### 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<T_{c}, where T_{c} 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 > T_{c}. 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 < T_{c} 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 language | English |
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Pages (from-to) | 108-128 |

Number of pages | 21 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 81 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1975 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*81*(1), 108-128. https://doi.org/10.1016/0378-4371(75)90039-4