One-dimensional XY model: Ergodic properties and hydrodynamic limit

A. G. Shuhov, Iouri M. Soukhov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We prove theorems on convergence to a stationary state in the course of time for the one-dimensional XY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps the XY dynamics onto a group of Bogoliubov transformations on the CAR C*-algebra over Z1. The role of stationary states for Bogoliubov transformations is played by quasifree states and for the XY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensional XY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of "normal modes," which is described by a hyperbolic linear differential equation of second order. For the XX model this equation reduces to a first-order transfer equation.

Original languageEnglish (US)
Pages (from-to)669-694
Number of pages26
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - Nov 1 1986

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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