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

General Formulation

• 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, \[\begin{align*} G^{\sigma}_{\vb*{k}}(\omega) &\approx \qty{\frac{1-\langle n_{-\sigma}\rangle}{\omega - T_0 - (E_{\vb*{k}} - T_0)(1-\langle n_{-\sigma}\rangle)} + \frac{\langle n_{-\sigma}\rangle}{\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle}}. \\ \rho_\sigma(\omega) &= \frac{1}{N} \sum_{\vb*{k}} \begin{cases} (1-\langle n_{-\sigma}\rangle) \delta[\omega - T_0 - (E_{\vb*{k}} - T_0)(1 - \langle n_{-\sigma}\rangle)] \\ \mbox{} + \langle n_{-\sigma} \rangle \delta[\omega - T_0 - U - (E_{\vb*{k}} - T_0)\langle n_{-\sigma} \rangle]. \end{cases} \end{align*}{}\]

Local Magnetic Moments in Metals

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: \[ \Re(\sigma_{\alpha\beta}) = \frac{\pi \beta}{v} \sum_{nm} e^{-\beta E_n} \bra{n} j_\beta \ket{m} \bra{m} j_\alpha \ket{n} \delta(E_n - E_m). \]