The Thermodynamics of Quantum Systems and Generalizations of Zamolodchikov's C-theorem

TL;DR

This work defines a finite-temperature generalization of Zamolodchikov's $C$-theorem by introducing a thermodynamic $C$-function, $C(eta,g,a)$, derived from the free energy that is monotone with temperature in regimes where quantum fluctuations dominate. Using a thermodynamic renormalization-group framework with variables like $t=(eta\, ext{Λ})^{-1}$ and $ ilde{eta}(ullet)$, the authors show that monotonicity can be viewed as a consequence of a temperature-driven RG flow, though it cannot be derived from thermodynamics alone and depends on quantum versus classical crossovers. They illustrate the approach with a free massive scalar field, the 1+1D Ising model, and the 2+1D non-linear sigma model, finding monotonic behavior in the quantum regime and non-monotonic behavior in classical regimes, especially near phase transitions. The results highlight a quantum-classical crossover as the mechanism behind (non)monotonicity and suggest that a universal higher-dimensional $C$-theorem is unlikely, while providing an experimentally accessible thermodynamic probe via the specific heat.

Abstract

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 monotonic behavior is absent.