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.

arXiv comment9 pages, 6 figures

References in corpus (12)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Long-range interacting rotators: connection with the mean--field approximation
• Phase transitions and configuration space topology
• A Theorem on the origin of Phase Transitions
• Canonical solution of a system of long-range interacting rotators on a lattice
• 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
• Energy landscape and phase transitions in the self-gravitating ring model
• Quasi-stationary trajectories of the HMF model: a topological perspective

Cited by in corpus (9)

• Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Phase transitions detached from stationary points of the energy landscape
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model
• Exploring the energy landscape of XY models
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• A toric deformation method for solving Kuramoto equations
• Counting equilibria of the Kuramoto model using birationally invariant intersection index