Low temperature behavior of finite-size one-dimensional Ising model and the partition function zeros

TL;DR

This paper investigates how boundary conditions affect the low-temperature behavior of finite one-dimensional Ising models, revealing different partition function zero distributions and size-dependent growth of the expansion coefficient.

Contribution

It derives explicit formulas for the low-temperature expansion under open and periodic boundary conditions, linking them to partition function zero distributions.

Findings

Boundary conditions significantly influence low-temperature expansions.

Partition function zeros distribute differently for open vs. periodic boundaries.

Leading coefficient grows with system size under periodic boundary conditions.

Abstract

In contrast to the infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive explicit formula for the leading term of the expansion both under open and periodic boundary conditions, and show they are related to different distributions of partition function zeros on the complex temperature plane. In particular, when the periodic boundary condition is imposed, the leading coefficient of the expansion grows with size, due to the zeros approaching the origin.