Stationary RG Flow and Thermodynamics of Conformal Field Theories ∗

We argue that one can understand the relationship between critical free-energy densities of theories connected via non-perturbative renormalization group flows, in terms of the the number of fields coupled to the class of different theories which flow into the same stationary renormalization group trajectory. As an explicit example, we study the free-energy density of bosonic and fermionic theories possessing strongly coupled critical points in D = 3. We construct a stationary trajectory which interpolates between the free massless theory of N scalars and a class of interacting theories including both bosons and fermions. The free-energy along this trajectory remains constant, but the degrees of freedom coupled to the underlying theories which flow into it are different. The difference in the degrees of freedom underlying two distinct points of the stationary trajectory, provides a natural explanation for the rational relationship between the critical free-energy densities of the free massless theory of N scalars and the O(N) vector model.