Functional renormalization group approach to phonon modified criticality: anomalous dimension of strain and non-analytic corrections to Hooke's law

TL;DR

This paper uses a functional renormalization group approach to analyze how critical elasticity interacts with Ising criticality, revealing non-analytic phonon behavior and corrections to Hooke's law near fixed points.

Contribution

It introduces a FRG framework with fixed volume constraints to study phonon-modified criticality and derives non-analytic corrections to elastic behavior.

Findings

Fixed points R and S have finite negative anomalous dimension of strain fluctuations.

Phonon energy dispersion exhibits non-analytic momentum dependence proportional to $k^{1-y_{*}/2}$.

Stress-strain relations near fixed points show linear behavior with non-analytic corrections to Hooke's law.

Abstract

We study the interplay between critical isotropic elasticity and classical Ising criticality using a functional renormalization group (FRG) approach which is implemented such that the volume is fixed during the entire renormalization group flow. For dimensions slightly smaller than four we use a simple truncation of the FRG flow equations to recover the fixed points of the constrained Ising model: the Gaussian fixed point G, the Ising fixed point I, the renormalized Ising fixed point R, and the spherical fixed point S. We show that the fixed points R and S are both characterized by a finite anomalous dimension $y_{\ast}<0$ of strain fluctuations, implying that the energy dispersion of longitudinal acoustic phonons exhibits a non-analytic momentum dependence proportional to $k^{1-y_{\ast}/2}$ for small momentum $k$. We also derive and solve flow equations for the free energy at constant strain and compute stress-strain relations in the vicinity of the fixed points. As a result, we reaffirm that Ising criticality, controlled by the fixed point I, is preempted by a bulk instability. Beyond that, we find that the stress-strain relation at R and S remains linear to leading order (Hooke's law), as long as the interaction between strain and Ising fluctuations is sufficiently weak. However, the finite anomalous dimension of strain fluctuations $y_{\ast}$ gives rise to non-analytic corrections to Hooke's law.