# One-dimensional q-state Potts model with multi-site interactions

## TL;DR

This paper analyzes a one-dimensional q-state Potts model with multi-site interactions, deriving exact partition and correlation functions, and maps it onto a two-dimensional model to explore critical behavior.

## Contribution

It provides exact solutions for the 1D Potts model with multi-site interactions and reveals its mapping to a 2D model, highlighting critical phenomena along the self-duality line.

## Key findings

- Exact partition function and correlation function at zero field
- Self-duality of the system in a field
- Mapping to a 2D Potts model with helical boundary conditions

## Abstract

A one-dimensional (1D) $q$-state Potts model with $N$ sites, $m$-site interaction $K$ in a field $H$ is studied for arbitrary values of $m$. Exact results for the partition function and the two-point correlation function are obtained at $H=0$. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions $K$ and $H$, on a cylinder with helical boundary conditions (BC). The 2D system has a length $N/m$ and a transverse size $m$. Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, $(e^{qK}-1)(e^{qH}-1)=q$, when $N/m\to\infty$ and $m\to\infty$.