HF_notes

Hartree-Fock in Real-Space

We start with the most general Hamiltonian defined on a tight-binding basis:

\begin{matrix} (1) & \begin{matrix} \hat{H} = \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} t_{r, R, α, β, s, s^{'}} c_{r + R, α, s}^{†} c_{r, β, s^{'}}^{} + \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} V_{r, R, α, β, s, s^{'}} {\hat{n}}_{r + R, α, s} {\hat{n}}_{r, β, s^{'}} \end{matrix} \end{matrix}

$\alpha, \beta$ $s, s'$ $t$ :

\begin{matrix} (2) & \begin{matrix} {[\sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} t_{r, R, α, β, s, s^{'}} c_{r + R, α, s}^{†} c_{r, β, s^{'}}^{}]}^{†} = \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} t_{r, R, α, β, s, s^{'}}^{*} c_{r, β, s^{'}}^{†} c_{r + R, α, s}^{} \Rightarrow t_{r, - R, α, β, s, s^{'}}^{*} = t_{r, R, β, α, s^{'}, s}^{} \end{matrix} \end{matrix}

$V$ :

\begin{matrix} (3) & \begin{matrix} {[\sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} V_{r, R, α, β, s, s^{'}} {\hat{n}}_{r + R, α, s} {\hat{n}}_{r, β, s^{'}}]}^{†} & = \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} V_{r, R, α, β, s, s^{'}}^{*} {\hat{n}}_{r, β, s^{'}} {\hat{n}}_{r + R, α, s} \\ = \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} V_{r, R, α, β, s, s^{'}}^{*} {\hat{n}}_{r + R, α, s} {\hat{n}}_{r, β, s^{'}} \\ \Rightarrow V_{r, R, α, β, s, s^{'}}^{*} = V_{r, R, α, β, s, s^{'}} \end{matrix} \end{matrix}

We want to do Hartree-Fock constrained to some bands. The Hartree-Fock decomposition of the interaction term is:

\begin{aligned} {\hat{n}}_{r + R, α, s} {\hat{n}}_{r, β, s^{'}} & = c_{r + R, α, s}^{†} c_{r + R, α, s}^{} c_{r, β, s^{'}}^{†} c_{r, β, s^{'}}^{} \\ \approx ⟨ c_{r + R, α, s}^{†} c_{r + R, α, s}^{} ⟩ c_{r, β, s^{'}}^{†} c_{r, β, s^{'}}^{} + ⟨ c_{r, β, s^{'}}^{†} c_{r, β, s^{'}}^{} ⟩ c_{r + R, α, s}^{†} c_{r + R, α, s}^{} & Hartree \\ - ⟨ c_{r + R, α, s}^{†} c_{r, β, s^{'}}^{} ⟩ c_{r, β, s^{'}}^{†} c_{r + R, α, s}^{} - ⟨ c_{r, β, s^{'}}^{†} c_{r + R, α, s}^{} ⟩ c_{r + R, α, s}^{†} c_{r, β, s^{'}}^{} & Fock \end{aligned}

The correlation functions can be obtained by the following calculation:

\begin{matrix} (4) & \begin{matrix} c^{†} H c = c^{†} U Λ U^{†} c = γ^{†} Λ γ \\ \Rightarrow ⟨ (c^{†})^{T} (c^{})^{T} ⟩ = ⟨ (γ^{†} U^{†})^{T} (U γ)^{T} ⟩ = ⟨ U^{*} (γ^{†})^{T} γ^{T} U^{T} ⟩ = U^{*} ⟨ (γ^{†})^{T} γ^{T} ⟩ U^{T} = U^{*} diag (n_{F} (λ_{i})) U^{T} \end{matrix} \end{matrix}

In the end we get a Hamiltonian

\begin{matrix} (5) & \begin{matrix} {\hat{H}}_{H F} = \sum_{\begin{matrix} r, R, \\ α, β, s, s^{'} \end{matrix}} h_{R, α, β, s, s^{'}} c_{r + R, α, s}^{†} c_{r, β, s^{'}}^{} \end{matrix} \end{matrix}

Real-Space Band Projection Operator

$\alpha$ $\beta$ be composite indices. In terms of bras and kets, the Hamiltonian is

\begin{matrix} (6) & \hat{H} = \sum_{k, α, β} h_{α β} (k) | u_{α} (k) ⟩ ⟨ u_{β} (k) | = \sum_{k, n} ε_{n} (k) | u_{n} (k) ⟩ ⟨ u_{n} (k) | \end{matrix}

$\ket{u_n(k)} = \sum_{\alpha} U_{n\alpha} \ket{u_{\alpha}(k)} \Rightarrow \ket{u_{\alpha}(k)} = \sum_{n} U^*_{\alpha n} \ket{u_n(k)}$ . The momentum eigenstates satisfy

\begin{matrix} (7) & ⟨ u_{n} (k) | u_{m} (k^{'}) ⟩ = δ_{n m} δ (k - k^{'}) \end{matrix}

$n$ th band. The projector on to this band in momentum space is

\begin{matrix} (8) & {\hat{P}}_{n} = \sum_{k} | u_{n} (k) ⟩ ⟨ u_{n} (k) | \end{matrix}

If we have some translationally invariant operator it has a representation in momentum space and original orbital basis as:

\begin{aligned} {\hat{H}}_{I} & = \sum_{k, α, β} α_{α β} (k) | u_{α} (k) ⟩ ⟨ u_{β} (k) | \\ = \sum_{k, n, m} (\sum_{α, β} U_{α n}^{*} α_{α β} (k) U_{β m}) | u_{n} (k) ⟩ ⟨ u_{m} (k) | \\ = \sum_{k, n, m} β_{n m} (k) | u_{n} (k) ⟩ ⟨ u_{m} (k) | \end{aligned}

$p$ th band is

\begin{aligned} {\hat{P}}_{p} {\hat{H}}_{I} {\hat{P}}_{p} & = \sum_{n m} \sum_{k, k^{'}, k^{″}} | u_{p} (k) ⟩ ⟨ u_{p} (k) | β_{n m} (k^{'}) | u_{n} (k^{'}) ⟩ ⟨ u_{m} (k^{'}) | | u_{p} (k^{″}) ⟩ ⟨ u_{p} (k^{″}) | \\ = \sum_{n m} \sum_{k, k^{'}, k^{″}} β_{n m} (k^{'}) | u_{p} (k) ⟩ δ_{p n} δ (k^{'} - k) δ_{m p} δ (k^{″} - k^{'}) ⟨ u_{p} (k^{″}) | \\ = \sum_{k} β_{p p} (k) | u_{p} (k) ⟩ ⟨ u_{p} (k) | \\ = \sum_{k} (\sum_{α, β} U_{α p}^{*} α_{α β} (k) U_{β p}) | u_{p} (k) ⟩ ⟨ u_{p} (k) | \end{aligned}

On the other hand the projector has the representation in real space

\begin{aligned} {\hat{P}}_{n} = \sum_{k} | u_{n} (k) ⟩ ⟨ u_{n} (k) | & = \sum_{k, α, β} U_{n α} U_{n β}^{*} | u_{α} (k) ⟩ ⟨ u_{β} (k) | \\ = \sum_{α β} \sum_{R, R^{'}} (\sum_{k} U_{n α} (k) U_{n β}^{*} (k) e^{i k (R - R^{'})}) | u_{α} (R) ⟩ ⟨ u_{β} (R^{'}) | \end{aligned}

Conservation of Time Reversal Symmetry

Now we will be explicit about the distinction between orbital and spin indices. In momentum space, the constraint on a TR symmetric hamiltonian is

\begin{matrix} (9) & ⟨ u_{α, s} (k) | \hat{H} | u_{β, s^{'}} (k) ⟩ = h_{α β s s^{'}} (k) = h_{α β - s - s^{'}}^{*} (- k) = (⟨ u_{α, - s} (- k) | \hat{H} | u_{β, - s^{'}} (- k) ⟩)^{*} \end{matrix}

$s, s'$ $\alpha, \beta$ are orbital indices.

\begin{matrix} (10) & \begin{matrix} Θ \hat{H} (k) Θ^{- 1} Θ | u_{α, s} (k) ⟩ = Θ ϵ_{α, s} | u_{α, s} (k) ⟩ \\ \Rightarrow {\hat{H}}^{*} (- k) Θ | u_{α, s} (k) ⟩ = ϵ_{α, s} Θ | u_{α, s} (k) ⟩ \end{matrix} \end{matrix}

Hartree-Fock in Momentum Space

$V_{(k\alpha)(k'\beta)q}$ $k$ $k'$ $q$ $\alpha$ $\beta$ . This is not completely general, as it doesn't include spin-flip or orbital transitions. Hartree-Fock procedes as usual:

\begin{aligned} \sum_{k α, k^{'} β} \sum_{q} V_{(k α) (k^{'} β) q} c_{k^{'} - q β}^{†} c_{k^{'} β}^{} c_{k + q α}^{†} c_{k α}^{} & \approx \sum_{k α, k^{'} β} \sum_{q} V_{(k α) (k^{'} β) q} ⟨ c_{k^{'} - q β}^{†} c_{k^{'} β}^{} ⟩ c_{k + q α}^{†} c_{k α}^{} \\ + \sum_{k α, k^{'} β} \sum_{q} V_{(k α) (k^{'} β) q} ⟨ c_{k + q α}^{†} c_{k α}^{} ⟩ c_{k^{'} - q β}^{†} c_{k^{'} β}^{} \\ - \sum_{k α, k^{'} β} \sum_{q} V_{(k α) (k^{'} β) q} ⟨ c_{k^{'} - q β}^{†} c_{k α}^{} ⟩ c_{k + q α}^{†} c_{k^{'} β}^{} \\ - \sum_{k α, k^{'} β} \sum_{q} V_{(k α) (k^{'} β) q} ⟨ c_{k + q α}^{†} c_{k^{'} β}^{} ⟩ c_{k^{'} - q β}^{†} c_{k α}^{} \end{aligned}

We need the creation and annihilation operators not inside averages to have consistent indices. We have:

\begin{array}{r} {\hat{V}}^{H F} = \sum_{k α, k^{'} β} \sum_{q} (V_{(k α) (k^{'} β) q} + V_{(k^{'} β) (k α) - q}) ⟨ c_{k^{'} - q β}^{†} c_{k^{'} β}^{} ⟩ c_{k + q α}^{†} c_{k α}^{} \\ - \sum_{k α, k^{'} β} \sum_{q} (V_{(k^{'} β) (k α) q} + V_{(k α) (k^{'} β) - q}) ⟨ c_{k - q α}^{†} c_{k^{'} β}^{} ⟩ c_{k^{'} + q β}^{†} c_{k α}^{} \end{array}

The Hartree term is then

\begin{matrix} (11) & V_{q^{+} (k) α, k α}^{H} = \sum_{k^{'} β} (V_{(k α) (k^{'} β) q} + V_{(k^{'} β) (k α) - q}) C_{q^{-} (k^{'}) β, k^{'} β} \end{matrix}

$q^{\pm}(k) = k\pm q$ . The Fock term is

\begin{matrix} (12) & V_{q^{+} (k) β, k α}^{F} = \sum_{k^{'} β} (V_{(k α) (k^{'} β) - q} + V_{(k^{'} β) (k α) q}) C_{q^{-} (k^{'}) α, k^{'} β} \end{matrix}

The Fourier transform of the Hubbard interaction is

\begin{aligned} U \sum_{R} {\hat{n}}_{R ↓} {\hat{n}}_{R ↑} & = U \sum_{R} c_{R ↓}^{†} c_{R ↓} c_{R ↑}^{†} c_{R ↑} \\ = U \sum_{k, l, m, n} \sum_{R} e^{i (- k + l - m + n) R} c_{k ↓}^{†} c_{l ↓}^{} c_{m ↑}^{†} c_{n ↑}^{} \\ = U \sum_{k, l, m, n} δ (- k + l - m + n) c_{k ↓}^{†} c_{l ↓}^{} c_{m ↑}^{†} c_{n ↑}^{} \\ = U \sum_{k, k^{'}, q} c_{k^{'} - q ↓}^{†} c_{k^{'} ↓}^{} c_{k + q ↑}^{†} c_{k ↑}^{} \end{aligned}

$V_{(k\alpha)(k'\beta)q} = U \delta_{\alpha\beta}$ .

The Random Phase Approximation

We want to compute the susceptibilty in the random phase approximation. The bare susceptibility (Lindhard function) is

\begin{matrix} (16) & χ^{0} (q, i q_{n}) = \sum = \frac{1}{V} \sum_{k α} \frac{n_{F} (ε_{α} (k)) - n_{F} (ε_{α} (k + q))}{i q_{n} + ε_{α} (k) - ε_{α} (k + q)} \end{matrix}

The full correlation function is given by the Dyson equation

\begin{matrix} (17) & χ (q, i q_{n}) = - \frac{χ_{0} (q, i q_{n})}{1 - W (q) χ_{0} (q, i q_{n})} \end{matrix}

$W(q)$ is the interaction.