Geometry and molecular dynamics of the Hamiltonian mean-field model in a magnetic field

The Hamiltonian mean-field model is investigated in the presence of a field. The self-consistent equations for the magnetization and the energy per particle are derived, and the field effect on the caloric curve is presented. The analytical geometric approach to Hamiltonian dynamics, under the hypothesis of quasi-isotropy, allows us to calculate the field effect on the energy-dependent microcanonical mean Ricci curvature and its fluctuations. Notably, the method proved suitable to identify that stable and metastable solutions of the Lyapunov exponent exhibit intriguing distinct curvature behavior very close to the critical point at extremely low field values. In addition, finite-size molecular dynamics (MD) simulations are used to observe the evolution of the magnetization and their components, including the stability properties of the solutions. Most importantly, comparison of finite-size MD calculations of the Lyapunov exponent and related properties with those via the geometric approach unveil the sensible dependence of these microcanonical quantities on energy, number of particles, and field, before a quasisaturation behavior at high fields. Finally, relaxation properties from out-of-equilibrium initial conditions are discussed in light of MD simulations.