Green's Function Formalism¶

Definition¶

The imaginary-time single-particle Green's function at finite temperature \(T = 1/(k_B\beta)\) is defined as:

\[ G^{p,q}_{ij\sigma}(\mathbf{R}-\mathbf{R}', \tau - \tau') = -\langle T_\tau\, c^p_{i\sigma}(\mathbf{R}, \tau)\, c^{\dagger\, q}_{j\sigma}(\mathbf{R}', \tau') \rangle \]

where \(T_\tau\) is the imaginary-time ordering operator, \(c^p_{i\sigma}\) annihilates an electron at site \((\mathbf{R}, p)\) with orbital \(i\) and spin \(\sigma\), and \(\langle \cdots \rangle\) denotes the grand-canonical thermal average. The indices \(p, q\) label basis sites and \(i, j\) label orbitals.

(Anti-)Periodicity¶

For \(0 < \tau < \beta\), the Green's function satisfies:

\[ G(-\tau) = \xi\, G(\beta - \tau), \qquad \xi = \begin{cases} -1 & \text{(fermion)} \\ +1 & \text{(boson)} \end{cases} \]

This boundary condition restricts the Matsubara frequencies to odd (fermionic) or even (bosonic) values.

Matsubara Frequency Representation¶

The Fourier transform to Matsubara frequency is:

\[ G^{pq}_{ij\sigma}(\mathbf{k}, i\omega_n) = \frac{1}{N_\mathbf{k}} \int_0^\beta d(\tau-\tau') \sum_{\mathbf{R},\mathbf{R}'} G^{pq}_{ij\sigma}(\mathbf{R}-\mathbf{R}', \tau-\tau')\, e^{i(\mathbf{k}\cdot(\mathbf{R}-\mathbf{R}') - \omega_n \tau)} \]

with Matsubara frequencies:

\[ i\omega_n = \begin{cases} i(2n+1)\pi/\beta & \text{(fermion)} \\ i\,2n\pi/\beta & \text{(boson)} \end{cases} \]

Lehmann (Spectral) Representation¶

The Green's function admits a spectral representation:

\[ G(\tau) = -\int_{-\infty}^{\infty} \frac{e^{-\omega\tau}}{1 + \xi\, e^{-\beta\omega}}\,\rho(\omega)\,d\omega \]

where \(\rho(\omega)\) is the spectral function. In Matsubara frequency this becomes:

\[ G(i\omega_n) = \int_{-\infty}^{\infty} \frac{\rho(\omega)}{i\omega_n - \omega}\,d\omega \]

The spectral function connects directly to experiment: \(A(\omega) = -\frac{1}{\pi}\mathrm{Im}\,G^R(\omega)\), where \(G^R\) is the retarded Green's function obtained by analytic continuation \(i\omega_n \to \omega + i0^+\).

Dyson Equation¶

The interacting Green's function \(G\) is related to the non-interacting \(G_0\) through the Dyson equation:

\[ G^{pq}_{ij\sigma}(\mathbf{k}, i\omega_n) = G^{pq}_{0\,ij\sigma}(\mathbf{k}, i\omega_n) + \sum_{r,s}\sum_{k,l} G^{pr}_{0\,ik\sigma}(\mathbf{k}, i\omega_n)\,\Sigma^{rs}_{kl\sigma}(\mathbf{k}, i\omega_n)\,G^{sq}_{lj\sigma}(\mathbf{k}, i\omega_n) \]

where \(\Sigma\) is the irreducible self-energy. In QAssemble, the self-energy includes Hartree (\(\Sigma^H\)), Fock (\(\Sigma^F\)), and optionally GW correlation (\(\Sigma^{C,GW}\)) contributions depending on the level of theory.

Lattice Fourier Transform¶

For periodic systems, transformations between momentum \(\mathbf{k}\) and real space \(\mathbf{R}\) are:

\[ G^{pq}_{ij\sigma}(\mathbf{k}) = \sum_{\mathbf{R}} G^{pq}_{ij\sigma}(\mathbf{R})\, e^{-i\mathbf{k}\cdot(\mathbf{R}+\tau_p-\tau_q)}, \qquad G^{pq}_{ij\sigma}(\mathbf{R}) = \frac{1}{N_\mathbf{k}} \sum_{\mathbf{k}} G^{pq}_{ij\sigma}(\mathbf{k})\, e^{i\mathbf{k}\cdot(\mathbf{R}+\tau_p-\tau_q)} \]

where \(\tau_p\), \(\tau_q\) are basis vectors. These transforms apply equally to Green's functions, self-energies, and interaction matrices.

QAssemble Implementation¶

Class	Description
`GreenBare`	Non-interacting \(G_0\) from \(H_0\): \(G_0(i\omega_n) = (i\omega_n - H_0)^{-1}\)
`GreenInt`	Interacting \(G\) via Dyson equation; manages \(\mu\) and density \(n = -G(\tau = \beta^-)\)
`FLatDyn`	Base class for dynamic (frequency-dependent) fermionic lattice quantities
`BLatDyn`	Base class for dynamic bosonic lattice quantities

References¶

A. A. Abrikosov, L. P. Gorkov & I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, 1975).
A. L. Fetter & J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover, 2003).
G. D. Mahan, Many-Particle Physics, 3rd ed. (Springer, 2000).