# Green function dependence on temperature

Consider the retarded Green function for fermions $$G_{ij}^r(t)=-\frac{i}{\hbar} \theta(t)\langle[c_i(t),c^\dagger_j(0)]_{+}\rangle$$ They can be understood as the $$(i,j)$$ entry of the matrix $$G^r(t)$$. The operators evolve in time as they are in the Heisenberg picture. The operator average means $$\langle A \rangle = \mathrm{Tr} \rho A$$ where the density matrix has temperature dependence. $$\rho = \frac{1}{Z} e^{-\beta H}.$$ In this answer, it is shown that the Fourier transform of $$G^r(t)$$ is $$G^r(E) = (E + i \eta - H)^{-1}.$$ The LHS depends on temperature but the RHS does not?

In the Heisenberg picture, the operators evolve according to the Heisenberg equation $$i\hbar \frac{d}{dt} A = [A, H]_-$$ This governs the time evolution of the annihilation operator \begin{align*} i\hbar \frac{d}{dt} c_j(t) &= [c_j(t), H]_- \\ &= \sum_{mn} H_{mn} [c_j(t), c_m^\dagger(t) c_n(t)]_- \\ &= \sum_{mn} H_{mn} (c_j(t) c_m^\dagger(t) c_n(t) - c_m^\dagger(t) c_n(t) c_j(t)) \\ &= \sum_{mn} H_{mn} ( (-c_m^\dagger(t) c_j(t) + \delta_{mj}) c_n(t) + c_m^\dagger(t) c_j(t) c_n(t)) \\ &= \sum_{mn} H_{mn} \delta_{mj} c_n(t) \\ &= \sum_{n} H_{jn} c_n(t) \end{align*} To considering all $$j$$ at once, define the notation \begin{align*} \vec{c}(t) = \begin{pmatrix}c_1(t) \\ c_2(t) \\ \vdots \\ c_N(t) \end{pmatrix} \end{align*} Then this is a system of coupled differential equations, which can be written in a Schrodinger-equation-like form. \begin{align*} i\hbar \frac{d}{dt} \vec{c}(t) = H \vec{c}(t) \end{align*} The solution can be obtained by the integrating factor method, and is \begin{align*} \vec{c}(t) = e^{-\frac{i}{\hbar} H t} \vec{c}(0) \equiv U(t) \vec{c}(0) \end{align*} Or individually, \begin{align*} c_i(t) = \sum_{q} U_{iq}(t) c_q(0) \end{align*} where $$U_{lq}(t)$$ are the entries of the unitary matrix $$U(t)$$. By taking the hermitian conjugate, we get, similarly, \begin{align*} c^\dagger_j(t) = \sum_{p} U^*_{jp}(t) c^\dagger_p \end{align*} We are now in a position to evaluate the electron Green function. The main thing is the anti-commutator at different times. \begin{align*} [c_i(t), c_j(0)]_+ &= \sum_{q} U_{iq}(t) [c_q(0), c_j(0)]_+ \\ &= \sum_{q} U_{iq}(t) \delta_{qj} \\ &= U_{ij}(t) \end{align*} The crucial fact is that the above anti-commutator is always a multiple of the identity operator. Thus, \begin{align*} G^r_{ij}(t) &= - \frac{i}{\hbar} \theta(t) \langle [c_i(t), c_j(0)^\dagger]_+ \rangle \\ &= - \frac{i}{\hbar} \theta(t) \mathrm{Tr}(\rho U_{ij}(t)) \\ &= - \frac{i}{\hbar} \theta(t) U_{ij}(t) \end{align*} has no dependence on temperature.
• You assumed that $H=\sum_{m,n} H_{mn}c_m^+c_n$, i.e. that electrons are free. It is a key assumption. One could also add that retarded and advanced Green functions $G^R$ and $G^A$ are independent on the temperature for free electrons but not other Green functions $G^<$, $G^>$ and $G^c$. Commented Apr 22, 2022 at 10:11