Scientific Results

Geometrical and topological study of the Kosterlitz–Thouless phase transition in the XY model in two dimensions

Year: 2021

Authors: Bel-Hadj-Aissa G., Gori M., Franzosi R., Pettini M.

Autors Affiliation: Dipartimento di Scienze Fisiche, della Terra e dell’Ambiente (DSFTA), University of Siena, Via Roma 56, 53100 Siena, Italy
QSTAR & CNR—Istituto Nazionale di Ottica, Largo Enrico Fermi 2, I-50125 Firenze, Italy
Quantum Biology Lab, Howard University, 2400 6th St NW, Washington, DC 20059, United States of America
Aix-Marseille University, Marseille, France, CNRS Centre de Physique Th ́eorique UMR7332, 13288 Marseille, France

Abstract: Phase transitions do not necessarily correspond to a symmetry- breaking phenomenon. This is the case of the Kosterlitz–Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a tran- sition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topo- logical concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the tran- sition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets.

Journal/Review: JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT

Volume: 2021      Pages from: 1  to: 33

KeyWords: classical phase transitions, dynamical processes, numerical simula- tions
DOI: 10.1088/1742-5468/abda27

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