Phase transitions and topology changes in configuration space

Abstract

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly "strong" topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.

37 pages, REVTeX4, 9 ps/eps figures, to appear in Journal of Statistical Physics (scheduled for June 2003)

References in corpus (11)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Saddles in the energy landscape probed by supercooled liquids
• Geometric approach to the dynamic glass transition
• Geometry of dynamics, Lyapunov exponents and phase transitions
• Phase Transitions in "Small" systems
• Role of saddles in mean-field dynamics above the glass transition
• Topology and phase transitions: a paradigmatic evidence
• Topological aspects of geometrical signatures of phase transitions
• Topological origin of the phase transition in a mean-field model
• Phase transitions as topology changes in configuration space: an exact result
• Hamiltonian dynamics and geometry of phase transitions in classical XY models

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