A Connection Between Complex-Temperature Properties of the 1D and 2D Spin $s$ Ising Model

TL;DR

This paper explores the complex-temperature phase diagrams of the 1D and 2D spin s Ising models, revealing a mathematical connection between their properties and extending understanding of their phase boundaries.

Contribution

It provides an exact determination of the 1D Ising model's complex-temperature phase diagram for arbitrary spin s, highlighting structural similarities with the 2D model.

Findings

The 1D phase diagram has 4s^2 infinite regions separated by boundary curves.

Boundary curves extend from the origin to infinity along specific angles.

A boundary line exists on the negative real axis if and only if s is half-integer.

Abstract

Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin $s$ and show that in the $u_s=e^{-K/s^2}$ plane (i) it consists of $N_{c,1D}=4s^2$ infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles $\theta_n = (1+2n)\pi/(4s^2)$, for $n=0,..., 4s^2-1$; (iii) of these curves, there are $N_{c,NE,1D}=N_{c,NW,1D}=[s^2]$ in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real $u_s$ axis if and only if $s$ is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin $s$ Ising model.