Thermodynamics

Self-consistent finite $\Phi$-derivable theories allow to get direct access to thermodynamic properties such as entropy, free energy, and specific heat.

The grand potential is defined in terms of Green’s functions, self-energies, and a $\Phi$-functional as ¹ $$ \begin{align*} \Omega &= \frac{1}{\beta} \left\{ \Phi[G] - Tr\{\ln\left[-G^{-1}\right]\} - Tr\{\Sigma G\} \right\}. \end{align*} $$ In practice, the evaluation direct evaluation of the second term in this form is complicated by the slow decay of Green’s functions as a function of frequency. Therefore, by defining $$ \begin{align} G^{-1} &= (i\omega_n + \mu) S - F - \Sigma_c\left[G\right],\\ G_{HF}^{-1} &= (i\omega_n + \mu) S - F ,\\ G_{0}^{-1} &= (i\omega_n + \mu) S - H_{0}, \end{align} $$ in terms of Matsubara frequencies $i\omega_n$, the chemical potential $\mu$, the overlap matrix $S$, the non-interacting $(U=0)$ Hamiltonian $H_0$ and the Fock matrix $F=H_0 + \Sigma^\infty$, we can evaluate the logarithmic term as² $$ \begin{align} Tr\{\ln\left[-G^{-1}\right]\} = Tr\{\ln\left[-G_{HF}^{-1}\right]\} + Tr\{\ln\left[1 - G_{HF} \Sigma\right]\} \end{align} $$ The $\Phi$-functional is expressed as $$ \begin{align} \Phi\left[G\right] = \sum_{n = 1}^{\infty} \Phi^{(n)}\left[G\right] = \sum_{n = 1}^{\infty}\frac{1}{2n} Tr{\Sigma^{(n)}\left[G\right] G}, \end{align} $$ where $\Sigma^{(n)}\left[G\right]$ is the total n-th order self-energy part. Standard textbook relations yield the entropy, specific heat, total energy and free energy as thermodynamic derivatives $$ \begin{align*} &S = -\frac{\partial \Omega}{\partial T},\\ &C_V = T \frac{\partial S}{\partial T}, \\ &E = T^2\frac{\partial ln Z}{\partial T}, \\ &F = \Omega + \mu N \end{align*} $$

Alternatively, the entropy can be evaluated from the Gibbs-Duhem relation $\Omega = E - TS - \mu N$; the specific heat from its definition $C_V = \frac{\partial E}{\partial T}$; and the energy from the Galitskii-Migdal formula.³ When the fully self-consistent conserving theories are applied different approach to evaluate thermodynamic quantities will produce the same result ^{4 5}

Grand potential in the second order approximation

Within the second-order perturbation theory (GF2), the self-energy is approximated as $\Sigma \approx \Sigma^{[2]} = \Sigma^{\infty} + \Sigma^{(2)}$, and using Eq. 5:

$$\Phi \approx \Phi^{[2]} = \Phi^{(1)} + \Phi^{(2)} = \frac{1}{2} Tr\{\Sigma^{\infty}\gamma\} + \frac{1}{4} Tr\{\Sigma^{(2)} G\},$$

where $\gamma = G(\tau=0^-)$ is the density matrix. Using Eqs. 1-3,5 with $\Sigma^{\infty} = F - H_{0}$, such that

$$ Tr\{\Sigma\left[G\right] G\}= Tr\{\Sigma^{\infty}\gamma\} + Tr\{\Sigma^{(2)} G\}. $$

Combining Eq.1-Eq.5 we obtain the second order approximation of the grand potential as $$ \begin{align*} \Omega^{[2]} &= \frac{1}{\beta} \left\{ \Phi^{[2]}[G] - Tr\{\ln\left[-G^{-1}\right]\} - Tr\{\Sigma^{[2]} G\} \right\} =\nonumber\\ &\frac{1}{\beta} \left\{ -\frac{1}{2}Tr\{\Sigma_{\infty}\gamma\} - \frac{3}{4}Tr\{\Sigma^{(2)} G\} - Tr\{\ln\left[-G_{HF}^{-1}\right]\} - Tr\{\ln\left[1 - G_{HF} \Sigma^{(2)}\right]\} \right\} \end{align*} $$

Grand potential in the GW approximation

Similarly to the second-order perturbation theory (GF2), the GW self-energy is approximated as $\Sigma \approx \Sigma^{[GW]} = \Sigma^{\infty} + \Sigma^{(GW)}$, and the GW approximation to the $\Phi$-functional is:

$$ \begin{align} \Phi \approx \Phi^{[GW]} &= -\frac{1}{2}\mathrm{Tr} \{\Sigma^{\infty}\gamma\} - \frac{1}{4}\mathrm{Tr}\{(\boldsymbol{{U\Pi}})^{2}\} + ... \\ &= \Phi^{\infty} + \tilde{\Phi}^{(GW)} \end{align} $$

where $\boldsymbol{U}$ are the bare Coulomb integrals $\boldsymbol{\Pi}$ is the non-interacting polarization function (see scGW for more details). We define the first term in Eq. 7 as the contribution of the static self-energy $\Phi^{\infty}$. We attribute the rest coming from the dynamical $GW$ self-energy diagrams as $\tilde{\Phi}^{(GW)}$ as

$$ \begin{align} \tilde{\Phi}^{(GW)}& = -\frac{1}{2}\sum_{m=2}^{\infty}\left[ \frac{1}{n} \mathrm{Tr}\{(\boldsymbol{{U\Pi}})^{n}\} \right] = \frac{1}{2}\left(\mathrm{Tr}\{(\boldsymbol{{U\Pi}})\} - \sum_{m=1}^{\infty}\left[ \frac{1}{m} \mathrm{Tr}\{(\boldsymbol{{U\Pi}})^{m}\} \right] \right) = \nonumber \\ &\frac{1}{2}\left(\mathrm{Tr}\{(\boldsymbol{{U\Pi}})\} + \ln\left[\mathbf{1} - \mathrm{Tr}\{(\boldsymbol{{U\Pi}})\}\right] \right) \end{align} $$

Combining Eq.1-Eq.5 and Eq.7 we obtain the GW approximation of the grand potential as

$$ \begin{align*} &\Omega^{[GW]} = \frac{1}{\beta} \left\{ \Phi^{[GW]}[G] - Tr\{\ln\left[-G^{-1}\right]\} - Tr\{\Sigma^{[GW]} G\} \right\} =\nonumber\\ \frac{1}{\beta} &\left\{ -\frac{1}{2}Tr\{\Sigma_{\infty}\gamma\} - Tr\{\Sigma^{(GW)} G\} + \tilde{\Phi}^{(GW)} - Tr\{\ln\left[-G_{HF}^{-1}\right]\} - Tr\{\ln\left[1 - G_{HF} \Sigma^{(2)}\right]\} \right\} \end{align*} $$