The thermodynamics of quantum systems and generalizations of Zamolodchikov's C-theorem

In this paper we examine the behavior in temperature of the free energy on quantum systems in an arbitrary number of dimensions. We define from the free energy a function C of the coupling constants and the temperature, which in the regimes where quantum fluctuations dominate, is a monotonically increasing function of the temperature. We show that at very low temperatures the system is controlled by the zero-temperature infrared stable fixed point while at intermediate temperatures the behavior is that of the unstable fixed point. The C-function displays this crossover explicitly. This behavior is reminiscent of Zamolodchikov's C-theorem of field theories in 1 + 1 dimensions. Our results are obtained through a thermodynamic renormalization-group approach. We find restrictions on the behavior of the entropy of the system for a C-theorem-type behavior to hold. We illustrate our ideas in the context of a free massive scalar field theory, the one-dimensional quantum Ising model and the quantum non-linear sigma model in two space dimensions. In regimes in which the classical fluctuations are important the monotonie behavior is absent.