One-dimensional XY model: Ergodic properties and hydrodynamic limit

A. G. Shuhov, Iouri M. Soukhov

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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
Volume45
Issue number3-4
DOIs
StatePublished - Nov 1 1986

Fingerprint

Hydrodynamic Limit
XY Model
One-dimensional Model
hydrodynamics
Jordan
Stationary States
Normal Modes
Linear differential equation
C*-algebra
algebra
differential equations
theorems
First-order
Motion
Theorem

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

@article{53036f7b10cc49079be2d46be6ffb498,
title = "One-dimensional XY model: Ergodic properties and hydrodynamic limit",
abstract = "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.",
author = "Shuhov, {A. G.} and Soukhov, {Iouri M.}",
year = "1986",
month = "11",
day = "1",
doi = "10.1007/BF01021090",
language = "English (US)",
volume = "45",
pages = "669--694",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "3-4",

}

One-dimensional XY model : Ergodic properties and hydrodynamic limit. / Shuhov, A. G.; Soukhov, Iouri M.

In: Journal of Statistical Physics, Vol. 45, No. 3-4, 01.11.1986, p. 669-694.

Research output: Contribution to journalArticle

TY - JOUR

T1 - One-dimensional XY model

T2 - Ergodic properties and hydrodynamic limit

AU - Shuhov, A. G.

AU - Soukhov, Iouri M.

PY - 1986/11/1

Y1 - 1986/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34250123666&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250123666&partnerID=8YFLogxK

U2 - 10.1007/BF01021090

DO - 10.1007/BF01021090

M3 - Article

AN - SCOPUS:34250123666

VL - 45

SP - 669

EP - 694

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -