# How to compute thermodynamic magnitudes with the Green's function?

I'm studying the SYK model and there seems two equivalent approaches for solving it. One is the diagrammatic expansion in the large $$N$$ limit, where we get self-consistent equations (in imaginary time)

\begin{aligned} G(i\omega)=\frac{1}{i\omega-\Sigma(i\omega)} && \Sigma(\tau)=J^2G(\tau)^3 \end{aligned}

The other approach involves solving the path integral in order to obtain the generating functional $$\mathcal{Z}$$. It is my understanding that diagrammatic and path integral approaches should be equal, in the end. However, I only know how to derive thermodynamic magnitudes with the partition function. we can calculate the free energy $$F$$ as an expansion in the euclidean action and once we know $$F$$ it's just using regular statistical mechanics to get the internal energy $$U$$, the specific heat $$c_V$$, etc.

How do we connect Green's functions (in imaginary or real time, I don't know yet) to thermodynamic magnitudes?

In the SYK model one can write down the path integral form of the (disorder averaged) partition function $$\langle Z\rangle_J$$ and using some tricks find an action in terms of two bi-local fields $$\widetilde{G}(\tau,\tau')$$ and $$\widetilde{\Sigma}(\tau,\tau')$$. I think this procedure is best explained in the paper Ads$$_2$$ holography and the SYK model by Gábor Sárosi.
What is so interesting about this form of the action is that in the large $$N$$ limit the theory becomes classical and one can thus use EFT tools. When finding the the extrema of this action using EFT one finds the Schwinger-Dyson equations (self-consistency equations you state). In the saddle point of the action $$\widetilde{G}(\tau,\tau')$$ and $$\widetilde{\Sigma}(\tau,\tau')$$ correspond directly to the two-point function and the self-energy of the model. This connects the Green's functions to the thermodynamics of the model since you can write $$\langle Z\rangle_J=e^{-NI[G,\Sigma]}$$.
Free energy can be computed using perturbative diagrammatic approach by summing over all the connected bubble diagrams in the theory. Basically, this is due to the fact that the path integral, $$\mathcal Z$$, is sum over all the bubble diagrams (connected as well as disconnected), and a simple argument shows that the exponential of the sum over connected diagrams in fact generates all these diagrams (I think this is discussed somewhere in Weinberg's QFT Vol. 2. I am sorry I don't immediately have access to the book to be able to provide the exact reference, I will update the post once I get access to the book).
When you carefully sum over all the diagrams, then you will typically end up with diagrams involving exact propagators (for example $$G(\tau)$$ in SYK model) and the expression will be same as the path integral approach expression that you mentioned in your question. In context of SYK model and the SYK-like tensor models this is discussed in Witten's paper.