Thermodynamics

Finite-temperature Green’s functions are useful because the same self-consistent calculation that gives the density matrix, self-energy, and spectra also gives access to electronic thermodynamics. This is especially important for metals, small-gap materials, and competing phases where the electronic entropy or Helmholtz free energy can decide which solution is thermodynamically stable.¹

The calculation is performed in the grand canonical ensemble, with inverse temperature $\beta=1/(k_B T)$ and chemical potential $\mu$. The basic object is the Matsubara Green’s function $G(i\omega_n)$, where $\omega_n=(2n+1)\pi/\beta$ for fermions. The density matrix is obtained from the equal-time limit of $G$, and the particle number is

where $S$ is the one-particle overlap matrix and $\mathrm{Tr}_{1}$ denotes the trace over one-particle indices, including spin, orbital, and crystal momentum when present.

Dyson equation and self-energy split

In an atomic-orbital or Bloch-orbital basis, the interacting Green’s function is obtained from the Dyson equation

It is useful to separate the self-energy into a static Hartree-Fock part and a dynamic correlation part,

and define the Fock matrix $F=H_0+\Sigma^\infty$. Then

This split matters for thermodynamics because the logarithm in the grand potential converges slowly if it is evaluated directly from $G^{-1}$. Using

one evaluates the logarithmic contribution as

The first term is a finite-temperature one-particle expression, while the second contains the decaying correlation correction. The self-energy in the second logarithm is the dynamic part $\Sigma^c$, not the full $\Sigma$.

Luttinger-Ward functional

For a conserving, self-consistent approximation, the grand potential is written in terms of the Luttinger-Ward $\Phi$ functional as²³⁴

Here $\mathrm{Tr}$ denotes the full Matsubara and one-particle trace in the convention used for $\Phi$ and $\Sigma G$. Some papers absorb the factor $1/\beta$ into the definition of $\mathrm{Tr}$; the formula above keeps it explicit.

The functional $\Phi[G]$ is the sum of closed, linked skeleton diagrams. Its key property is

up to the same trace convention. As a consequence, $\Omega[G]$ is stationary at a self-consistent solution of the Dyson equation. This stationarity is the reason $\Phi$-derivable approximations such as self-consistent GF2 and self-consistent GW are conserving and thermodynamically consistent.⁵

For a two-body interaction, the skeleton expansion may be written as

This identity is especially convenient in perturbative approximations because the same self-energy diagrams used in the Dyson equation determine the thermodynamic functional.

Thermodynamic quantities

Once $\Omega$ is known, the grand partition function is

The Helmholtz free energy, internal energy, entropy, and specific heat are related by

so that

Equivalently, at fixed external parameters one may use derivative definitions such as

For fixed particle number, the electronic heat capacity is

The internal energy is commonly evaluated from the finite-temperature Galitskii-Migdal expression. With the static self-energy included in $F$ and the dynamic part denoted by $\Sigma^c$, the molecular form used in the GF2 thermodynamics paper is¹⁶

For periodic systems, the one-particle trace is supplemented by the Brillouin-zone average. In a fully self-consistent $\Phi$-derivable calculation, evaluating $E$ by this formula, by thermodynamic integration, or by derivatives of $\Omega$ gives the same result within numerical accuracy.⁴

GF2 grand potential

In self-consistent second-order perturbation theory (GF2), the self-energy is

where $\Sigma^{(2)}$ is built from the fully interacting $G$. The corresponding truncated $\Phi$ functional is

The self-energy trace separates in the same way,

Combining these identities with the logarithmic split gives

This form is the practical GF2 expression: the static Hartree-Fock contribution, the dynamic second-order contribution, and the two logarithmic terms are evaluated from the converged Green’s function and self-energy.

GW grand potential

Self-consistent GW is also $\Phi$-derivable. Its self-energy is split as

where $\widetilde{\Sigma}^{GW}$ is generated by the screened interaction $W$ or, equivalently, by the RPA bubble series. In the Gaussian-density-fitting formulation used in the finite-temperature scGW paper, the dynamic part of the GW $\Phi$ functional can be written in terms of the auxiliary polarization matrix $\widetilde{P}_0^{q}(i\Omega_n)$ as⁴

The logarithm is a matrix logarithm before the auxiliary-space trace is taken. Expanding it gives the bubble series with the first-order term removed:

The full GW functional is then

with the sign following the scGW convention used for the periodic implementation. Inserting $\Phi_{GW}$ and $\Sigma^{GW}$ into the same Luttinger-Ward expression for $\Omega[G]$ defines the GW grand potential. The thermodynamic quantities $A$, $E$, $S$, and $C_V$ are then obtained from the identities above.

The practical lesson is the same for GF2 and GW: thermodynamic quantities should be evaluated from the self-consistent Green’s function and the $\Phi$ functional belonging to the same approximation. Mixing a self-energy from one approximation with a thermodynamic functional from another generally destroys the stationarity and consistency that make the calculation useful.