An Exact Solution to a Three Dimensional Ising Model and Dimensional Reductions

TL;DR

This paper presents an exact solution to a three-dimensional Ising model using high temperature expansion, maps complex high-dimensional models to solvable lower-dimensional variants, and explores dimensional reductions in related spin systems.

Contribution

It introduces a novel high temperature expansion approach to solve complex high-dimensional Ising models and demonstrates dimensional reductions without compactification.

Findings

Mapped 3D Ising models to 2D variants

Provided algebraic recursive series solution for 3D Ising

Mapped 3D Ising model to a single spin 1/2 particle

Abstract

A high temperature expansion is employed to map some complex anisotropic nonhermitian three and four dimensional Ising models with algebraic long range interactions into a solvable two dimensional variant. We also address the dimensional reductions for anisotropic two dimensional XY and other models. For the latter and related systems it is possible to have an effective reduction in the dimension without the need of compactifying some dimensions. Some solutions are presented. This framework further allows for some very simple general observations. It will be seen that the absence of a ``phase interference'' effect plays an important role in high dimensional problems. A very forbidding purely algebraic recursive series solution to the three dimensional nearest neighbor Ising model will be given. In the aftermath, the full-blown three dimensional nearest neighbor Ising model is exactly mapped onto a single spin 1/2 particle with nontrivial dynamics. All this allows for a formal high dimensional Bosonization.