Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins

Abstract

We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing {\it interacting classical spin systems} in the thermodynamic limit, including the occurrence of a phase transition, using topology arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is to show that, in the context of the studied classical models, the Euler characteristic $\chi (E)$ embeds the necessary features for a correct description of several magnetic thermodynamic quantities of the systems, such as the magnetization, correlation function, susceptibility, and critical temperature. Despite the classical nature of the studied models, such quantities are those that do not violate the laws of thermodynamics [with the proviso that Van der Waals loop states are mean field (MF) artifacts]. We also discuss the subtle connection between our approach using the Euler entropy, defined by the logarithm of the modulus of $\chi (E)$ per site, and that using the {\it Boltzmann} microcanonical entropy. Moreover, the results suggest that the loss of regularity in the Morse function is associated with the occurrence of unstable and metastable thermodynamic solutions in the MF case. The reliability of our approach is tested in two exactly soluble systems: the infinite-range and the short-range $XY$ models in the presence of a magnetic field. In particular, we confirm that the topological hypothesis holds for both the infinite-range ($T_c\ne 0$) and the short-range ($T_c=0$) $XY$ models. Further studies are very desirable in order to clarify the extension of the validity of our proposal.

References in corpus (13)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Negative Absolute Temperature for Motional Degrees of Freedom
• Dynamical band flipping in fermionic lattice systems: An ac-field-driven change of the interaction from repulsive to attractive
• Phase transitions and configuration space topology
• Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices
• Phase transitions and topology changes in configuration space
• Atomic Gases at Negative Kinetic Temperature
• Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials
• 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
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• On the mean Euler characteristic and mean Betti numbers of the Ising model with arbitrary spin

Cited by in corpus (1)

• Topological Theory of Phase Transitions