Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

The Hamiltonian dynamics associated with classical, planar, Heisenberg $\mathrm{XY}$ models is investigated for two- and three-dimensional lattices. In addition to the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively different information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of Newtonian dynamics suggests consideration of other observables of geometric meaning tightly related to the largest Lyapunov exponent. The numerical computation of these observables—unusual in the study of phase transitions—sheds light on the microscopic dynamical counterpart of thermodynamics, also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces ${\Sigma }_E$ of phase space can be naturally established. In this framework, an approximate formula is worked out determining a highly nontrivial relationship between temperature and topology of ${\Sigma }_E.$ From this it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of ${\Sigma }_E.$ This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.