What is free energy in the context of a quantum field theory? I was reading the papers Large $N$ behavior of mass deformed ABJM theory and New 3D ${\cal N}=2$ SCFT's with $N^{3/2}$ scaling. These papers talk about the free energy in the context of quantum field theory. I have an idea of what thermodynamic free energy is (related to the work done by the system). But what is free energy in the context of a quantum field theory?
 A: The definition of the free energy in QFT is the same as in many-body QM
$$
\exp(-\beta F) = {\rm Tr}\left[\exp(-\beta H)\right]
$$
where $H$ is the Hamiltonian of the system and $\beta=1/(k_BT)$. Note that this definition implies the standard thermodynamic identities. If you have a box of this stuff (described by $H$) coupled to a heat bath, then the isothermal change of $F$ is $dF=-pdV$, etc. There is a euclidean path integral representation of $Z={\rm Tr}[\exp(-\beta H)]$
$$
Z = \int_{S_1\times R^3}{\cal D\phi} \;\exp(-S_E)
$$
where the size of the circle $S_1$ is equal to $\beta$, ${\cal D}\phi$ is the path integral measure in the QFT, and $S_E$ is the euclidean action. We have to impose periodic/anti-periodic boundary conditions for bosons/fermions along the $S_1$.  
A: The theories on question are local QFTs in 3d. You can put such a theory on any compact 3-manifold $M$, and then you can compute observables such as the partition function $Z[M]$ or correlation functions $\langle O_1 \dotsm O_n \rangle_M$. All these observables will in any case depend on the couplings of the original theory and any parameters you use to define $M$, for instance its size $R$. In the papers in question the three-sphere $M=S^3$ is used. We always have in mind that we're tuning to a critical point, such that the couplings of the original theory are completely fixed.
Typically this procedure is a little bit ambiguous, in the sense that in the Lagrangian you can turn on new couplings:
$$\mathcal{L} \mapsto \mathcal{L} + \text{cosmological constant} + \text{Ricci scalar} + \ldots$$
that don't exist in flat space. If you measure the partition function for $M=S^3$, you find that 
$$\ln Z[M] = a (\Lambda R)^3 + b \Lambda R - F + \ldots$$
for some dimensionless coefficients $a,b,f$. (Here $\Lambda$ is the UV cutoff, and all couplings are measured in units of $\Lambda$.) We can set $a,b = 0$ by tuning the cosmological constant and the Ricci scalar. Once you're at the critical point, the partition function $Z[M]$ is therefore a pure number, $e^{-F}$, and often this $F$ is called the free energy of a 3d CFT.
A: You are talking about Self Consistent Field Theories.
It is a key tool for describing the phase behavior of block polymers.
The lowest free energy state is the thermodynamic equilibrium.
http://pscf.cems.umn.edu/scft/background
In a thermodynamic system, the internal energy (U) is a function of entropy (S) and volume. Now since entropy is not easy to measure, we need to use temperature (T).
F = U - T x S
https://books.google.hu/books?id=nnuW_kVJ500C&pg=PA262&lpg=PA262&dq=free+energy+in+the+context+of+a+quantum+field+theory?+scft&source=bl&ots=vrzrkFVQ5Y&sig=ACfU3U28DZt1k6j8pEsupvE-iF5IpQnI8g&hl=en&sa=X&ved=2ahUKEwi-iJP5-M3jAhWFUxUIHQRkAWwQ6AEwBnoECAkQAQ#v=onepage&q&f=false


In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature and volume (isothermal, isochoric). The negative of the change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which volume is held constant. If the volume were not held constant, part of this work would be performed as boundary work. This makes the Helmholtz energy useful for systems held at constant volume. Furthermore, at constant temperature, the Helmholtz energy is minimized at equilibrium.


https://en.wikipedia.org/wiki/Helmholtz_free_energy
So basically, at constant temperature, free energy is the minimized energy at equilibrium, and at constant volume, it is the maximum amount of work.
