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

Abstract

The stationary points of the Hamiltonian H of the classical XY chain with power-law pair interactions (i.e., decaying like r^{-α} 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 Szegö-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 α between zero and one. For α 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.

9 pages, 6 figures

References in corpus (6)

• Phase transitions and configuration space topology
• Partial equivalence of statistical ensembles and kinetic energy
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Phase transitions induced by saddle points of vanishing curvature
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems

Cited by in corpus (4)

• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Exploring the energy landscape of XY models
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Counting equilibria of the Kuramoto model using birationally invariant intersection index