Hydrodynamics of an n-component phonon system

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 $\text{O}$(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 $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ 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 $\text{z =}\text{1}\text{2}\text{d}$. 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 $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.