Intuition about non-equilibrium Green functions

Question

In the following paper by Fotso and Freericks, the definitions of the Green functions are given as \begin{align} G^<(t,t') &= i \langle c^\dagger(t') c(t) \rangle \\ G^R(t,t') &= -i \theta(t - t') \langle \{ c(t), c^\dagger(t')\} \rangle. \end{align}

There is the following remark

In particular, the retarded Green’s function is related to the quantum states while the lesser Green’s function is directly related to how those states are occupied by fermions.

I also read that these Green functions are related to fluctuation and dissipation. Can anyone give a brief explanation of the remark, as to why we can interpret the Green functions in this manner?

Answer 1

The Green's functions give information about the degree of correlation in our system through the Fermionic field operators $c, c^{\dagger}$ which create/destroy electrons in the system at a given time.

The lesser Green's function contains information about how the states are occupied because of the ordering of the operators- at equal times we are averaging the operator $c^{\dagger}(t)c(t)= n(t)$, the occupation number as a function of time.

$G^R$, also called the causal Green's function because of the theta function ensuring that time $t$ be greater than $t'$, encodes the density of states. In non-equilibrium we have a weird time-dependent and frequency dependent density of states (for non-equilibium details see the formalism section of my paper here: https://doi.org/10.1103/PhysRevB.104.155104), but consider equilibrium for simplicity. In this case one can show that the interacting density of states (also called the spectral function) $A(\omega)$ is given by \begin{align} A(\omega)= -\frac{1}{\pi}\text{Im}\int_{-\infty}^{\infty} \,dt\ e^{i\omega t}G^R(t). \end{align}

The relation to fluctuation-dissipation is given by a statement of the fluctuation-dissipation theorem that says the ratio of the lesser Green's function to the imaginary part of the causal one (in frequency space) is proportional to the Fermi occupation function, in particular \begin{align} \frac{G^<(\omega)}{\text{Im}G^R(\omega)}=-2if_T(\omega)\end{align} where $f_T(\omega)$ is the Fermi function at temperature $T$.