Hydrodynamics of an n-component phonon system

L. Sasvári, F. Schwabl, P. Szépfalusy

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Abstract

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations O(n). For n ≥ 2 a continuous symmetry is broken for T<Tc, where Tc is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the 1 2n (n - 1) angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are 1 2n (n - 1) additional independent diffusive modes for T > Tc. In the ordered phase we find 2 (n - 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < Tc require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent z = 1 2d. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent z ̃ = φ/2ν, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.

Original languageEnglish
Pages (from-to)108-128
Number of pages21
JournalPhysica A: Statistical Mechanics and its Applications
Volume81
Issue number1
DOIs
Publication statusPublished - 1975

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ASJC Scopus subject areas

  • Statistics and Probability
  • Condensed Matter Physics

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