One-dimensional Ising model with multispin interactions

TL;DR

This paper analyzes a one-dimensional Ising model with multispin interactions, deriving its partition function, correlation functions, and mapping it to a two-dimensional model, revealing a critical singularity in the thermodynamic limit.

Contribution

It introduces a method to solve the 1D Ising chain with multispin interactions and maps it onto a 2D model, uncovering critical behavior in the limit.

Findings

Partition function derived for finite chains with various boundary conditions.

System exhibits self-duality in an external field.

A 2D critical singularity emerges in the thermodynamic limit.

Abstract

We study the spin-$1/2$ Ising chain with multispin interactions $K$ involving the product of $m$ successive spins, for general values of $m$. Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions (BC) and we calculate the two-spin correlation function. When placed in an external field $H$ the system is shown to be self-dual. Using another change of spin variables the one-dimensional (1D) Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions $K$ and $H$. The 2D system, with size $m\times N/m$, has the topology of a cylinder with helical BC. In the thermodynamic limit $N/m\to\infty$, $m\to\infty$, a 2D critical singularity develops on the self-duality line, $\sinh 2K\sinh 2H=1$.