Strong spin-phonon interaction. II. Multimode linear chain

The strong spin-phonon Hamiltonian in a multimode linear chain is studied, the Zeeman energy of the spins being treated by second-order perturbation theory on the $N$-fold configurationally degenerate ground state of the lattice plus spin-phonon Hamiltonian. An analytic procedure is developed to calculate the matrix elements of the secular determinant, taking account of all intermediate states with all possible distributions of phonon populations among the $O\mathrm{}\left(N\right)\mathrm{}$ modes of the chain. Diagonalization of secular determinant yields a set of collective excitations within the ground-state multiplet characterized by a rest mass and maximum speed, for which we give explicit expressions. The results are compared with those of single-mode models previously studied and the limits of validity of the perturbative approach are examined. The relevance of the present model to some problems of solid-state physics and quantum electronics is discussed, and approximate evaluation of the main parameters of the dynamical ground state is performed.