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arXiv:2410.01797

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

Abigail Plummer

Published Oct 2, 2024• 0 citations
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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 $ε=4-D$. 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.

DOI
10.48550/arxiv.2410.01797

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The stability of a quantum critical point in the O(N)O(N)O(N) universality class with respect to an elastic coupling, that preserves O(N)O(N)O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension d=3−εd=3-εd=3−ε. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for 1≤N≤41\leq N \leq 41≤N≤4, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length ξξξ. For N>4N > 4N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached the sound velocity is found to vanish only asymptotically as ξ→∞ξ\to \inftyξ→∞ such that the crystal remains microscopically stable for any finite value of ξξξ. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in d=4−εd=4-εd=4−ε, where the crystal always remains microscopically stable for finite ξξξ.