Perturbative renormalization and thermodynamics of quantum crystalline membranes

TL;DR

This paper uses perturbative renormalization group techniques to analyze the quantum and thermal properties of free-standing crystalline membranes, revealing logarithmic corrections to elastic properties and anomalous stress-strain relations at zero and finite temperatures.

Contribution

It provides a systematic derivation of logarithmic corrections and scaling laws for quantum crystalline membranes using an effective renormalizable model within the RG framework.

Findings

Logarithmic corrections to bending rigidity and Young's modulus at zero temperature.

Anomalous stress-strain relation replacing Hooke's law under weak tension.

Negative thermal expansion coefficient with logarithmic temperature dependence.

Abstract

We analyze the statistical mechanics of a free-standing quantum crystalline membrane within the framework of a systematic perturbative renormalization group (RG). A power-counting analysis shows that the leading singularities of correlation functions can be analyzed within an effective renormalizable model in which the kinetic energy of in-plane phonons and subleading geometrical nonlinearities in the expansion of the strain tensor are neglected. For membranes at zero temperature, governed by zero-point motion, the RG equations of the effective model provide a systematic derivation of logarithmic corrections to the bending rigidity and to the elastic Young modulus derived in earlier investigations. In the limit of a weakly applied external tension, the stress-strain relation at $T = 0$ is anomalous: the linear Hooke's law is replaced with a singular law exhibiting logarithmic corrections. For small, but finite temperatures, we use techniques of finite-size scaling to derive general relations between the zero-temperature RG flow and scaling laws of thermodynamic quantities such as the thermal expansion coefficient $\alpha$, the entropy S, and the specific heat C. A combination of the scaling relations with an analysis of thermal fluctuations shows that, for small temperatures, the thermal expansion coefficient $\alpha$ is negative and logarithmically dependent on $T$, as predicted in an earlier work. Although the requirement $\lim_{T \to 0} \alpha = 0$, expected from the third law of thermodynamics is formally satisfied, $\alpha$ is predicted to exhibit such a slow variation to remain practically constant down to unaccessibly small temperatures.