Advanced Solid State Physics (II)
Strongly Correlated Systems
Solid State Physics
Wannier Functions
- Bloch to Wannier:
\[
\psi_{n\vb*{k}} = N^{-1/2} \sum_{\vb*{r}} e^{i\vb*{k}\cdot \vb*{l}} a_n(\vb*{r} - \vb*{l}).
\]
- Annihilation operators:
\[
c_{n\vb*{k}} = \frac{1}{\sqrt{N}} \sum_{\vb*{l}} c_{n\vb*{l}} e^{-i\vb*{k}\cdot \vb*{l}}.
\]
- Orthogonality:
\[
\int \dd{\vb*{r}} a^*_{n}(\vb*{r} - \vb*{l}) a_{n'}(\vb*{r} - \vb*{l}') = \delta_{nn'}\delta_{ll'}.
\]
- Completeness:
\[
\sum_{n} \sum_{l} a^*_n(\vb*{r} - \vb*{l}) a_n(\vb*{r}' - \vb*{l}) = \delta(\vb*{r} - \vb*{r}').
\]
Statistical Mechanics
- The trace on a grand canonical ensemble is defined by
\[
\langle \cdots \rangle = \frac{\operatorname{Tr}{\qty[e^{-\beta(H-\mu N)} \cdots]}}{\operatorname{Tr}{\qty[e^{-\beta(H-\mu N)}]}}.
\]
Green's Functions
- Retarded Green's function:
\[
\color{orange}\langle\langle A(t); B(t') \rangle\rangle = -i\Theta(t-t') \langle \qty{A(t), B(t')} \rangle.
\]
- Fourier transform:
\[
\langle\langle A|B \rangle\rangle (\omega + i\epsilon) = \int_{-\infty}^{+\infty} \langle\langle A(t); B(0)\rangle\rangle e^{i(\omega + i\epsilon) t}.
\]
- Denoted by \(G_{\mathrm{r}}(\omega)\).
- Spectral theorem:
\[
\color{red} \langle BA \rangle = \frac{i}{2\pi} \int_{-\infty}^{+\infty} \dd{\omega} \frac{\qty{\langle\langle A|B\rangle\rangle(\omega + i\eta) - \langle\langle A|B\rangle\rangle(\omega - i\eta)}}{e^{\beta(\omega - \mu)} - 1}.
\]
- \[
\langle BA \rangle = \sum \operatorname*{Res}_{\omega \in \mathbb{R}} \langle \langle A|B \rangle \rangle(\omega) f(\omega).
\]
- Fluctuation-Dissipation Theorem: for canonical ensembles,
\[
\color{red} \langle BA \rangle = \frac{1}{2\pi} \int_{-\infty}^{\infty} \dd{\omega} \frac{\qty{-2\Im G_{\mathrm{r}}(\omega)}}{e^{\beta\hbar\omega} \pm 1}.
\]
- Equation of motion:
\begin{gather*}
i\dv{}{t} \langle\langle A(t);B(t') \rangle\rangle = \delta(t-t') \langle \qty{A(t),B(t)} \rangle + \langle\langle [A(t),H]; B(t')\rangle\rangle \\
\Updownarrow \\
\color{red}\langle\langle A|B\rangle\rangle(\omega) = \begin{cases}
\langle \qty{A,B} \rangle + \langle\langle [A,H]|B \rangle\rangle(\omega), \\
\langle \qty{A,B} \rangle - \langle\langle A|[B,H] \rangle\rangle(\omega).
\end{cases}
\end{gather*}
- Population of electrons:
\[\begin{align*}
\color{darkcyan} n_\sigma &= \frac{1}{N} \sum_j \langle c^\dagger_{j\sigma} c_{j\sigma}\rangle \\
&\color{darkcyan}= \int_{-\infty}^{+\infty} \dd{\omega} f(\omega) \rho_\sigma(\omega).
\end{align*}{}\]
- Local density of states:
\begin{align*}
\color{orange} \rho_\sigma(\omega) &\color{orange}= \frac{i}{2\pi N} \sum_j \qty{G^\sigma_{jj}(\omega + i\eta) - G^\sigma_{jj}(\omega - i\eta)} \\
&\color{orange}= \frac{i}{2\pi N} \sum_{\vb*{k}} \qty{G^\sigma_{\vb*{k}}(\omega + i\eta) - G^\sigma_{\vb*{k}}(\omega - i\eta)}.
\end{align*}
Green's Function of the Field Operator
- The Green's functions solve the equation
\[
(-i\hbar\partial_t + H) G(\vb*{r},t;\vb*{r}',t') = -\hbar \delta(\vb*{r} - \vb*{r}')\delta(t-t').
\]
- The field operator is defined by
\[
\psi^\dagger_\sigma(\vb*{r},t) = \sum_{\vb*{p},\sigma} u_{\vb*{p}}(\vb*{r}) a^\dagger_{\vb*{p},\sigma}.
\]
- \[
G_{nn} = \frac{1}{2\pi} \int \dd{E} \bra{n}G\ket{n} e^{-iEt/\hbar} = \begin{cases}
-ie^{-iE_nt/\hbar}, & \text{if } t>0, \\
0, & \text{if } t\le 0.
\end{cases}
\]
- The Green's function is given by
\begin{align*}
G_\sigma(\vb*{r},t;\vb*{r}',t') &= -i \operatorname{T}\langle \psi_\sigma(\vb*{r}, t) \psi^\dagger_\sigma(\vb*{r}',t') \rangle \\
&= \Theta(t-t') G^>_\sigma(\vb*{r},t;\vb*{r}',t') + \Theta(t'-t) G^<_\sigma(\vb*{r},t;\vb*{r}',t').
\end{align*}
- \[
G^> = -i \langle \psi_\sigma(\vb*{r},t) \psi^\dagger_\sigma(\vb*{r}',t') \rangle.
\]
- \[
G^< = i \langle \psi^\dagger_\sigma(\vb*{r'},t') \psi_\sigma(\vb*{r},t) \rangle.
\]
- Retarded Green's function:
\[
G^{\mathrm{R}}_\sigma(\vb*{r},t;\vb*{r}',t') = -i\Theta(t-t')\langle \qty{\psi_\sigma(\vb*{r},t), \psi^\dagger_\sigma(\vb*{r}',t)} \rangle.
\]
- Non-zero only for \(t>t'\).
- Advanced Green's function:
\[
G^{\mathrm{A}}_\sigma(\vb*{r},t;\vb*{r'},t') = i\Theta(t'-t)\langle \qty{\psi_\sigma(\vb*{r},t), \psi^\dagger_\sigma(\vb*{r}',t')} \rangle.
\]
- Non-zero only for \(t<t'\).
Hubbard Model
- Assumptions:
- The system admits translational invariance.
- Hamiltonian: single conduction band,
\[
\color{orange} H = \underbrace{\sum_{i,j} \sum_\sigma T_{ij} c^\dagger_{i\sigma} c_{j\sigma}}_{H_0} + \frac{U}{2} \sum_i \sum_\sigma n_{i\sigma}n_{i, -{\sigma}}.
\]
- Creation and annihilation in the Wannier basis.
- \[
H_0 = \sum_{\vb*{k},\sigma} E_{\vb*{k}} c^\dagger_{\vb*{k}\sigma} c_{\vb*{k}\sigma}.
\]
- \[
T_{ij} = N^{-1} \sum_{\vb*{k}} e^{i\vb*{k}\cdot (\vb*{R}_i - \vb*{R}_j)} E_{\vb*{k}}.
\]
- \[
T_0 = T_{ii}.
\]
- \[
T_1 = T_{i,i+1} < 0.
\]
- \(i+1\) denotes the neighbours of \(i\).
- \(T_1 \sim \Delta\) (width of the band):
\[
E_{\vb*{k}} \approx T_0 + T_1 \sum_{\text{n. n.}} e^{-i\vb*{k}\cdot \vb*{R}_{\mathrm{n}}}.
\]
- \[
U = \bra{ii} v\ket{ii} = e^2 \int \frac{a^*(\vb*{r} - \vb*{R}) a^*(\vb*{r'} - \vb*{R}) a(\vb*{r} - \vb*{R}_i)a(\vb*{r}' - \vb*{R}_i)}{\abs{\vb*{r} - \vb*{r}'}}\dd{\vb*{r}}\dd{\vb*{r}'}.
\]
Zero Band Width
- Hamiltonian: \(T_{ij} = T_0 \delta_{ij}\),
\[
H = T_0 \sum_{i,\sigma} n_{i\sigma} + \frac{1}{2}U\sum_{i,\sigma} n_{i\sigma}n_{i,-\sigma}.
\]
- Single-particle Green's function:
\[
\color{orange} G^\sigma_{ij}(\omega) = \langle\langle c_{i\sigma} | c^\dagger_{j\sigma}\rangle\rangle(\omega).
\]
\[
\delta(\omega - \omega_0) = -\frac{1}{\pi} \lim_{\epsilon\rightarrow 0} \Im \frac{1}{\omega - \omega_0 + i\epsilon}.
\]
- Solution (exact):
\[
\color{darkcyan} G^\sigma_{ij}(\omega) = \delta_{ij} \qty{\frac{1-\langle n_{i,-\sigma} \rangle}{\omega - T_0} + \frac{\langle n_{i,-\sigma}\rangle}{\omega - T_0 - U}}.
\]
- Local density of states:
\[
\color{darkcyan} \rho_\sigma(\omega) = (1-\langle n_{-\sigma}\rangle)\delta(\omega - T_0) + \langle n_{-\sigma}\rangle \delta(\omega - T_0 - U).
\]
Small Band Width
- Hamiltonian: \(T_{ij} \neq 0\).
- Green's function in Bloch basis:
\[
G^{\sigma}_{\vb*{k},\vb*{k}'} = \langle\langle c_{\vb*{k}\sigma} | c^\dagger_{\vb*{k},-\sigma} \rangle\rangle = \delta_{\vb*{k},\vb*{k}'} G^\sigma_{\vb*{k}}(\omega).
\]
- Bloch basis to Wannier basis:
\[
G^{\sigma}_{ij}(\omega) = \frac{1}{N} \sum_{\vb*{k}} e^{i\vb*{k}\cdot (\vb*{R}_i - \vb*{R}_j)} G^\sigma_{\vb*{k}}(\omega).
\]
- Solution (approximate):
\begin{align*}
\color{darkcyan} G^{\sigma}_{\vb*{k}}(\omega) &= \frac{\omega - T_0 - U(1 - \langle n_{-\sigma} \rangle)}{(\omega - E_{\vb*{k}})(\omega - T_0 - U) + \langle n_{-\sigma} \rangle U(T_0 - E_{\vb*{k}})} \\
&\color{darkcyan}= \frac{A^{(1)}_{\vb*{k}\sigma}}{\omega - E^{(1)}_{\vb*{k}\sigma}} + \frac{A^{(2)}_{\vb*{k}\sigma}}{\omega - E^{(2)}_{\vb*{k}\sigma}}.
\end{align*}
- \[
\left.\begin{array}{l}
E^{(1)}_{\vb*{k}\sigma} \\ E^{(2)}_{\vb*{k}\sigma}
\end{array}\right\} = \frac{1}{2}\qty{E_{\vb*{k}} + U + T_0 \mp \sqrt{(E_{\vb*{k}} - U - T_0)^2 + 4U\langle n_{-\sigma}\rangle (E_{\vb*{k}} - T_0)}}.
\]
- \[
\left.\begin{array}{l}
A^{(1)}_{\vb*{k}\sigma} \\ A^{(2)}_{\vb*{k}\sigma}
\end{array}\right\} = \frac{1}{2} \qty{1 \mp \frac{E_{\vb*{k}} - U - T_0 + 2U\langle n_{-\sigma} \rangle}{\sqrt{(E_{\vb*{k}} - U - T_0)^2 + 4U\langle n_{-\sigma} \rangle (E_{\vb*{k}} - T_0)}}}.
\]
- Approximation:
\[
[n_{i,-\sigma}c_{i,\sigma}] = \underbrace{\cdots}_{\text{linear in } c_{i\sigma}} + \cancelto{0}{\sum_{\vb*{l}} T_{i\vb*{l}} (c_{i,-\sigma}^\dagger c_{l,-\sigma} - c^\dagger_{l,-\sigma} c_{i,-\sigma}) c_{i\sigma}}.
\]
- Local density of states:
\[
\color{darkcyan} \rho_\sigma(\omega) = \frac{1}{N} \sum_{\vb*{k}} \qty{A^{(1)}_{\vb*{k}\sigma} \delta[\omega - E^{(1)}_{\vb*{k}\sigma}] + A^{(2)}_{\vb*{k}\sigma}\delta[\omega - E^{(2)}_{\vb*{k}\sigma}]}.
\]
- More accurate approximation:
- Split (insulator):
\[
\frac{\Delta}{U} < \frac{2}{\sqrt{3}}.
\]
- Doesn't split (metal):
\[
\frac{\Delta}{U} > \frac{2}{\sqrt{3}}.
\]
\(U=0\), no correlation,
\[
\rho_\sigma(\omega) = D(\omega).
\]
\(U\ge \Delta\), \(U \gg \abs{E_{\vb*{k}} - T_0}\), strong correlation,
\[\begin{align*}
G^{\sigma}_{\vb*{k}}(\omega) &\approx \qty{\frac{1-\langle n_{-\sigma}\rangle}{\omega - T_0 - (E_{\vb*{k}} - T_0)(1-\langle n_{-\sigma}\rangle)} + \frac{\langle n_{-\sigma}\rangle}{\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle}}. \\
\rho_\sigma(\omega) &= \frac{1}{N} \sum_{\vb*{k}} \begin{cases}
(1-\langle n_{-\sigma}\rangle) \delta[\omega - T_0 - (E_{\vb*{k}} - T_0)(1 - \langle n_{-\sigma}\rangle)] \\
\mbox{} + \langle n_{-\sigma} \rangle \delta[\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle].
\end{cases}
\end{align*}{}\]
Phenomenology of Local Magnetic Moments
- Notation:
- \(d\) denotes an impurity state,
- \(\vb*{k}\) denote a band state,
- \(i\) and \(j\) denote nearest neighbour lattice sites.
- Anderson Hamiltonian:
\[
\color{orange} H^{\mathrm{A}} = \sum_{\vb*{k}\sigma} E_{\vb*{k}} n_{\vb*{k}\sigma} + \sum_\sigma E_d n_{d\sigma} + U n_{d\uparrow} n_{d\downarrow} + \sum_{\vb*{k}\sigma} V_{\vb*{k}d}(a^\dagger_{\vb*{k}\sigma} a_{d\sigma} + a^\dagger_{d\sigma}a_{\vb*{k}\sigma}).
\]
- Electrons in the pure metal:
\[
\sum_{\vb*{k}\sigma} E_{\vb*{k}} n_{\vb*{k}\sigma}.
\]
- In the presence of a magnetic field:
\[
\color{orange}E_{\vb*{k}\sigma} = E_{\vb*{k}} + \sigma \mu_{\mathrm{B}} H.
\]
- Unperturbed impurity atom: correlation not included,
\[
\sum_\sigma E_d n_{d\sigma}.
\]
- In the presence of a magnetic field:
\[
\color{orange}E_{d\sigma} = E_{d} + \sigma \mu_{\mathrm{B}} H.
\]
- Correlation between electrons in the impurity atom:
\[
U n_{d\uparrow} n_{d\downarrow}.
\]
- Coulomb energy between electrons: \[
U = \bra{dd} V_{\mathrm{ee}} \ket{dd} = \int \dd{\vb*{r}_1} \dd{\vb*{r}_2} \abs{\phi_d(\vb*{r}_1)}^2 \frac{e^2}{\abs{\vb*{r}_1 - \vb*{r}_2}} \abs{\phi_d(\vb*{r}_2)}^2.
\]
- s-d mixture:
\[
\sum_{\vb*{k}\sigma} V_{\vb*{k}d}(a^\dagger_{\vb*{k}\sigma} a_{d\sigma} + a^\dagger_{d\sigma}a_{\vb*{k}\sigma}).
\]
- \(V_{\vb*{k}d}\) describes the magnitude of mixing between s and d states:
\[
V_{\vb*{k}d} = \bra{\phi_d(\vb*{r})} H_0 \ket{\phi_{\vb*{k}}(\vb*{r})},
\]
where \(H_0\) is the single-electron Hamiltonian.
- Remarks:
- The interaction of localized electrons with eletrons on other sites are not included. This term may be described by integrals of the form
\[
\bra{i}\bra{j} V_{\mathrm{ee}} \ket{k}\ket{l},
\]
which is one order of magnitude smaller than \(U\).
- The Coulomb repulsion \(U\) favors the formation of local magnetic moments because it inhibits double occupation.
- Phillips: \(E\) is the energy of the impurity site relative to the Fermi level.
- The Anderson model is governed by serveral parameters: \(\epsilon_{\vb*{k}}\), \(U\), and \(\Gamma\).
- \(\Gamma\) characterize the transition rate of \(\vb*{k} \rightarrow d\) and is refered to as the hybridization energy:
\[
\frac{1}{\tau} = 2\pi \frac{\abs{V_{\vb*{k}d}}^2 N(\epsilon_d)}{\hbar} = \frac{2\Gamma}{\hbar}.
\]
- If \(U\gg \epsilon_d \gg \Gamma\), then the system supports local moment formation.
- If \(U\gg \Gamma \gg \epsilon_d\), then the occupation state undergoes rapid flucutations and the system is not magnetic.
- If \(\Gamma \gg U\), then the impurity level is broadened and is occupied with spin up and spin down electrons with equal probability, resulting in a non-magnetic state. This state is termed as localized spin fluctuation.
Density of States of Impurity
- For local moment to form we demand
\[
\langle n_{d\sigma} \rangle \neq \langle n_{d,-\sigma} \rangle.
\]
- Green's function:
- \[
\color{darkcyan} \langle\langle a_{\vb*{k}\sigma} | a^\dagger_{\vb*{k}'\sigma}\rangle\rangle = \frac{\delta_{\vb*{k}\vb*{k}'}}{\omega - E_{\vb*{k}\sigma}} + \frac{V_{\vb*{k}d}V_{\vb*{k}'d}}{(\omega - E_{\vb*{k}\sigma})(\omega - E_{\vb*{k}'\sigma})}\langle\langle a_{d \sigma} | a^\dagger_{\sigma}\rangle\rangle.
\]
- \[
\color{darkcyan} \langle\langle a_{d\sigma} | a_{d\sigma}^\dagger \rangle\rangle(\omega \pm i\epsilon) = \frac{1}{\omega - E_{d\sigma} - U\langle n_{d,-\sigma}\rangle \pm i\Gamma}.
\]
- \[
i\Gamma = \sum_{\vb*{k}} \frac{\abs{V_{\vb*{k}\sigma}}^2}{\omega - E_{\vb*{k}\sigma} + i\epsilon} \approx -i\pi \abs{V_{\vb*{k}d}}^2 \rho^{(0)}(\omega).
\]
- Approximation: Hatree-Fock approximation,
\begin{gather*}
\langle\langle n_{d,-\sigma} a_{d\sigma} | a^\dagger_{d\sigma}\rangle\rangle(\omega) \approx \langle n_{d,-\sigma}\rangle \langle\langle a_{d\sigma} | a^\dagger_{d\sigma}\rangle\rangle(\omega) \\
\Updownarrow \\
\frac{U}{2}\sum_\sigma n_{d\sigma}n_{d,-\sigma} \approx U\sum_\sigma \langle n_{d,-\sigma}\rangle n_{d\sigma} - U\langle n_{d\uparrow} \rangle\langle n_{d\downarrow} \rangle.
\end{gather*}
- The Coulomb interaction shift the energy level by a real number.
- The mixing between \(d\) and \(\vb*{k}\) moves the poles of the Green's function off the real axis and therefore broadens the energy level.
- The d states do decay.
\[
G_{\mathrm{r}}(t) = -i\Theta(t) e^{-i(E_{d\sigma} + U\langle n_{d,-\sigma} \rangle)} e^{-\Gamma t}.
\]
- The impurity density of states:
\[
\color{darkcyan} \rho_{d\sigma}(\omega) \sim \frac{1}{\pi} \frac{\Gamma}{(\omega - E_{d\sigma} - U\langle n_{d,-\sigma} \rangle)^2 + \Gamma^2}.
\]
- \(\Gamma \sim \mathrm{FWHM}\).
- Susceptibility:
\[
\chi = \chi_{\mathrm{P}} + \underbrace{\lim_{H\rightarrow 0} \frac{\mu\sigma_{\mathrm{B}}}{H} \langle a^\dagger_{d\sigma} a_{d\sigma}\rangle}_{\chi_{\mathrm{I}}}.
\]
- \(\chi_{\mathrm{P}}\) is the Pauli susceptibility.
- Population of d state:
\[
\langle a^\dagger_{d\sigma} a_{d\sigma}\rangle = \int_{-\infty}^\infty \dd{\omega} f(\omega) \rho_{d\omega}(\omega).
\]
- \(T=0\):
\[
\color{darkcyan} \langle n_{d\sigma} \rangle = \frac{1}{\pi} \arccot \qty[\frac{E_d - E_{\mathrm{F}} + U\langle n_{d,-\sigma} \rangle + (\operatorname{sign} \sigma)\mu_{\mathrm{B}}H}{\Gamma}].
\]
- For \(U\rho_{d\sigma}(E_{\mathrm{F}}) < 1\):
\[
\chi_{\mathrm{I}} = 2\mu_{\mathrm{B}}^2 \frac{1}{\displaystyle \frac{\pi \Gamma}{\sin^2 n_0 \pi} - U}.
\]
- Assumption: nonmagnetic solution under \(H=0\)
\[
\langle n_{d\uparrow} \rangle = \langle n_{d\downarrow} \rangle.
\]
- \(\Gamma\) to DoS:
\[
\rho_{d\sigma}(E_{\mathrm{F}}) = \frac{1}{\pi} \frac{\sin^2 \pi n_0}{\Gamma}.
\]
- For \(U\rho_{d\sigma}(E_{\mathrm{F}}) > 1\): nonmagnetic solution unstable.
- Magnetic ground state exists only if
- \[\color{red}U\rho_{d\sigma}(E_{\mathrm{F}}) > 1.\]
- \[\color{red}
0 < \frac{E_{\mathrm{F}} - E_{\mathrm{d}}}{U} < 1.
\]
Linear Response
Monochromatic Perturbation
- Perturbation:
\[
H_{\mathrm{e}}(t) = B e^{-i\omega t + \eta t}.
\]
- Heisenberg picture:
\[
H'_{\mathrm{e}}(t) = B(t) e^{-i\omega t + \eta t},
\]
where
\[
B(t) = e^{iHt/\hbar} B e^{-iHt/\hbar}.
\]
- Green's function:
\[
G_{\mathrm{r}} = -\frac{i}{\hbar}\Theta(t-t')\langle [A(t), B(t')] \rangle.
\]
- Spatial formulation:
\[
\Delta A = \int_{-\infty}^{\infty} G_{\mathrm{r}}(t-t') e^{-i\omega t + \eta t'} \dd{t'}.
\]
- Frequency domain formulation:
\[
\Delta A = G_{\mathrm{r}}(\omega) e^{-i\omega t + \eta t}.
\]
where
\[
G_{\mathrm{r}}(\omega) = \int_{-\infty}^{\infty} G_{\mathrm{r}}(t) e^{i\omega t - \eta t} \dd{t}.
\]
Example: Conductivity
- Current density operator:
\[
j_\alpha(\vb*{r}) = \frac{1}{2m} \sum_i e_i\qty{\vb*{p}_{i\alpha}\delta(\vb*{r} - \vb*{r}_i) + \delta(\vb*{r} - \vb*{r}_i) p_{i\alpha}},
\]
or
\[
j_\alpha(\vb*{q}) = \frac{1}{2m} \sum_i e_i\qty[\vb*{p}_{i\alpha} e^{i\vb*{q}\cdot \vb*{r}_i} + e^{i\vb*{q}\cdot \vb*{r}_i}\vb*{p}_{i\alpha}].
\]
DC Conductivity
- DC conductivity:
\[
\Re(\sigma_{\alpha\beta}) = \frac{\pi \beta}{v} \sum_{nm} e^{-\beta E_n} \bra{n} j_\beta \ket{m} \bra{m} j_\alpha \ket{n} \delta(E_n - E_m).
\]