Variational approximation

Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical physics and data science.

Suppose we have some complicated distribution $P(x)$, from which we cannot infer the partition function and any other expectation values directly, and we approximate it by some parametric family of distributions $Q(x, theta)$, where $theta$ -denotes the set of tunable parameters.

The objective function, used to measure, how well this function appoximates the initital distribution is variational free energy, defined by:
$$
beta tilde{F} (theta) = sum_x Q(x, theta) ln frac{Q(x, theta)}{exp[-beta E(x, J)]}
$$
where $J$ denotes some set of couplings. The genuine free energy is :
$$
beta F = -ln Z(beta, J)
$$
There holds following identity:
$$
beta tilde{F} (theta) – beta F = D_{KL} (Q Vert P) qquad D_{KL} (Q Vert P)= sum_x Q(x, theta) ln frac{Q(x, theta)}{P(x)}
$$
Where $D_{KL}$ is Kullback-Leibler divergence, which is always positive. This technique is actually a mean field theory, where in some sense we have integrated out the fluctuations.

Renormalization group, and $c$-theorem

On the other hand, from the point of view of conformal field theory we have $c$-theorems in various dimensions about the existence of some function, that monotenously decreases along the RG-flow. In a nutshell, this means, that the number of degrees of freedom in the $UV$-theory exceeds the number of $IR$-theory, which is clear from the physical intuition, because at high energies all degrees of freedom are excited, whereas, in the $IR$ – massive degrees of freedom will be inactive.

As far as I understand, by far there is rigorous proof of the $c$-theorem in the 2D by Zamolodchikov. There is also some evidence and list of conjectures for the $2+1$ dimensional case, and the $a$-theorem in $3+1$.

In this video http://pirsa.org/displayFlash.php?id=11110115 Zohar Komargodski speaks about the RG flow properties in various dimensions, and he has mentioned, that the one can suppose, that the free energy $F$ could be naively thought of a function, that decreases along the RG-flow, but, however, there are counterexamples.

The Question

What are the concrete examples of theories, for which the free energy doesn’t decrease monotonically along the RG-flows, and how to connect this with the fact that in the formulas above for approximation of initial distribution $P(x)$ by a simplier one, one can take as $Q(x)$ – distribution of the theory, obtained from the high-energy, by integrating out massive degrees of freedom?

Thanks for any comments and clarifications!