Topological Origin of the Phase Transition in a Mean-Field Model

Lapo Casetti; E. G. D. Cohen; Marco Pettini

doi:10.1103/PhysRevLett.82.4160

Topological Origin of the Phase Transition in a Mean-Field Model

Lapo Casetti, E. G. D. Cohen, and Marco Pettini

Phys. Rev. Lett. 82, 4160 – Published 24 May 1999

Abstract

We argue that the phase transition in the mean-field $XY$ model is related to a particular change in the topology of its configuration space. The nature of this topological change can be discussed on the basis of elementary Morse theory using the potential energy per particle $V$ as a Morse function. The value of $V$ where such a topological change occurs equals the thermodynamic value of $V$ at the phase transition and the number of (Morse) critical points grows very fast with the number of particles $N$ . Furthermore, as in statistical mechanics, the way the thermodynamic limit is taken is crucial in topology.

Received 30 October 1998

DOI:https://doi.org/10.1103/PhysRevLett.82.4160

Authors & Affiliations

Lapo Casetti^1,*, E. G. D. Cohen^2,†, and Marco Pettini^3,4,‡

¹Istituto Nazionale per la Fisica della Materia (INFM), Unità di Ricerca del Politecnico di Torino, Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
²The Rockefeller University, 1230 York Avenue, New York, New York 10021-6399
³Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy
⁴and Istituto Nazionale per la Fisica della Materia (INFM), Unità di Ricerca di Firenze, Firenze, Italy

^*Electronic address: lapo@polito.it
^†Electronic address: egdc@rockvax.rockefeller.edu
^‡Also at INFN, Sezione di Firenze, Firenze, Italy. Electronic address: pettini@arcetri.astro.it