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

TL;DR

This paper investigates the temperature dependence of free energy in quantum systems across dimensions, introducing a $C$-function that behaves monotonically under quantum fluctuations, akin to Zamolodchikov's $C$-theorem, using a thermodynamic RG approach.

Contribution

It generalizes the $C$-theorem to quantum systems in arbitrary dimensions, establishing conditions for a monotonic $C$-function based on thermodynamic principles.

Findings

The $C$-function is monotonic at high temperatures dominated by quantum fluctuations.

At low temperatures, the system approaches the zero-temperature fixed point.

Classical fluctuation regimes do not exhibit monotonic $C$-behavior.

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.