Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.

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 ferro- magnetic 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.