Topology and Phase Transitions: Paradigmatic Evidence

1999/11/16 by Roberto Franzosi, Marco Pettini, Lionel Spinelli · 4 citations
Mathematics · Physics and Astronomy · #Combinatorics #Computation #Equipotential #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Invariant (physics) #Lattice (music) #Mathematical physics #Mathematics #Phase transition #Physics #Quantum mechanics #Sigma #Sigma model #Theoretical and Computational Physics #Topology (electrical circuits) #cond-mat.stat-mech #hep-th #math-ph #math.MP

paper · pdf · doi:10.1103/physrevlett.84.2774

published as Phys.Rev.Lett.84:2774-2777,2000 · 4 pages, 4 figures

arxiv created 1999/11/16 · openalex publication_date 2000/03/27 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We report upon the numerical computation of the Euler characteristic \ensuremathχ (a topologic invariant) of the equipotential hypersurfaces \ensuremathΣ_v of the configuration space of the two-dimensional lattice \ensuremathφ⁴ model. The pattern \ensuremathχ(\ensuremathΣ_v) versus v (potential energy) reveals that a major topology change in the family \ensuremathΣ_v_v\ensuremath∈R is at the origin of the phase transition in the model considered. The direct evidence given here---of the relevance of topology for phase transitions---is obtained through a general method that can be applied to any other model.

Citations

• Elementary Topics in Differential Geometry
• Geometric Measure Theory

Cited by

• Phase Transitions Detached from Stationary Points of the Energy Landscape
• Saddle index properties, singular topology, and its relation to thermodynamic singularities for aϕ4mean-field model
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• Topology and phase transitions II. Theorem on a necessary relation