Ze Chen

Topological Insulator (I)


From the Hall Effect to Quantum Spin Hall Effect

The Hall resistance is given by \[ R_H = \frac{V_H}{I}. \] In the anomalous Hall effect (AHE), the Hall resistance may have in addition to the contribution from \(B\) a contribution from \(M\), i.e. \[ R_H = R_O B + R_A M. \] The AHE has extrinsic and intrinsic origin.

  • Extrinic origin: disorder-related spin-dependent scattering of carriers.
  • Intrinic origin: spin-dependent structure of conduction electrons. The spin-orbit effect induces a spin-dependent transverse force and deflects electrons of different spins to different directions.

The spin Hall effect results in spin accumulation on the lateral surfaces of a current-carrying sample, the signs of the spin orientations being opposite on two opposite boundaries.

  • Extrinsic: spin-dependent scattering of carriers.
  • Intrinsic: occurs even without impurity scattering.
  • Resonant spin Hall effect: a small current induces a finite spin current and polarization.

Schematic of the SHE

Schematic of the inverse SHE

The quantum Hall effect is the phenomenon that the in a strong magnetic field, the longitudinal conductance becomes zero while the quantum plateau of Hall conductance appears at \(\nu e^2/h\), where \(\nu = 1,2,\cdots\). When one Laudau level is fully filled, the filling factor is \(\nu = 1\).

In a sample of higher mobility, there may be fractional quantum Hall effect where \(\nu = 1/3,2/3,1/5,2/5,...\) Such effect in a topological phase of composite fermions, which breaks time reversal symmetry.

In a lattice system of spinless electrons in a periodic magnetic flux, electrons may be driven to form a conducting edge channel by the periodic magnetic flux, leading to the quantum anomalous Hall effect, i.e. the QHE in the absence of an external field or landau levels.

The quantum spin Hall effect is a quantum version of the SHE or a spin version of the QHE, and can be regarded as a combination of two quantum anomalous Hall effects of spin-up and spin-down electrons with opposite chirality.

Hall Effect\(B\neq 0\)1879 AHE\(B=0\), \(M\neq 0\)1880 SHE\(B=0\), \(M=0\)2004-2006
   
QHE\(\sigma_H = n\dfrac{e^2}{h}\)1980/1982 QAHE\(\sigma_H = \dfrac{e^2}{h}\)not yet confirmed QSHE\(\sigma_S = \dfrac{e}{4\pi}\)2007

Topological Insulators as Generalization of QSHE

A topological insulator is a state of a quantum matter that behaves as an insulator in its interior while as a metal on its boundary.

  • Electron spins in the surface states are locked to their momenta.
  • In a weak topological insulator, the resultant surface states are not so stable to disorder or impurities.
  • In a strong topological insulator the surface states are protected by time reversal symmetry.

Some examples are listed as follows.

  • One-dimsional: polyacetylene.
  • Two-dimensional: \(\ce{HgTe/CdTe}{}\) quantum well and \(\ce{InAs/CdTe}{}\) quantum well.
  • Three-dimensional: \(\ce{Bi_{1-x}Sb_x}{}\), \(\ce{Bi2Te3}{}\), \(\ce{Bi2Te2Se}{}\).

The Dirac Equation

In the Dirac equation (see my note) we set \[ H = c\vb*{p}\cdot \alpha + mc^2 \beta. \] Under a transformation of mass \(m\rightarrow -m\), the equation is invariant if we replace \(\beta \rightarrow -\beta\). There is no topological distinction between particles with positive and negative masses.

The absence of a negatively charged electron that has a negative mass and kinetic energy would manifest itself as a positively charged particle which has an equal positive mass and positive energy.

  • In one dimension we take \[ \alpha_x = \sigma_x,\quad \beta = \sigma_z. \]
  • In two dimensions we take \[ \alpha_x = \sigma_x,\quad \alpha_y = \sigma_y,\quad \beta = \sigma_z. \]
  • In three dimensions we take \[ \alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix},\quad \beta = \begin{pmatrix} \mathbb{1}_{2\times 2} & 0 \\ 0 & \mathbb{1}_{2\times 2} \end{pmatrix}. \]

Solutions of Bound States

Jackiw-Rebbi Solution in One Dimension

Let \[ h(x) = -iv\hbar \partial_x \sigma_x + m(x) v^2 \sigma_z \] where \[ m(x) = \begin{cases} -m_1, & \text{if } x<0, \\ +m_2, & \text{otherwise}. \end{cases} \]

The solution for \(x>0\) is given by \[ \begin{pmatrix} \varphi_1(x) \\ \varphi_2(x) \end{pmatrix} = \begin{pmatrix} \varphi_1^+ \\ \varphi_2^+ \end{pmatrix} e^{-\lambda_+ x} \] where \[ \lambda_+ = \frac{\sqrt{m_2^2 v^4 - E^2}}{\nu \hbar} \] For \(x<0\) the solution is similar. The continuity condition allows a solution at \(E=0\).

More generally, for any \(m(x)\) that changes from negative to positive at two ends, we also have a solution \[ \varphi_\eta(x) \propto \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \eta i \end{pmatrix} \exp\qty[-\int^x \eta \frac{m(x')v}{\hbar}\dd{x'}], \] where \(\eta\) is a sign, taking value \(\pm 1\).

Two Dimensions

With \(p_z=0\) we find two solutions \[ \Psi_+(x,k_y) = \sqrt{\frac{v}{h}\frac{m_1m_2}{m_1+m_2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ i \end{pmatrix} e^{-\abs{m(x) vx}/\hbar + ik_y y} \] and \[ \Psi_-(x,k_y) = \sqrt{\frac{v}{h}\frac{m_1m_2}{m_1+m_2}}\begin{pmatrix} 0 \\ 1 \\ i \\ 0 \end{pmatrix} e^{-\abs{m(x) vx}/\hbar + ik_y y} \] with dispersion relations \[ \epsilon_{k,+} = v\hbar k_y \] and \[ \epsilon_{k,-} = -v\hbar k_y, \] respectively. The two states move with velocity \[ v_\pm = \frac{\partial \epsilon_{k,\pm}}{\hbar \partial k} = \pm v. \]

Quadratic Correction to the Dirac Equation

We introduce a correction in the Hamiltonian and get \[ H = v\vb*{p}\cdot \alpha + (mv^2 - B\vb*{p}^2)\beta. \]

  • \(p = 0\): the spin orientation is determined by \(mv^2 \beta\), i.e. the sign of \(m\).
  • \(p \rightarrow \infty\): determined by \(-B\vb*{p}^2 \beta\) or sign of \(B\).
    • \(mB>0\): as \(p\rightarrow \infty\), the spin rotate eventually to the opposite \(z\)-direction. This consists of two merons in the momentum space, a.k.a. skyrmion.
    • \(mB<0\): as \(p\rightarrow \infty\), the spin rotate back to the \(z\)-direction.
One Dimension

The Dirac equation in this case is given by \[ h(x) = vp_x \sigma_x + (mv^2 - Bp_x^2) \sigma_z, \] which admits solutions for \(x\ge 0\) under the Dirichlet condition \(\varphi(0) = 0\) \[ \varphi_\eta(x) = \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ i \end{pmatrix}\qty(e^{-x/\xi_+} - e^{-x/\xi_-}) \] where \[ \xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar}\qty(1 \pm \sqrt{1-4mB}). \]

  • For \(B\rightarrow 0\) we have
    • \(\xi_+ \rightarrow \abs{B}\hbar/v\),
    • and \(\xi_- = \hbar/mv\).
  • Further let \(m\rightarrow 0\) we have
    • \(\xi_- \rightarrow \infty\), where a topological phase transition occurs.

The four-component forms of the solutions are written as \[ \Psi_1 = \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ 0 \\ 0 \\ i \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}) \] and \[ \Psi_2 = \frac{C}{\sqrt{2}} \begin{pmatrix} 0 \\ \operatorname{sign}{B} \\ i \\ 0 \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}). \]

Two Dimensions: Helical Edge States

The exact solutions to \[ h_\pm = vp_x \sigma_x \pm vp_y\sigma_y + (mv^2 - Bp^2)\sigma_z \] is given by \begin{align*} \Psi_1 &= \frac{C}{\sqrt{2}} \begin{pmatrix} \operatorname{sign}{B} \\ 0 \\ 0 \\ i \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}) e^{+ip_y y/\hbar}; \\ \Psi_2 &= \frac{C}{\sqrt{2}} \begin{pmatrix} 0 \\ \operatorname{sign}{B} \\ i \\ 0 \end{pmatrix} \qty(e^{-x/\xi_+} - e^{-x/\xi_-}) e^{+ip_y y/\hbar}, \\ \end{align*} where the characteristic length is given by \[ \xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar}\qty(1\pm \sqrt{1-4mB + 4B^2p_y^2/v^2}). \]

In two dimensions, the Chern number for a system with Hamiltonian \[ H = \vb{d}(p)\cdot \sigma \] is given by \[ n_c = -\frac{1}{4\pi}\int \dd{\vb*{p}} \frac{\vb{d}\cdot \qty(\partial_{p_x}\vb{d}\times \partial_{p_y}\vb{d})}{d^3} \] where the integral runs over the first BZ. The result is given by \[ n_\pm = \pm\qty(\operatorname{sign}{m} + \operatorname{sign}{B})/2. \] The system is topologically nontrivial if \(n\neq 0\), i.e. \(m\) and \(B\) have the same sign.

Using the perturbation theory to the first order, the dispersion relation may be obtained to be \[ \epsilon_{p_y,\pm} = \pm vp_y. \]

Three Dimensions: Surface States

In three dimensions the Hamiltonian is given by \[ H = H_{1D}(x) + H_{3D}(x) \] where \[ H_{3D}(x) = vp_y \alpha_y + vp_z\alpha_z - B(p_y^2 + p_z^2)\beta. \]

The exact solutions are given by \[ \Psi_\pm = C \Psi_\pm^0 \qty(e^{-x/\xi_+} - e^{-x/\xi_-})\exp\qty[+i\qty(p_y y + p_z z)/\hbar] \] where \begin{align*} \Psi_+^0 &= \begin{pmatrix} \cos \frac{\theta}{2} \operatorname{sign}{B} \\ -i \sin \frac{\theta}{2} \operatorname{sign}{B} \\ \sin \frac{\theta}{2} \\ i \cos \frac{\theta}{2} \end{pmatrix}; \Psi_-^0 &= \begin{pmatrix} \sin\frac{\theta}{2}\operatorname{sign}{B} \\ i \cos \frac{\theta}{2} \operatorname{sign}{B} \\ -\cos \frac{\theta}{2} \\ i\sin \frac{\theta}{2} \end{pmatrix}, \tan\theta &= \frac{p_y}{p_z}, \end{align*} and the dispersion is given by \[ \epsilon_{p\pm} = \pm vp \operatorname{sign}{B}. \] The penetration depth is \[ \xi_\pm^{-1} = \frac{v}{2\abs{B}\hbar} \qty(1\pm \sqrt{1-4mB + 4B^2 p^2 /\hbar^2}). \]

Under the condition \(mB>0\),

  • In one dimension, there exists bound state of zero energe near the ends.
  • In two dimensions, there exists helical edge states nea the edge.
  • In three dimensions, there exists surface states near the surface.

Berry Phase

General Formalism
  • \(\vb*{R}\) a set of parameter of the Hamiltonian.
  • Schrödinger equation: \[ H(\vb*{R}) \ket{n(\vb*{R})} = E_n(\vb*{R}) \ket{n(\vb*{R})}. \]
  • Gauge freedom: an arbitrary phase factor \(e^{-i\theta(\vb*{R})}\),
    • or, in the case of degenerate states, a matrix. \[ \ket{\psi(t)} = e^{-i\theta(t)} \ket{n(\vb*{R}(t))}. \]
  • Smooth and singled-valued gauge may not be possible.
  • Under slowing moving \(\vb*{R}(t)\), \[ \ket{\psi(t)} = \exp\qty({\frac{1}{\hbar} \int_0^t E_n(\vb*{R}(t')) \dd{t'}}) e^{i\gamma_n} \ket{n(\vb*{R}(t))}. \]
    • Geometric phase: \[ \gamma_n = \int \dd{\vb*{R}}\cdot \vb*{A}_n(\vb*{R}) = -\Im \int \bra{n(\vb*{R})}\grad_{\vb*{R}} \ket{n(\vb*{R})} \dd{\vb*{R}}. \]
    • Berry connection: \[ \vb*{A}_n(\vb*{R}) = i\bra{n(\vb*{R})} \pdv{}{\vb*{R}} \ket{n(\vb*{R})}. \]
    • Gauge transformation: under \[ \ket{n(\vb*{R})} \rightarrow e^{i\xi(\vb*{R})} \ket{n(\vb*{R})}, \] the Berry connection transforms like \[ \vb*{A}_n(\vb*{R}) \rightarrow \vb*{A}_n(\vb*{R}) - \pdv{}{\vb*{R}} \xi(\vb*{R}). \]
      • Under such a transformation, the geometric phase is changed by \[ \gamma_n \rightarrow \gamma_n + \xi(\vb*{R}(0)) - \xi(\vb*{R}(T)). \]
  • Berry curvature: \[ \mathcal{F} = \dd{\mathcal{A}} = \qty(\pdv{\bra{\vb*{R}}}{R^\mu}) \qty(\pdv{\ket{\vb*{R}}}{R^\nu}) \dd{R^\mu} \wedge \dd{R^\nu}. \]
  • Geometric phase: \[ \gamma_n = -\Im \int \dd{\vb*{S}}\cdot \qty(\grad \bra{n(\vb*{R})}) \times \qty(\grad \ket{n(\vb*{R})}). \]
Numerical Calculation of Berry Phase
  • Gauge-independent formula: for singly degenerate level, \[ \gamma_n = -\iint \dd{\vb*{S}} \cdot \vb*{V}_n = -\iint \dd{\vb*{S}} \cdot \Im \sum_{m\neq n} \frac{\bra{n(\vb*{R})}(\grad_{\vb*{R}} H(\vb*{R})) \ket{m(\vb*{R})} \times \bra{m(\vb*{R})}(\grad_{\vb*{R}} H(\vb*{R})) \ket{n(\vb*{R})}}{(E_m(\vb*{R}) - E_n(\vb*{R}))^2}. \]
    • \[ \sum_n \gamma_n = 0. \]
  • Two-level systems as an example: near degeneracy \(\vb*{R}^*\), \[ \vb*{V}_+(\vb*{R}) = \Im \frac{\bra{+(\vb*{R})} (\grad H(\vb*{R}^*)) \ket{-(\vb*{R})} \times \bra{-(\vb*{R})}(\grad H(\vb*{R}^*))\ket{+(\vb*{R})}}{(E_+(\vb*{R}) - E_-(\vb*{R}))^2}. \]
Two-Level Systems
  • Generic Hamiltonian: \[ H = \vb*{\epsilon}(\vb*{R}) \mathbb{1}_{2\times 2} + \vb*{d}(\vb*{R}) \cdot \vb*{\sigma}. \]
    • Energy levels \[ E_{\pm} = \epsilon(\vb*{R}) \pm \sqrt{\vb*{d} \cdot \vb*{d}}. \]
Example: Two-Level Systems
  • \(\vb*{d}(\vb*{R})\) parametrized as \[ \vb*{d}(\vb*{R}) = d\cdot \vb*{n}(\theta,\phi). \]
  • Eigenstates: \[ \ket{-\vb*{R}} = \begin{pmatrix} \sin(\theta/2) e^{-i\phi} \\ -\cos(\theta/2) \end{pmatrix};\quad \ket{+\vb*{R}} = \begin{pmatrix} \cos(\theta/2) e^{-i\phi} \\ \sin(\theta/2) \end{pmatrix}. \]
    • Not well-defined at the poles.
  • Berry curvature: \[ F_{\theta\phi} = \frac{\sin\theta}{2}. \]
  • Example: \(d(\vb*{R}) = \vb*{R}\), under which \[ \vb*{V}_- = \frac{1}{2} \frac{\vb*{d}}{d^3}. \]
    • Berry curvature integrated over a closed surface encircling the origin yields the Chern number.
    • Degeneracy points in the parameter space act as sources and drains of the Berry curvature.
    • Dirac-Weyl fermion: taking a closed path in the \((k_x,k_y)\) plane by setting \(k_z = 0\). \(\gamma = \pm 2\pi\) if the curve encircles the origin. Otherwise \(\gamma = 0\).
Example: Spin in a Magnetic Field
  • Energy levels: \[ E_n(B) = Bn,\quad n = -S,-S+1,\cdots,+S. \]
  • Geometric phase: \[ \gamma_n = -\iint \dd{\vb*{S}} \cdot n\frac{\vb*{B}}{B^3}. \]

Hall Conductance

Linear Response
  • Current operator: \[ \vb*{j}(\vb*{q}) = \sum_{\vb*{k},\alpha,\beta} c^\dagger_{\vb*{k} + \vb*{q}/2}c_{\vb*{k} - \vb*{q}/2} \pdv{h^{\alpha\beta}_{\vb*{k}}}{\vb*{k}}. \]
  • External Hamiltonian: \[ H_{\mathrm{ext}} = \sum_{\vb*{q}} \vb*{j}(\vb*{q}) \cdot \vb*{A}(-\vb*{q}). \]
  • Total current operator: \[ \begin{align*} \vb*{J}(\vb*{x}, t) &= \frac{1}{2}\sum_i \qty[(\vb*{p}_i - e\vb*{A}) \delta(\vb*{x} - \vb*{x}_i) + \delta(\vb*{x} - \vb*{x}_i)(\vb*{p}_i - e\vb*{A})] \\ &= \vb*{j}(\vb*{x}) - ne\vb*{A}(\vb*{x}, t). \end{align*} \]
  • External Hamiltonian to the first order: \[ H_{\mathrm{ext}} = -e \int \dd{^3 \vb*{x}} \vb*{A}(\vb*{x},t) \vb*{j}(\vb*{x}). \]
  • Expectation value of current: \[ \bra{E_N(t)} J_i(x,t) \ket{E_N(t)} = \bra{E_N} j_i(x) \ket{E_N} + \int_{-\infty}^{\infty} \dd{t'} \int \dd{^3 x'} \sum_j R_{ij}(x-x',t-t') A_j(x',t'). \]
  • Response function (Zero temperature): \[ R_{ij}(x-x',t-t') = -i\Theta(t-t')\bra{E_N} [j_i(x,t), j_j(x',t')] \ket{E_N} - ne\delta_{ij} \delta(x-x')\delta(t-t'). \]
  • Reponse function (Nonzero temperature): \[ \bra{E_N} [j_i(x,t), j_j(x',t')] \ket{E_N} \rightarrow \frac{\tr\qty{e^{-\beta H_0}[j_i(x,t), j_j(x',t')]}}{Z}. \]
  • Formulation: \[ J_\alpha(\omega) = \sigma_{\alpha\beta}(\omega) E_\beta(\omega) = i\omega \sigma_{\alpha\beta}(\omega) A_\beta(\omega). \]
Zero-Temperature: the Fluctuation-Dissipation Theorem
  • Correlation functions: \[ \begin{align*} J_1(\omega) &= \int_{-\infty}^{\infty} \langle A(t) B(0) \rangle e^{i\omega t}\dd{t} = \frac{2\pi}{Z}\sum_{mn} e^{-\beta E_m} \bra{n}B\ket{m} \bra{m}A\ket{n}\delta(E_m - E_n + \omega), \\ J_2(\omega) &= \int_{-\infty}^{\infty} \langle B(0) A(t) \rangle e^{i\omega t}\dd{t} = \frac{2\pi}{Z}\sum_{mn} e^{-\beta E_n} \bra{n}B\ket{m} \bra{m}A\ket{n}\delta(E_m - E_n + \omega), \\ J_2(\omega) &= e^{-\beta \omega} J_1(\omega). \end{align*}{} \]
  • Retarded Green's function: \[ \begin{align*} G^R(t) &= -i\Theta(t) \langle[A(t), B(0)] \rangle \\ &= -i\Theta(t) \cdot \frac{1}{2\pi} \int_{-\infty}^{\infty} J_1(\omega) (1-e^{-\beta\omega}) e^{-i\omega t} \dd{\omega}. \end{align*}{} \]
    • Frequency domain: \[ G^R(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{J_1(\omega')}{\omega - \omega' + i\eta}(1-e^{-\beta \omega'}) \dd{\omega'}. \]
    • \[ \Im(G^R(\omega)) = -\frac{1}{2} (1-e^{-\beta\omega}) J_1(\omega). \]
  • Time-ordered Green's function: \[ \begin{align*} G^T(t) = -i\langle T[A(t)B(0)] \rangle. \end{align*}{} \]
    • Frequency domain: \[ G^T(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} J_1(\omega') \qty{\frac{1}{\omega - \omega' + i\eta} - \frac{e^{-\beta\omega'}}{\omega - \omega' - i\eta}} \dd{\omega'}. \]
    • \[ \begin{align*} \Re G^T(\omega) = \frac{1}{2\pi} \mathcal{P}\int_{-\infty}^{\infty} (1-e^{-\beta \omega'}) \frac{J_1(\omega')}{\omega - \omega'}. \end{align*}{} \]
    • \[ \Im G^T(\omega) = -\frac{1}{2}(1+e^{-\beta\omega}) J_1(\omega). \]
Finite Temperature: Matsubara Function
  • Temperature function \[ G^T(\sigma) = \langle T[A(\sigma)B(0)]\rangle = \Theta(\sigma)\langle A(\sigma) B(0) \rangle + \Theta(-\sigma)\langle B(0) A(\sigma) \rangle. \]
  • Imaginary time-evolution: \[ A(\sigma) = e^{\sigma H} A e^{-\sigma H}. \]
  • Periodicity: \[ G^T(\sigma-\beta) = G^T(\sigma). \]
  • Frequency domain formulation: \[ G^T(\omega_m) = -\frac{1}{\beta} \cdot \frac{1}{2\pi} \int_{-\infty}^{\infty} (1-e^{-\beta \omega'}) \frac{J_1(\omega')}{i\omega_m - \omega'}\dd{\omega'}. \] where \[ \omega_m = \frac{2\pi m}{\beta}. \]
  • Analytic continuation: \[ G^R(\omega) = -\beta G^T(\omega_m) |_{i\omega_m = \omega + i\eta}. \]
Formula for Hall Conductivity
  • Hall conductivity (in unit of \(e^2/h\)): \[ \begin{align*} \sigma_{ij\mathrm{Hall}} &= \int \frac{\dd{k_x} \dd{k_y}}{(2\pi)^2} \sum_{a=1}^m (-i)\qty(\bra{\partial_i(a,k)}\partial_j\ket{a,k} - \bra{\partial_j(a,k)\partial_i(a,k)}). \end{align*} \]
  • Summation over all filled bands.
  • In terms of the curvature tensor: \[ \sigma_{xy} = \frac{e^2}{h} \frac{1}{2\pi} \iint \dd{k_x} \dd{k_y} F_{xy}(k), \] where \[ F_{xy}(k) = \pdv{A_y(k)}{k_x} - \pdv{A_x(k)}{k_y} \] and \[ A_i = -i \sum_{\mathrm{filled bands}} \bra{a\vb*{k}} \pdv{}{k_i} \ket{a\vb*{k}}. \]
  • The integral is always a integer.

Time-Reversal Symmetry

Time-Reversal of Spinless Systems
  • TR of creation operators: \[ T c_j T^{-1} = c_j. \]
  • TR on band: if the system is time-reversal, then \[ T h(k) T^{-1} = h(-k). \]
  • Zero Hall conductance: \[ F_{ij}(-k) = -F_{ij}(k) \] and therefore the integral vanishes in the BZ.
Time-Reversal of Spinful Systems
  • The TR operator: \[ T = e^{-i\pi S_y} K. \]
    • For spin-\(1/2\) particles: \[ e^{-i\pi \sigma_y/2} = -i\sigma_y. \]
  • Scattering probability: \[ \bra{\psi} H \ket{T\psi} = 0. \]
Time-Reversal in Crystal
  • Hamiltonian spectrum: \[ H = \sum_{\vb*{k}} c^\dagger_{\vb*{k}\alpha\sigma} h^{\sigma\sigma'}_{\alpha\beta} c_{\vb*{k}\beta\sigma'}. \]
  • TR on annhilation operator: relative phase difference due to distinct numbers of particles, \[ \begin{align*} T c_{ja\sigma} T^{-1} &= i (\sigma^{y}_{\sigma'\sigma}) c_{ja\sigma'}, \\ T c^\dagger_{ja\sigma} T^{-1} &= c^\dagger_{ja\sigma'} i (\sigma^y)^T_{\sigma'\sigma}, \\ T c_{\vb*{k}a\sigma} T^{-1} &= i(\sigma^y)_{\sigma\sigma'} c_{-\vb*{k}a\sigma'}, \\ T c^\dagger_{\vb*{k} a\sigma} T^{-1} &= c^\dagger_{-\vb*{k}a\sigma'}i(\sigma^y)^T_{\sigma'\sigma}. \\ \end{align*}{} \]

Smalltalks

Topological Insulators
  • Trivial insulator: the insulators that, upon slowly turning off the hopping elements and the hybridization between orbitals on different sites, flows adiabatically into the atomic limit.

2021/2/21 18:51:01

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