High-dimensional Ising model

This article may contain an excessive amount of intricate detail that may interest only a particular audience. Please help by spinning off or relocating any relevant information, and removing excessive detail that may be against Wikipedia's inclusion policy. (November 2024) (Learn how and when to remove this message)

The Ising model is a prototypical model in statistical physics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. A model of this type can be defined on lattices in any number of dimensions. Techniques that are applicable for small dimensions are not always useful for larger dimensions, and vice versa.

In any dimension, the Ising model can be productively described by a locally varying mean field. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit. The accuracy of this approximation improves as the dimension becomes larger. A deeper understanding of how the Ising model behaves, going beyond mean-field approximations, can be achieved using renormalization group methods.