# Ising model

The **Ising model** is also known as the **Lenz-Ising** model. For a history of the Lenz-Ising model see Refs. 1 and 2.
The Ising model is commonly defined over an ordered lattice.
Each site of the lattice can adopt two states: either
UP (S=+1) or DOWN (S=-1).

The energy of the system is the sum of pair interactions between nearest neighbors.

where is the Boltzmann constant, is the temperature, indicates that the sum is performed over nearest neighbors, and indicates the state of the i-th site, and is the coupling constant.

## Contents

## 1-dimensional Ising model

- 1-dimensional Ising model (exact solution)

## 2-dimensional Ising model

Solved by Lars Onsager in 1944. Rudolf Peierls had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.

## 3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice:

- Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown
- Sorin Istrail "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87 - 96 (2000)

## ANNNI model

The **axial next-nearest neighbour Ising** (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.