# 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?

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 $$S_1$$.

• This is not the free energy used by the papers in question. – Hans Moleman Jul 25 at 18:51
• @HansMoleman How do you know? (What other free energy is there?) – Thomas Jul 25 at 20:45

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.

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