Stationary point approach to the phase transition of the classical XY chain with power-law interactions

TL;DR

This paper analyzes stationary points of the classical XY chain with power-law interactions, linking the behavior of the Hessian determinant at these points to phase transitions, especially for interaction exponents between zero and one.

Contribution

It provides an analytical computation of the Hessian determinant's asymptotic behavior for spinwave stationary points, connecting stationary point properties to phase transition detection.

Findings

Hessian determinant behavior indicates phase transitions for alpha between 0 and 1.

For alpha between 1 and 2, phase transitions are not reflected in the Hessian determinant.

The approach uses Toeplitz matrices and Szeg"o-type theorems to analyze stationary points.

Abstract

The stationary points of the Hamiltonian H of the classical XY chain with power-law pair interactions (i.e., decaying like r^{-{\alpha}} with the distance) are analyzed. For a class of "spinwave-type" stationary points, the asymptotic behavior of the Hessian determinant of H is computed analytically in the limit of large system size. The computation is based on the Toeplitz property of the Hessian and makes use of a Szeg\"o-type theorem. The results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of classical many-body Hamiltonian functions. In agreement with this relation, the exact phase transition energy of the model can be read off from the behavior of the Hessian determinant for exponents {\alpha} between zero and one. For {\alpha} between one and two, the phase transition is not manifest in the behavior of the determinant, and it might be necessary to consider larger classes of stationary points.