The Ising chain constrained to an even or odd number of positive spins

TL;DR

This paper analyzes the one-dimensional Ising chain with constraints on the number of positive spins, deriving exact thermodynamic properties and revealing how these constraints affect convergence and scaling behaviors.

Contribution

It introduces a generalized transfer matrix method to exactly compute properties of the constrained Ising chain, revealing new scaling functions and distributions.

Findings

Constraints slow down convergence to the thermodynamic limit

New scaling functions emerge at zero temperature and magnetic field

Constraints provide solutions to a stochastic voter model

Abstract

We investigate the statistical mechanics of the periodic one-dimensional Ising chain when the number of positive spins is constrained to be either an even or an odd number. We calculate the partition function using a generalization of the transfer matrix method. On this basis, we derive the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. We show that in general the constraints substantially slow down convergence to the thermodynamic limit. By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. We demonstrate how these results solve a stochastic version of the one-dimensional voter model.