Families of planar lattices with arbitrarily high $T_{\rm c}$ for the ferromagnetic Ising model

TL;DR

This paper constructs families of planar lattices with arbitrarily high critical temperatures for the ferromagnetic Ising model, demonstrating a logarithmic scaling with maximal coordination number and identifying optimal lattice families.

Contribution

It introduces explicit lattice constructions with high $T_c$, derives their asymptotic behavior, and proposes a conjectured optimal $T_c^*$ function for periodic tessellations.

Findings

$T_c$ scales as $A \, ext{ln} \, q_{max}$ with $A=2/\text{ln} 2$

Apollonian lattices saturate the conjectured optimal bound

Explicit expressions for $T_c$ are derived for constructed lattices

Abstract

We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, $T_{\rm c}$, for the classical ferromagnetic Ising model. Our approach is motivated by recently found exact bounds, which imply that large values of $T_{\rm c}$ require large values of the maximal coordination number of the lattice, $q_{\rm max}$. We create such lattices through iterative triangulation and derive explicit expressions for their $T_{\rm c}$. Furthermore, we show that $T_{\rm c}$ for these families scales asymptotically as $T_{\rm c}/J\sim A \ln q_{\rm max}$ with a universal prefactor $A=2/\ln 2$. We introduce a function $T_{\rm c}^*(q_{\rm max})$ that we conjecture to be optimal for all periodic tessellations of the plane. We show that the family of so-called Apollonian lattices, which are derived from the Triangular lattice through iterative triangulation, saturates this bound. The lattices discussed in this work are relevant for theoretical questions of optimality in network systems and may be realized experimentally in Coherent Ising Machines or topoelectric circuits in the future.