Spin systems on hypercubic Bethe lattices: A Bethe-Peierls approach

TL;DR

This paper extends the Bethe-Peierls approach to spin systems on hypercubic Bethe lattices, providing accurate phase diagrams that closely match those of real cubic lattices, especially at low and high temperatures.

Contribution

It introduces a Bethe-Peierls method for hypercubic Bethe lattices, bridging the gap between tree-like lattices and real cubic lattices for spin systems.

Findings

Phase diagrams closely match cubic lattices at low/high temperatures.

Improves upon conventional Bethe lattice with connectivity 2d.

Provides exact thermodynamic equations for these lattices.

Abstract

We study spin systems on Bethe lattices constructed from d-dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe-Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferromagnetic Ising models on hypercubic Bethe lattices with dimension d=2, 3, and 4. Our results are in good agreement with the results of the same models on d-dimensional cubic lattices, for low and high temperatures, and offer an improvement over the conventional Bethe lattice with connectivity k=2d.