Geometry of dynamics and phase transitions in classical lattice phi^4 theories

TL;DR

This study investigates the geometric and dynamical properties of classical lattice phi^4 models in three dimensions, revealing phase transitions and Lyapunov exponent behaviors through Hamiltonian dynamics and geometric analysis.

Contribution

It introduces a geometric approach to analyze phase transitions in classical lattice phi^4 theories, linking topological changes to observable dynamical phenomena.

Findings

Phase transition detected via time averages of thermodynamic observables.

Largest Lyapunov exponents show peculiar behavior at the transition point.

Geometric observables suggest topological changes in the underlying manifolds.

Abstract

We perform a microcanonical study of classical lattice phi^4 field models in 3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these systems that undergo a second order phase transition in the thermodynamic limit are here investigated. The microscopic Hamiltonian dynamics neatly reveals the presence of a phase transition through the time averages of conventional thermodynamical observables. Moreover, peculiar behaviors of the largest Lyapunov exponents at the transition point are observed. A Riemannian geometrization of Hamiltonian dynamics is then used to introduce other relevant observables, that are measured as functions of both energy density and temperature. On the basis of a simple and abstract geometric model, we suggest that the apparently singular behaviour of these geometric observables might probe a major topological change of the manifolds whose geodesics are the natural motions.