Structure of the Phonon Propagator for an Anharmonic Crystal

Abstract

The phonon propagator for an arbitrary crystal is the analytic continuation to the complex z plane of the Fourier coefficient of the imaginary time correlation function $D_{\mathrm{jj\prime }}(\mathbf{k}{\mathrm{;u)=〈TA}}_{\mathbf{k}j}{\mathrm{(u)A}}_{\mathbf{k}\mathrm{j\prime }}^{+}\text{(0)〉}$ , where A_kj is the field operator for phonons with wavevector k and polarization or branch index j. Considering D(k; u) as a 3r × 3r matrix whose elements are labeled by j and j′ (j,j′ = 1, 2, …, 3r), where r is the number of atoms in a primitive unit cell of the crystal, the restrictions imposed on the form of this matrix by the symmetry and structure of the crystal are determined here. In particular, it is proved that the element D_jj′(k; u) vanishes unless j and j′ label normal modes of vibration which transform according to the same row of the same irreducible multiplier representation of the point group of the wavevector k, G₀(k). As a corollary to this result it follows that if no two modes labeled by the wavevector k exist whose frequencies are different, but whose associated eigenvectors transform according to the same irreducible multiplier representation of G₀(k), the matrix D_jj′(k; u) is diagonal in j and j′.