Hamiltonian Dynamics and the Phase Transition of the XY Model

TL;DR

This paper introduces a Hamiltonian dynamics framework for the XY model, linking analytical derivations with numerical simulations to study phase transition behavior and thermodynamic properties.

Contribution

It develops a Hamiltonian dynamics approach for the XY model and derives a nonlinear dispersion relation that accurately predicts thermodynamic quantities near the phase transition.

Findings

Thermodynamical properties from microcanonical simulations match canonical Monte-Carlo results.

Derived dispersion relation agrees with numerical experiments up to near the transition temperature.

Identified dominant phonon propagation at low temperatures and topological defect formation above the transition.

Abstract

A Hamiltonian dynamics is defined for the XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte-Carlo results in the explored temperature region. The behavior of the magnetization and the energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature $T_{KT}\approx 0.855$.