Correlations and phase structure of Ising models at complex temperature

TL;DR

This paper explores the complex-temperature behavior of Ising models, revealing helimagnetic order near eigenvalue degeneracies and linking long-range order to partition function zeros, with implications for phase structure analysis.

Contribution

It provides new insights into the phase structure of Ising models at complex temperatures, especially the relationship between eigenvalues, order, and partition function zeros.

Findings

Helimagnetic order near eigenvalue degeneracies in 1D and ladder systems

Proliferation of partition function zeros correlates with long-range order

Rich structure of phase behavior in the complex temperature plane

Abstract

We investigate the spin-spin correlation functions of Ising magnets at complex values of the temperature, T. For one-dimensional chain and ladder systems, we show the existence of a kind of helimagnetic order in the vicinity of contours where the leading two eigenvalues of the transfer matrix become equal in magnitude. We analyse the development of long-range order as the two-dimensional limit is approached, and find that there is rich structure in much of the complex-T plane. In particular, and contrary to the work of Fisher on this problem, the development of long-range order is actually associated with a proliferation of partition function zeros in a certain finite region of that plane containing the real-temperature magnetically ordered phase. The thermodynamic consequences of this are also discussed.