Statistical mechanics of the flexural Ising model in $D$ dimensions

TL;DR

This paper investigates the critical behavior of a generalized elastic Ising model in D dimensions, revealing how spin-elastic coupling influences phase transitions and criticality, especially in physically relevant two-dimensional membranes in three-dimensional space.

Contribution

It derives renormalization group equations for the coupled membrane-Ising system and identifies the relevance of spin-elastic interactions in specific dimensions, extending understanding of critical phenomena in elastic systems.

Findings

Coupling is relevant for D=2, d=3, indicating strong interaction at criticality.

Renormalization group analysis shows relevance of spin-elastic coupling in certain dimensions.

Coupling becomes irrelevant when the difference between space and membrane dimensions exceeds twelve.

Abstract

A generalization of the compressible Ising model in which spins are hosted on an elastic $D$-dimensional lattice embedded in $d>D$ dimensions is studied. Two critical systems interact when temperature is tuned to the Ising transition point, as a freely-fluctuating thermalized crystalline membrane in its flat phase is already critical. Noting that the upper critical dimension of both the membrane and Ising model is $D_{uc}=4$, renormalization group recursion relations are found by expanding in $\epsilon=4-D$. The coupling between spin and elastic degrees of freedom is shown to be a relevant operator for the physical case of a two-dimensional membrane fluctuating in three-dimensional space ($D=2$, $d=3$), which suggests that the thermalized membrane and Ising systems become more strongly coupled at long wavelengths. The coupling is irrelevant when the difference between the space dimension and membrane dimension is greater than twelve.