum Band Insulator (under review).

Topological Insulators

Classification

• Strong TI: robust to perturbations that break all crystal symmetries.
  • 2D:
    • Chern insulator (no symmetry required).
    • Quantum spin Hall insulator (time-reversal symmetry).
  • 3D:
    • \(\mathbb{Z}_2\) TI with time reversal symmetry.

  TCI. Topological Crystal Insulator.

• TCI:
  • Weak TI: protected by translation symmetry.
  • Mirror Chern insulator.

Basics of Group Representation

Group Representation

• The number of non-equivalent IR of \(G\) equals to the number of conjugacy classes in \(G\).
• \[ \color{red} d_1^2 + \cdots + d_C^2 = \abs{G}. \]
• IRs of abelian groups are always one-dimensional.
• Inner Product of Characters: \[ \color{orange}(\chi_1,\chi_2) = \frac{1}{\abs{G}} \sum_{g\in G} \chi_1^*(g) \chi_2(g). \]
• Characters of IRs are orthonormal. \[ (\chi_i,\chi_j) = \delta_{ij}. \]
• Expansion into IRs: \begin{align*} \color{red}T &\color{red}= m_1 R_1 \oplus \cdots \oplus m_c R_c, \\ \color{red}m_j &\color{red}= (\chi_T, \chi_{i}). \end{align*}
• Conjugacy classes are strictly distinguishable symmetry operations.

Conjugacy classes of \(D_3\) includes \(E\), \(2C_3\), \(3C_2'\), each of which has \(1\), \(2\) and \(3\) elements.

Interlude: O(3)

Topological Matter School 2018 - Juan Luis Mañes - Lecture 4

• Character: \(P = \pm 1\),
  • \[ \chi_l(E) = 2l + 1. \]
  • \[ \chi_l(C_\theta) = \dfrac{\sin\qty(l+\frac{1}{2})\theta}{\sin \frac{\theta}{2}}. \]
  • \[ \chi_l(I) = (2l+1) P. \]
  • \[ \chi_l(\sigma) = \chi_l(C_\pi) P. \]
  • \[ \chi_l(S_\theta) = \chi_l(C_{\pi - \theta}) P. \]
• For each IR of \(\mathrm{SO}(3)\) there are two different IRs of \(\mathrm{O}(3)\) denoted \(D_l^\pm\).
  • The superscript \(\pm\) specify the behaviour under parity.
  • \(P=+1\): the representation is insensitive to parity, and therefore \(\rho(g) = \rho(g/\det g)\) (I am just guessing).
  • \(P=-1\): the representation is sensitive to parity, and therefore \(\rho(g) = (\det g)\rho(g/\det g)\) (again guessing).
• How does \(O(3)\) acts on \(Y_l^m\)?
  • \(Y_l^m\) are sensitive to parity, with \(P = (-1)^l\).

\(O(3)\) acts on an electron on the p-orbit by \[ D_{1}^- \otimes D_{1/2}^{+} = D_{1/2}^- \oplus D_{3/2}^-. \] Note that on both sides we have a \(-1\) parity.

When taking \(\mathrm{O}(3)\) as a subgroup of \(\mathrm{SU}(2)\), for \(j=l+1/2\) we should have \[ \rho(C_{\theta}) \neq \rho(C_{\theta + 2\pi}). \] Therefore, each conjugacy class of \(\mathrm{O}_3\) should split into \(C\) and \(\overline{C}\) (\(C\) multiplied by \(C_{2\pi}\) I guess).

• Double-valued representations have a bar on their notation. For example, \(\Gamma_1\) is a single-valued representation and \(\overline{\Gamma}_7\) is a double-valued one. Double conjugacy classes are also barred. For example, \(E\) denote the identity operator while \[\overline{E} = C_{2\pi} \cdot E.\]

Every two-dimensional point group could be embedded into a three-dimensional point group. We may either map \(E\) to \(E\) or \(m\) (reflection about the \(\mathbb{R}^2\) plane). In the latter case we may obtain a double-valued representation.

Interlude: Little Group

little group: \(G_{\vb*{k}}\) is the set of symmetries that leave \(\vb*{k}\) invariant. \[ \color{warning} G_{\vb*{k}} = \Set{g\in G|g\vb*{k} = \vb*{k} + \vb*{K}}, \] where \(\vb*{K}\) is a reciprocal lattice vector.

• The Bloch waves \(\qty{\psi_{\vb*{k}}^a}\) transform among themselves under \(G_{\vb*{k}}\).
  • This defines the small representation \(T_{\vb*{k}}\) of the little group \(G_{\vb*{k}}\).

\(G_{\vb*{k}}\) is infinite.

Little cogroup: \(\tilde{G}_{\vb*{k}}\) defined by \[ \tilde{G}_{\vb*{k}} = G_{\vb*{k}} / T \] where \(T\in G\) is the subgroup of lattice translations. The little cogroup is isomorphic to a point group.

Site symmetry groups: \(G_{\vb*{q}}\) defined by \[ G_{\vb*{q}} = \Set{g\in G|g\vb*{q} = \vb*{q}}, \] where \(g\) may contain a nonzero translation.

\(G_{\vb*{q}}\) is finite.

Crystal Symmetry

Symmetry Operations

Topological Matter School 2018 - Juan Luis Mañes - Lecture 2

• \(C_n\): rotations.
  • \(\chi_V(C_n) = 1 + 2\cos\theta_n\).
  • \(\chi_R(C_n) = 1 + 2\cos\theta_n\).
• \(\sigma\): reflection planes.
  • \(\chi_V(\sigma) = 1\).
  • \(\chi_R(\sigma) = -1\).
• \(I\): inversion.
  • \(\chi_V(I) = -3\).
  • \(\chi_R(I) = 3\).
• \(S_n\): roto-reflections.
  • \(\chi_V(S_n) = -1 + 2\cos\theta_n\).
  • \(\chi_R(S_n) = 1 - 2\cos\theta_n\).

Trace representation trace: \[ \chi_M(g) = n_{\mathrm{inv}}(g)\chi_V(g), \] where \(n_{\mathrm{inv}}(g)\) is the number of atoms fixed under this operation.

By \[ \qty{R,\vb*{l}} \] we denote a operation \[ \vb*{r} \rightarrow R\vb*{r} + \vb*{l}. \]

Mulliken Symbols

Wikibooks: Structural Biochemistry/Character Table.

Reflection σ.

Symmetry Operations and Character Tables.

• A: singly degenerate.

  • Only one orbital has that particular symmetry and energy.
  • Symmetric with respect to primary rotational axis.

• B: singly degenerate.

  • Only one orbital has that particular symmetry and energy.
  • Antisymmetric with respect to primary rotational axis.

• E: doubly degenerate.

  • Two orbitals has the same symmetry and energy

• T: triply degenerate.

  • Three orbitals has the same symmetry and energy.

• g: symmetric with respect to inversion center.

• u: antisymmetric with respect to inversion center.

• 1: symmetric with respect to perpendicular \(C_2\) axis.

• 2: antisymmetric with respect to perpendicular \(C_2\) axis.

• ': designates symmetric with respect to horizontal reflection plane.

• '': designates antisymmetric with respect to horizontal reflection plane. In particular, we have dimensions \begin{align*} \color{red}\dim A &\color{red}= \dim B = 1, \\ \color{red}\dim E &\color{red}= 2, \\ \color{red}\dim T &\color{red}= 3. \end{align*}

• \(\sigma_h\): a plane perpendicular to the axis; horizontal.

• \(\sigma_v\): a vertical mirror containing the main axis.

  • \(\sigma_d\): if such plane bisects the angle between two \(C_2\) axes.

Wyckoff Positions

For Wyckoff positions of a specific SG, look out the Bilbao Crystallographic Server.

A Wyckoff position is a set of points whose site symmetry groups are conjugate to each other. A Wyckoff position may contain multiple points.

Wyckoff positions are labelled \(n\alpha\), where \(n\) is the multiplicity of the position and \(\alpha\) is a letter that orders Wyckoff positions in a particular space group by ascending \(n\).

Maximal Wyckoff positions: those whose site-symmetry groups are not a proper subgroup of any other site-symmetry group.

\(\vb*{k}^\star\) denote all points \(\set{g\vb*{k}| g\in G}\) in the BZ.

IR of the space groups are labelled by \(\vb*{k}^\star\) and are induced from IRs of the little groups.

• Look up the Wyckoff positions of a given space group.

Group Representation and Crystal Symmetry

Representation of Point Group

Topological Matter School 2018 - Juan Luis Mañes - Lecture 1.

Point group:

• \(M\) for mechanical representation:
  • Write down the position vectors of all atoms: \[ u = \begin{pmatrix} \vb*{u}_1 & \cdots & \vb*{u}_n \end{pmatrix}^T. \]
  • For each symmetry operation, \(M\) could be represented by a matrix that consists of \(3\times 3\)-blocks.
  • Only diagonal terms (i.e. atom fixed under this operation) contribute to the trace.
• \(V\) for vector representation: how vectors transform under the elements of the symmetric group.
• \(R\) for rotational representation: how pseudovectors transform under the lements of the symmetry group.
• \(\mathrm{Vib}\) for vibrational representation.

Point group of \(\ce{CH4}{}\):

• Vector representation:
  • The four vertices mix under symmetry operations, therefore triply degenerate.

\begin{align*} M &= A_1 \oplus E \oplus 3T_2 \oplus T_1, \\ V &= T_2, \\ R &= T_1, \\ \mathrm{Vib} &= M-V-R = A_1 \oplus E \oplus 2T_2. \end{align*} Since \(\dim \mathrm{Vib} = 9\) we have \(9\) vibration modes, \(1\) non-degenerate, \(1\) doubly and \(2\) triply degenerate.

Point group of \(\ce{CH_3D}{}\):

\(C_{3v}{}\)	\(E\)	\(2C_3\)	\(3C'_2\)
\(A_1\)	\(1\)	\(1\)	\(1\)
\(A_2\)	\(1\)	\(1\)	\(-1\)
\(E\)	\(2\)	\(-1\)	\(0\)

• Vector representation:
  • The lower three vertices mix under symmetry operations, therefore doubly degenrate; plus
  • The upper vertex in invariant under any symmetry operations.

\begin{align*} M &= 4A_1 \oplus A_2 \oplus 5E, \\ V &= A_1 \oplus E, \\ R &= A_2 \oplus E, \\ \mathrm{Vib} &= M - V - R = 3A_1 \oplus 3E. \end{align*} Since \(\dim \mathrm{Vib} = 9\) we have \(9\) vibration modes, \(3\) non-degenerate and \(3\) doubly degenerate.

Molecular Vibrations

Topological Matter School 2018 - Juan Luis Mañes - Lecture 3.

• For \(T_d\) symmetry, \[ \mathrm{Vib} = A_1(q_1) \oplus E(q_2,q_3) \oplus T_2(q_4,q_5,q_6) \oplus T_2(q_7,q_8,q_9). \]
• The potential energy (under the normalization\(M = \mathbb{1}\)) is given by \[\begin{align*} U_{ij}q_iq_j &= aq_1 + b(q_2^2+q_3^3) + c(q_4^2+q_5^2+q_6^2) \\ &\phantom{{}={}} d(q_7^2 + q_8^2 + q_9^2) + 2e(q_4 q_7 + q_5 q_8 + q_6 q_9). \end{align*}{}\]
• After permuting \(\qty{q_4,\cdots,q_6}\), we find \[ \begin{pmatrix} a & & & & & & & & \\ & b & & & & & & & \\ & & b & & & & & & \\ & & & c & e & & & & \\ & & & e & d & & & & \\ & & & & & c & e & & \\ & & & & & e & d & & \\ & & & & & & & c & e \\ & & & & & & & e & d \end{pmatrix}. \]
• We find \(4\) sequences, \(1\) non-degenerate, \(2\) doubly and \(2\) triply degenerate.
• In general, if \[ \color{red} \mathrm{Vib} = \cdots \oplus m_i R_i \oplus \cdots, \] then there are \(m\) different frequencies, each \(\dim R_i\)-degenerated.

Degeneration Splitting

Topological Matter School 2018 - Juan Luis Mañes - Lecture 4.

From character tables

\(C_{3v}{}\)	\(E\)	\(2C_3\)	\(3C'_2\)
\(A_1\)	\(1\)	\(1\)	\(1\)
\(A_2\)	\(1\)	\(1\)	\(-1\)
\(E\)	\(2\)	\(-1\)	\(0\)

\(T_d\)	\(E\)	\(8C_3\)	\(3C'_2\)	\(6S_4\)	\(6\sigma_d\)
\(A_1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(A_2\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)
\(E\)	\(2\)	\(-1\)	\(2\)	\(0\)	\(0\)
\(T_2\)	\(3\)	\(0\)	\(-1\)	\(1\)	\(-1\)
\(T_1\)	\(3\)	\(0\)	\(-1\)	\(-1\)	\(1\)

we find the subduction table

\(T_d\)	\(A_1\)	\(A_2\)	\(E\)	\(T_2\)	\(T_1\)
\(C_{3v}{}\)	\(A_1\)	\(A_2\)	\(E\)	\(A_1\oplus E\)	\(A_2\oplus E\)

Triply degenerate level of a \(T_d\) system will split into two levels, one if which double degenerate.

In the central field approximation \[ H\ket{nlm} = E_{nl}\ket{nlm}, \] each level has a degeneracy \(2(2l+1)\). In crystal, the symmetry is reduced from \(\mathrm{O}(3)\) to the local point group.

Weak spin-orbit: from the character table \(\mathrm{O}(3)\) to \(C_{3v}{}\)

\(C_{3v}{}\)	\(E\)	\(2C_3\)	\(3C'_2\)
\(A_1\)	\(1\)	\(1\)	\(1\)
\(A_2\)	\(1\)	\(1\)	\(-1\)
\(E\)	\(2\)	\(-1\)	\(0\)

\(O(3)\)	\(E\)	\(2C_3\)	\(3C'_2\)
\(D_0^+\)	\(1\)	\(1\)	\(1\)
\(D_1^-\)	\(3\)	\(0\)	\(1\)
\(D_2^+\)	\(5\)	\(-1\)	\(1\)
\(D_3^-\)	\(7\)	\(1\)	\(1\)

we find \begin{align*} s &\rightarrow a_1, \\ p &\rightarrow a_1 \oplus e, \\ d &\rightarrow a_1 \oplus 2e, \\ f &\rightarrow 2a_2 \oplus a_2 \oplus 2e. \end{align*}

Strong spin-orbit: from the character table \(\mathrm{SU}(2)\) to \(C_{3v}{}\)

\(C_{3v}{}\)	\(E\)	\(\overline{E}{}\)	\(2C_3\)	\(2\overline{C}_3\)	\(3C'_2\)	\(3\overline{C}'_2\)
\(A_1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(A_2\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)
\(^1\overline{E}{}\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(i\)	\(-i\)
\(^2\overline{E}{}\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-i\)	\(i\)
\(E\)	\(2\)	\(2\)	\(-1\)	\(-1\)	\(0\)	\(0\)
\(\overline{E}_1\)	\(2\)	\(-2\)	\(1\)	\(-1\)	\(0\)	\(0\)

\(O(3)\)	\(E\)	\(\overline{E}{}\)	\(2C_3\)	\(2\overline{C}_3\)	\(3C'_2\)	\(3\overline{C}'_2\)
\(D_{1/2}^-\)	\(2\)	\(-2\)	\(1\)	\(-1\)	\(0\)	\(0\)
\(D_{3/2}^-\)	\(4\)	\(-4\)	\(-1\)	\(1\)	\(0\)	\(0\)

we find \begin{align*} D^{-}_{1/2} &= \overline{E}_1, \\ D^{-}_{3/2} &= \overline{E}_1 \oplus {}^1\overline{E} \oplus {}^2\overline{E}. \end{align*}

Example: Group p2

Wallpaper Group.

Group p2:

• Point group: \(C_2\).
• The group p2 contains coutains four centers of order 2 (180°), but no reflections or glide reflections.

The rotation centers in a p2 crystal are also Wyckoff positions:

• ◆: 1a, symmetry group \(\qty{C_{2z}, \vb*{0}}\), where \(C_{2z}\) denote a \(C_2\) rotation around \(1a\).
• ◆: 1b, symmetry group \(\qty{C_{2z}, \hat{\vb*{x}}}\),
• ◆: 1c, symmetry group \(\qty{C_{2z}, \hat{\vb*{y}}}\),
• ◆: 1d, symmetry group \(\qty{C_{2z}, \hat{\vb*{x}} + \hat{\vb*{y}}}\),
• General position: 2e, which consists of two points \((x,y)\) and \((-x,-y)\). Symmetry group trivial.

Note that the symmetry group at 1a, 1b, 1c and 1d are isomorphic but not conjugate to each other.

1a, 1b, 1c and 1d are maximal Wyckoff positions while 2e is not.

Little groups of points of high symmetry in the BZ (\(\Gamma,X,Y,M\)) equal to the full group \(G\). The little groups thereof are \(C_2\).

Since there are two IRs to \(C_2\), the band representations are given by \[ \rho_{G,1\alpha,A/B}(g) = \begin{cases} e^{-i\vb*{k}\cdot \vb*{t}}, & \text{if } g = \qty{E|\vb*{t}}, \\ \pm e^{i\vb*{k}\cdot (\vb*{t} - 2\vb*{q}_\alpha)}, & \text{if } g = \qty{C_2|\vb*{t}}. \end{cases} \] Different representations are distinguished by their \(\qty{C_2|\vb*{0}}\) eigenvalues.

Band Representation

• \(G\) denotes the space group.
• \(G_{\vb*{q}}\) is the site-symmetry group of some site \(\vb*{q} = \vb*{q}_1\).
• \(\qty{q_{\alpha}}\) are the sites in the Wyckoff position of \(\vb*{q}\) in the primitive cell, where \(\alpha = 1,2,\cdots,n\).
• \(g_\alpha\in G\) such that \(\vb*{q}_\alpha = g_\alpha \vb*{q}\).
• \(G\) admits the coset decomposition \[ G=\bigcup_{\alpha=1}^n g_\alpha(G_{\vb*{q}} \ltimes T). \]
• A representation of \(G_{\vb*{q}}\) induces a representation of \(G\) by \[ [\rho_G(g)]_{(i,\alpha,\vb*{t}), (j,\beta,\vb*{t}')} = \qty[\tilde{\rho} \qty(g^{-1}_\alpha \qty{E|\vb*{t}} g \qty{E|\vb*{t}'}^{-1} g_\beta)]_{ij}. \]
  • This representation is denoted by \[ \rho_{\vb*{q}} \uparrow G. \]
• Band representation: a direction sum of representations, each induced from a representation of the site-symmetry group of a Wyckoff position. \[ \color{orange} \bigoplus_{\vb*{q}} (\rho_{\vb*{q}} \uparrow G). \]
  • \(\vb*{q}\) runs through representative site in all Wyckoff positions.

Wannier Basis

• A Wannier function \(W_{i\alpha}(\vb*{r})\) carries three indices:
  • \(i = 1,\cdots,\dim(\rho)\) spans the dimension of the representation. I guess this denote the orbital.
  • \(\alpha\) labels sites of a Wyckoff position (I guess) in the primitive cell.
  • \(\vb*{t}\) is a lattice vector.
• Wannier functions of at the same site of different orbitals are related by \[ gW_{i1}(\vb*{r}) = [\rho(g)]_{ji}W_{j1}(\vb*{r}). \]
  • \(g\in G_{\vb*{q}}\).
  • Wannier functions provide a representation of \(G_{\vb*{q}}\).
• Wannier functions of the same orbital \(i\) at different \(\vb*{q}_\alpha\) are defined by \[ W_{i\alpha} = g_\alpha W_{i1}(\vb*{r}) = W_{i1}(g_\alpha^{-1}\vb*{r}). \]
• Wanier functions are extended outside the primitive cell by \[ \qty{E|\vb*{t}} W_{i\alpha}(\vb*{r}) = W_{i\alpha}(\vb*{r} - \vb*{t}). \]
• \(W_{i\alpha}(\vb*{r} - \vb*{t})\) forms a set of \[ \dim(\rho) \times n \times \mathcal{N} \] functions, where \(\mathcal{N}\) denotes the number of unit cells.
• Now the Wannier functions provide a representation of \(G\).
  • With the decomposition \[ G = \bigcup_{\alpha = 1}^n g_\alpha(G_{\vb*{q}}\ltimes T) \] we write for a general \(g\in G\) that \[ gg_\alpha = \qty{E | \vb*{t}_{\beta\alpha}} g_\beta g_{\vb*{q}}. \]
    • \(g_\beta\) denote the representative element in the coset decomposition.
    • \(\vb*{t}_{\beta\alpha} = g\vb*{q}_\alpha - \vb*{q}_\beta\).
    • \(g_{\vb*{q}}\in G_{\vb*{q}}\).
  • Wannier functions provide a representation by \[ \color{orange} \rho_G(g) W_{i\alpha}(\vb*{r} - \vb*{t}) = \sum_{j=1}^{\dim(\rho)} [\rho(g)]_{ji}W_{j\beta}(\vb*{r} - R\vb*{t} - \vb*{t}_{\beta\alpha}). \]
    • \(\rho\) is a representation of \(G_{\vb*{q}}\).
    • I guess \(R\) is the \(\mathrm{O}(3)\) part of \(g\).
• Fourier transformed Wannier functions: \[ a_{i\alpha}(\vb*{k},\vb*{r}) = \sum_{\vb*{t}} e^{i\vb*{k}\cdot \vb*{t}} W_{i\alpha}(\vb*{r} - \vb*{t}). \]
  • Isn't this just a Bloch wave generated by orbital \(i\) of \(\alpha\) atoms?
• Momentum space Wannier functions provide a representation by \[ \color{orange} \rho_G(g) a_{i\alpha}(\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{j=1}^{\dim(\rho)} [\rho(g)]_{ji} a_{j\beta}(R\vb*{k}, \vb*{r}). \]
  • Represented by a \(\mathcal{N}\times \mathcal{N}\) matrix of blocks \((\vb*{k},\vb*{k}')\), each \((i\alpha,j\beta)\) of size \[ n\dim(\rho)\times n\dim(\rho). \]
  • Most of the blocks are zero, unless \(\vb*{k}' = R\vb*{k}\).
  • Such nonzero blocks are denoted by \(\rho^{\vb*{k}}_G(g)\).
• The above matrix \(\rho_G(g)\) is a block diagonal matrix in \(\vb*{k}\) if and only if \[ g\in G_{\vb*{k}}. \]
• Band representation: a function takes \(\vb*{k}\) to a representation \[ \rho_G \downarrow G_{\vb*{k}}. \]
  • This subduced representation is reducible (I guess).
  • Under this definition, \(\rho_G(g)\) does not make sense for every \(g\), inconsistent with parts of the text remaining.
• Two band representations \(\rho_G\) and \(\sigma_G\) are equivalent if and only if there exists a unitary matrix valued tunction \(S(\vb*{k}, 1, g)\) that
  • is smooth in \(\vb*{k}\),
  • is continuous in \(t\), and
  • for all \(g\in G\), \[ S(\vb*{k},0,g) = \rho^{\vb*{k}}_G(g),\quad S(\vb*{k},1,g) = \sigma_G^{\vb*{k}}(g). \]
  • In other words, equivalent representations are homotopy when regarded as functions of \(\vb*{k}\).

  EBR. Elementary Band Representation.

• Elementary band representation: a band representation that is not equivalent to a direct sum of two or more other band representations.
  • Otherwise the representation is composite.
  • EBRs are not IRs of the space group.
  • There are a finite number of EBRs, indexed by IRs of maximal Wyckoff positions.

Electronic Bands

Topological Matter School 2018 - Juan Luis Mañes - Lecture 5.

• \(\vb*{k}_R\) denote a reciprocal vector. \(\mathcal{T}\) denotes the translation group.
• Representations of the translation group:
  • All IRs of the translation group are one-dimensional.
  • \[ \tau_{\vb*{k}}(\vb*{t}) = e^{i\vb*{k}\cdot \vb*{t}}. \]
  • \[ \vb*{k} + \vb*{k}_R \sim \vb*{k}. \]
• Tight binding approximation: \[ \psi_{\vb*{k}}^a(\vb*{r}) = \sum_{\vb*{t}\in \mathcal{T}} \phi^a(\vb*{r} - \vb*{r}_a - \vb*{t}) e^{i\vb*{k}\cdot \vb*{t}}. \]
• Band structure obtained by diagonalizing \[ \bra{\psi_{\vb*{k}}^a} H\ket{\psi_{\vb*{k}}^b} = H^{ab}(\vb*{k})\delta_{\vb*{k},\vb*{k}'}. \]
• There are \(N_o\) orbitals in total.

Character of the little group of \(p_z\)-orbital: \[ \color{red} \chi_T(g,\vb*{k}) = \sum_{i=1}^A \varphi_i(g,\vb*{k}) \theta_i(g) \chi_\tau(g). \]

• \(A\): number of atoms in the primitive cell.
• \(\theta_i(g)\): \(1\) if atom \(i\) is invariant under \(g\), zero otherwise.
• \(\varphi_i(g,\vb*{k})\): additional phase.
• \(\chi_\tau(g)\): the character of the IR of the orbitals.

The atoms in graphene all belong to the same Wyckoff position. There are two atoms per unit cell in graphene, \(8\) orbitals in total: \[ \qty{\phi^1_s, \phi^1_{p_x}, \phi^1_{p_y}, \phi^1_{p_z},\phi^2_s, \phi^2_{p_x}, \phi^2_{p_y}, \phi^2_{p_z}}. \] \((s,p_x,p_y)\) are even while \(p_z\) is odd under \(\sigma_h\). We discard \(\sigma_h\) and work with \(C_{6v}\) only: \[ C_{6v} = \qty{E, C_2, C_3^\pm, C_6^\pm, \sigma_{di}, \sigma_{vi}}. \] The site symmetry group for each kind of atom is \[ C_{3v} = \qty{E, C_3^\pm, \sigma_{di}}. \] We have \[ D_1^- = A_1(p_z) \oplus E(p_x,p_y) \] and therefore \(p_z \rightarrow a_1\).

We introduce two Bloch waves here: \[ \psi_{\vb*{k}}^a(\vb*{r}) = \sum_{\vb*{t}\in \mathcal{T}} \phi_{p_z}(\vb*{r} - \vb*{r}_a - \vb*{t}) e^{i\vb*{k}\cdot \vb*{t}} \] where \(a = 1,2\) denotes the two kinds of atoms. \(\phi^1_{\vb*{k}}\) and \(\psi^2_{\vb*{k}}\) defines the 2-dimensional representation \(T_{\vb*{k}}\) of the little group. Now we obtain the characters of little groups at points of high symmetry.

• \[ T_\Gamma = A_1 \oplus B_2 = \Gamma_1 \oplus \Gamma_3. \]
• \[ T_K = K_3. \]
• \[ T_M = M_1 \oplus M_2. \]
• If \(k\) is on a symmetry line, then \(T_{\vb*{k}}\) is a subgroup of the little groups for the endpoints.

We could associate the names of IRs with bands, since \(G\) acts on them.

• At each \(\vb*{k}\) we obtain a representation of the little group. If no degeneracy occurs, we label the bands by the names of the IRs.
• If degeneracy occurs, then there must be a IR of more than one-dimension. We label the intersection of the bands by this IR.

Topological Systems

Let \(\vb*{k}_1\) be a point near a high-symmetry point \(\vb*{k}_0\). We have \[ G_{\vb*{k}_1} \subset G_{\vb*{k}_0} \] and the compatibility relations \[ \color{orange} \rho_1 = \rho_0 \downarrow G_{\vb*{k}_1}. \]

• Quasiband representation: any little group representations satisfying these compatibility relations.
  • Band representations are also quasiband representations, but not vice versa.
  • Quasiband representation may lack exponentially localized Wannier functions. Such quasiband representations are topological bands.
• Symmetry indicated topological bands: topological bands that can be distinguished by their little group representations.
  • Symmetry indicated topological bands \(\Leftrightarrow\) nonnegative integer vectors \(\vb*{v}\) that are not expressible as nonnegative integer sums of EBR vectors \(\vb*{e}_a\).
  • Stable symmetry-indicated topological bands: \(\vb*{v}\) cannot be expressed as any integer (positive or not) linear combination of EBR vectors.
  • Fragile symmetry-indicated topological bands: \(\vb*{v}\) can only be expressed as a linear combination of EBR vectors with at least one negative coefficient.
• Let \(\rho_{i[\vb*{k}]}\) denote the \(i\)th IR of the little group at \(\vb*{k}\).
  • \([\vb*{k}]\) denotes the symmetry-equivalent class of \(\vb*{k}\).
• We form a vector space \(V_G\) spanned by \[ \vb*{v} = \bigoplus_{i,[\vb*{k}]} n_{i[\vb*{k}]} \rho_{i[\vb*{k}]}. \]
  • Every quasiband representation corresponds to a \(\vb*{v}\) whose \(n_{i[\vb*{k}]}\) satisfy the compatibility relations.
  • Each EBR \(\rho_{\vb*{k}}^a\) corresponds to a vector \(\vb*{e}_a\) in that space.
  • Each atomic-limit band structure can be identified with a vector \[ \vb*{a} = \bigoplus_{a} n_a\vb*{e}_a. \]
  • We organize \(\vb*{e}_a\)'s into a matrix \(\vb*{A}\). The problem of expanding a quasiband representation into EBR is formulated as \[ \vb*{v} = \vb*{A} \vb*{n}. \]
  • We demand in the above expressions that \(n\)'s be non-negative.

After a linear transformation, \(\vb*{v} = \vb*{A}\vb*{n}\) could be reformulated into \[ \vb*{v}' = \vb*{D} \vb*{n}', \] where \(\vb*{D}\) is a diagonal matrix of size \(n\times m\).

• Trivial bands are the integer solutions.
• The space \(\mathcal{V}_G\) of symmetry indicated topological bands is isomorphic to \[ \color{darkcyan} \mathcal{V}_G \approx \bigoplus_i \mathbb{Z}_{d_i}. \]
• \(\mathcal{V}_G\) is the symmetry indicator group for the space group \(G\).

The \(\vb*{k}\) points of p2 fall into five categories:

• \(C_2\), which admits two irreducible representations, denoted \(A\) and \(B\): \(\Gamma\), \(X\), \(Y\) and \(M\).
• Trivial: GP. Therefore, \(V_G\) has nine elements in its basis: \[ \Gamma_\pm, X_\pm, Y_\pm, M_\pm, GP. \] Compatibility relations: since lines connecting different \(\vb*{k}\)'s are trivial, the only compatibility relation is that the dimension of the representation at each \(\vb*{k}\) be the same. \[ n_{\Gamma+} + n_{\Gamma-} = n_{X+} + n_{X-} = n_{Y+} + n_{Y-} = n_{M+} + n_{M-} = n_{GP}. \]
• Quasiband representation: \[ \vb*{v} = (n_{\Gamma+}, N - n_{\Gamma+}, n_{X+}, N-n_{X+}, n_{Y+}, N-n_{Y+}, n_{M+}, N-n_{M+}, N). \]
• EBR: \begin{align*} \vb*{e}_{1aA} &= (1,0,1,0,1,0,1,0,1), \\ \vb*{e}_{1aB} &= (0,1,0,1,0,1,0,1,1), \\ \vb*{e}_{1bA} &= (1,0,0,1,1,0,0,1,1), \\ \vb*{e}_{1bB} &= (0,1,1,0,0,1,1,0,1), \\ \vb*{e}_{1cA} &= (1,0,1,0,0,1,0,1,1), \\ \vb*{e}_{1cB} &= (0,1,0,1,1,0,1,0,1), \\ \vb*{e}_{1dA} &= (1,0,0,1,0,1,1,0,1), \\ \vb*{e}_{1dB} &= (0,1,1,0,1,0,0,1,1). \end{align*} We have \[ D = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}. \]
• \(0\) of multiplicity \(3\): we have only \(5\) degrees of freedom.
• \(1\) of multiplicity \(4\): four of the dimension are trivial.
• \(2\) of multiplicity \(1\): topologically nontrivial part, \(\mathbb{Z}_2\) symmetry.
• Indicator: \[\nu = n_{\Gamma+} + n_{M+} - n_{X+} - n_{Y+} \operatorname{mod} 2 \in \mathbb{Z}_2.\]

Example: Group P1̅1'

Symmetries:

• Three linearly independent lattice translations.
• Inversion.
• Time reversal.

Wyckoff positions:

• 1a to 1h, corresponding to the eight inversion centers in the unit cell, \[ \frac{1}{2}(n,m,l), \] where each of \(n,m,l\) takes value \(0\) or \(1\). They are also called time-reversal invariant momenta.

EBRs: 16 in total.

• Induced from a Kramers pair of orbitals with either \(+1\) or \(-1\) inversion eigenvalue at one of these eight Wyckoff positions.

The Indicator group of P1̅1' is \(\mathbb{Z}_4\), and \[ \nu_4 = \sum_{\mathrm{TRIMs} \vb*{k}_i} n_{\vb*{k}_i -} \operatorname{mod} 4, \] where \(n_{\vb*{k}_i -}\) the multiplicity of IRs at the TRIM \(\vb*{k}_i\) with negative inversion eigenvalue.

Disconnected Elementary Band Representations

• Suppose \(\rho_{\vb*{k}}\) is disconnected.
• \(\rho_{\vb*{k}}\) could be written as a sum of quasiband representations \[ \rho_{\vb*{k}} = \rho_{1\vb*{k}} \oplus \rho_{2\vb*{k}}. \]
• \(\rho_1\) and \(\rho_2\) are quasiband representations but cannot be both band representations.
  • Topologically nontrivial.

In the Kane-Mele model of the p6mm group, the four-bane EPR induced from \(p_z\) orbitals at honeycomb sites (2b Wyckoff position) may split into two disconnected components (in the topological phase), each of which has a nontrivial \(\mathbb{Z}_2\) index.

Band Representation and Its Applications

Band Representation

• What do we need to construct a band representation? We need to pick a site \(\vb*{q}\), which belongs to a Wyckoff position of multiplicity \(n\), and an \(d_q\)-dimensional representation \(\rho\) of \(G_{\vb*{q}}\).
  • It's the Chemists' habit to let the representation act on Wannier functions, i.e. \[ g_{\vb*{q}} W_{i1}(\vb*{r}) = [\rho(g_{\vb*{q}})]_{ji} W_{j1}(\vb*{r}),\quad \text{for}\quad g\in G_{\vb*{q}}. \]
    • Now we have the linear space where this representation works on: \[ V = \set{W_{i1}(\vb*{r}) | i = 1,\cdots,d_q}. \]
  • I think (just guessing) that the Wannier functions here are just those simple basis functions like \(x\), \(y\), and \(z\) for \(p\)-orbitals and \(xy\) for \(d\)-orbitals, etc, not the calculated Wannier functions obtained after we know the exact wavefunctions.
  • Now we extend the representation to the whole point group by defining the action of representative elements as \[ W_{i\alpha}(\vb*{r}) = g_\alpha W_{i1}(\vb*{r}), \] where \[ \vb*{q}_\alpha = g_\alpha \vb*{q}_1. \]
  • We extend the representation further to the whole space group by \[ \qty{E|\vb*{t}} W_{i\alpha}(\vb*{r}) = W_{i\alpha}(\vb*{r} - \vb*{t}). \]
  • Basis of our huge representation: \[ \qty{W_{i\alpha}(\vb*{r} - \vb*{t})} = \bigcup_{\alpha,\vb*{t}} \qty{g_\alpha | \vb*{t}} V, \] i.e. Wannier function of orbital \(\alpha\) at cell \(\vb*{t}\) and the site labelled by \(i\).
  • Dimension: \[ n\times d_q \times N_1 N_2 N_3. \]
• This is exactly what we had for induced representations. Let figure it out.
  • Which space does \(g\) move \(W\) to? i.e. \[ g \cdot \qty{g_\alpha | \vb*{t}_\alpha} W_{i1}(\vb*{r}) = \qty{g_{?}|\vb*{t}_?} g_{\vb*{q}} W_{i1}(\vb*{r}). \]
  • The representation is determined once the \(?\) is determined.
• With the Wannier functions we recover the Bloch functions (not necessarily the real Bloch functions: just toy Bloch functions), and let \(g\in G\) acts on it: \[ a_{i\alpha}(\vb*{k}, \vb*{r}) = \sum_{\vb*{t}} e^{i\vb*{k}\cdot \vb*{t}} W_{i\alpha}(\vb*{r} - \vb*{t}). \]
  • This amounts to a change of basis in out huge linear space \[ \bigcup_{\alpha,\vb*{t}} \qty{g_\alpha|\vb*{t}} V. \]
  • Why do we need this when our huge set of Wannier functions already serve as a good representation? Because our huge representation is obviously reducible. By doing the Fourier transform, our basis functions are made into eigenfunctions of translations, i.e. pre-diagonalization \[ \qty{E|\vb*{t}} a_{i\alpha}(\vb*{k}, \vb*{r}) = e^{-i\vb*{k}\cdot \vb*{t}} a_{i\alpha}(\vb*{k}, \vb*{r}). \]
• Band representation: the band representation \(\rho_G\) induced from the \(d_q\)-dimensional representation \(\rho\) of the site-symmetry group \(G_{\vb*{q}}\) of a particular point \(\vb*{q}\) whose orbit contains the sites \(\qty{\vb*{q}_\alpha = g_\alpha \vb*{q}}\) in the unit cell, is defined by the action \[ \color{orange} [\rho_G(g)a]_{i\alpha}(\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{i'=1}^{d_{\vb*{q}}} \rho_{i'i}(g_\beta^{-1} \qty{E\vert - \vb*{t}_{\beta\alpha}}gg_\alpha) a_{i'\beta}(R\vb*{k},\vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \sum_{i'=1}^{d_{\vb*{q}}} \rho_{i'i}(g_{\vb*{q}}) a_{i'\beta}(R\vb*{k},\vb*{r}), \]
  • \[ g = \qty{R|\vb*{v}}. \]
  • \[ gg_\alpha = \qty{E|\vb*{t}_{\beta\alpha}} g_\beta g_{\vb*{q}}, \]
  • \[ \vb*{t}_{\beta\alpha} = h\vb*{q}_\alpha - \vb*{q}_\beta. \]
  • Block form: \(\rho\) connects \[ \vb*{k'} = R\vb*{k} \] with \(\vb*{k}\).
  • \[ \rho^{\vb*{k}}_G(g)_{j\beta,i\alpha} = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \rho_{ji}\qty(g_\beta^{-1}\qty{E|-\vb*{t}_{\beta\alpha}}gg_\alpha). \]
  • At each \(\vb*{k}\) we have a \(n\times d_q\)-dimensional representation of the space group.
• Equivalent band representations: two band representations \(\rho_G\) and \(\sigma_G\) are equivalent if and only if there exists a unitary matrix-valued function \(S(\vb*{k}, t, g)\) that is smooth in \(\vb*{k}\) and continuous in \(t\) such that
  • \(S(\vb*{k},t,g)\) defines a band representation by \[ S(\vb*{k},t,g)_{j\beta, i\alpha} = \tau^{\vb*{k}}_G(g)_{j\beta,i\alpha}. \]
  • \[ S(\vb*{k},0,g) = \rho^{\vb*{k}}_G(g). \]
  • \[ S(\vb*{k},1,g) = \sigma^{\vb*{k}}_G(g). \]
• If two band representations are equivalent, then their subduced representation at little groups \(G_{\vb*{k}}\) (small representations) are equivalent as well.
  • Equivalence of small representations could be verified by characters.
  • This is only a necessary condition.

Small Representations

• The little group of \(\vb*{k}\): \[ G_{\vb*{k}} = \set{g=\qty{R|\vb*{v}} | R\vb*{k} = \vb*{k} + \vb*{K}, g\in G}. \]
• \(\qty{R|\vb*{v}}\) and \(\qty{R|\vb*{v}'}\) differs only by a factor of \(e^{-i\vb*{k}\cdot \vb*{t}}\).
• Small representation: \[ \rho_G \downarrow G_{\vb*{k}} \] restricted to those elements in \(G_{\vb*{k}}\), and retain only the block \[ \rho^{\vb*{k}}_G(g)_{j\beta,i\alpha} = e^{-i(R\vb*{k})\cdot \vb*{t}_{\beta\alpha}} \rho_{ji}\qty(g_\beta^{-1}\qty{E|-\vb*{t}_{\beta\alpha}}gg_\alpha). \]
• Dimension: \[ n\times d_{q}. \]
• Character: \[ \chi^{\vb*{k}}_G(g) = \sum_\alpha e^{-i(R\vb*{k})\cdot \vb*{t}_{\alpha\alpha}}\tilde{\chi}\qty[\rho(g_\alpha^{-1}\qty{E|-\vb*{t}_{\alpha\alpha}}gg_\alpha)], \] where \[ \tilde{\chi}[\rho(g)] = \begin{cases} \chi[\rho(g)], & \text{if } g\in G_{\vb*{q}}, \\ 0, & \text{otherwise}. \end{cases} \] For the trace to be nonzero, we demand \[ g_\alpha^{-1}\qty{E|\vb*{t}_{\alpha\alpha}} g g_\alpha = g_{\vb*{q}} \] for some carefully picked \(g_\alpha\) and \(\vb*{t}_{\alpha\alpha}\).
• Since \[ \rho^{\vb*{k}}_G = (\rho \uparrow G) \downarrow G_{\vb*{k}} \] is a representation of \(G_{\vb*{k}}\), it may be decomposed into a direct sum of IRs of \(G_{\vb*{k}}\) as \[ (\rho \uparrow G) \downarrow G_{\vb*{k}} = \bigoplus_i m_i^{\vb*{k}} \sigma_i^{\vb*{k}}. \] The coefficients may be obtained directly from the characters.

Qusaiband Representation

• Quasiband representation: any solution to the compatibility relations.
• Topological quasiband representation: a QBR that is not a (composite or elementary) band representation.

Elementary Band Representation

• Composite band representation: a band representation that is equivalent to a direct sum of other band representations.
• Elementary band representation: a band representation that is not composite.
• All elementary band representations could be induced from IRs of the maximal site symmetry groups.
• A band representation \(\rho_G\) is elementary if and only if it can be induced from an IR \(\rho\) of a maximal site-symmetry group \(G_{\vb*{q}}\), with only a few exceptions.

Classification of Elementary representations

• (Fully filled) Connected EBR ⬌ Exponentially localized Wannier orbitals that respect to the symmetries of crystals ⬌ Topologically trivial.
• Fractionally filled EBR ⬌ Protected (semi-)metals.
• Disconnected EBR ⬌ Topological bands ⬌ At least one group of bands is a weak, strong, or crystalline topological insulator.
• Isolated bands not equivalent to a band representation ⬌ strong, weak, or crystalline topological insulator.

Topological Bands

• Bands in the atomic limit: bands that can be induced from localized Wannier functions consistent with the crystalline symmetry of the SG.
• Topological bands: bands not in the atomic limit.

Hidden Obstructed Atomic Limits

• Bands in the obstructed atomic limit: a set of bands that possess symmetric, localized Wannier functions that reside on a Wyckoff position distinct from the Wyckoff position distinct from the Wyckoff position of the underlying atoms and cannot be smoothly defoemed to the ionic position.

IR-equivalent bands: two groups of bands with the same IRs at all high-symmetry points in the BZ.

It is possible that two such bands transforms identically under all symmetries at each point in the BZ but differ by a topologically nontrival global gauge transformation. Such band representations are called obstructed atomic limits.

In F222 each EBR induced from the 4a position is IR-equivalent to that induced from the 4b position. The basis functions are related by \[ \ket{w_b} = e^{i(k_2+k_3-k_1)/2} \ket{w_a}. \] If \(H(t)\) has occupied band 4a at \(t=0\) and 4b at \(t=1\), then a gap-closing phase transition must occur at some intermediate \(t\).

Examples

Position 1a in p6mm

Example copied from Building blocks of topological quantum chemistry: Elementary band representations.

• Little group: \(C_{6v}{}\).
• Sites: \[ \qty{q_\alpha} = \qty{q_1}. \]
• Band representation: \[ [\rho_G(\qty{R|\vb*{t}})a]_i(\vb*{k}, \vb*{r}) = e^{-i(R\vb*{k})\cdot \vb*{t}} \sum_{i'} [\Gamma(R)]_{i' i}a_{i'}(R\vb*{k}, \vb*{r}). \]

Position 2b in p6mm

• Little group: \(C_{3v}{}\).
• Sites: \[ \qty{q_{\alpha}} = \qty{C_2 q_1, q_1}. \]
  • \[ g_1 = \qty{E | \vb*{0}}. \]
  • \[ g_2 = \qty{C_2, \vb*{0}}. \]
• We pick a specific IR of the little group: \(\overline{\Gamma}_6\), which is the spin-1/2 representation and describes the behavior of the \(p_z\) orbital.
  • In particular, the orbital index \(i\) takes two values, and the site index \(\alpha\) takes two values, giving rise to a four-dimensional representation at each site \(\vb*{k}\).
• We are now able to compute for each \(\vb*{k}\) the matrix \[ \rho^{\vb*{k}}_G(g)_{j\beta,i\alpha} \] and restrict \(g\) to \(g\in G_{\vb*{k}}\). This furnishes a representation of \(G_{\vb*{k}}\), which might be reducible. \[ \begin{align*} \rho^{\Gamma} &= \overline{\Gamma}_8 \oplus \overline{\Gamma}_9, \\ \rho^K &= \overline{K}_4 \oplus \overline{K}_5 \oplus \overline{K}_6, \\ \rho^M &= \overline{M}_5 \oplus \overline{M}_5. \end{align*}{} \]
• Suppose that we are now given a group of bands that
  • transforms at \(\Gamma\) according to \[ \overline{\Gamma}_8 \oplus \overline{\Gamma}_9; \]
  • transforms at \(K\) according to \[ \overline{K}_4 \oplus \overline{K}_5 \oplus \overline{K}_6; \]
  • transforms at \(M\) according to \[ \overline{M}_5 \oplus \overline{M}_5. \] Then how are they possibly connected on the line joining these points? The little group of each point on the line are isomorphic to \(C_s\), all generated by a mirror combined with translations. From the character tables we obtain the compatibility relations
  • from \(\Gamma\): \[ \begin{align*} \overline{\Gamma}_8 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4, \\ \overline{\Gamma}_9 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4; \end{align*} \]
  • from \(K\): \[ \begin{align*} \overline{K}_4 & \rightarrow \overline{\Gamma}_3, \\ \overline{K}_5 & \rightarrow \overline{\Gamma}_4, \\ \overline{K}_6 & \rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4; \\ \end{align*} \]
  • from \(M\): \[ \begin{align*} \overline{M}_5 &\rightarrow \overline{\Gamma}_3 + \overline{\Gamma}_4. \end{align*} \]
• The compatibility relations per se allow different connections between.
  • In fig (a) the lower band is equivalent to the \(\overline{\Gamma}_9\) representation induced from the 1a position. However, the upper band is not equivalent to any band representation and is therefore topological.
    • The Wannier function of the upper band is given by (see Wannier representation of Z₂ topological insulators) \[ \ket{p_z \uparrow + \downarrow} \] on the \(A\) sites, and \[ \ket{p_z \uparrow - \downarrow} \] on the \(B\) sites.
    • The Wannier functions does not respect the symmetries (for example, \(C_{3v}{}\)) of the 2b positions.
  • In fig (b), the bands are connected and therefore nontopological.
• Only with a given crystal with its bands calculated could we know which connection scheme do the bands adopt and thereby find whether the band is topological or not.

Miscellaneous

Little group and little cogroup?

• Little group is a space group while the little cogroup is the point group thereof.

What do we mean when we say a time-reversal symmetric representation?

• I don't know. I am making this up. Under time-reversal we have \[ \psi_{\vb*{k},\sigma} \rightarrow \psi_{-\vb*{k},-\sigma}. \] I guess that, for \(\rho\) being a time-reversal symmetric IR of a site symmetry, \(\rho(T)\) does not connect two different degenerate spaces.
  • What the heck is that?

Band representation and quasiband representation?

• Both satisfy the compatibility relations. However, band representations allows Wannier functions that exponentially decay while quasiband representations may or may not.

What do we mean when we say %#@&^! is protected by $&*<# symmetry?

• As long as the symmetry exists, the property is there disregarding perturbations.