In the continuum limit, we map a Heisenberg chain model, in which the spins are explicitly coupled to the lattice degrees of freedom, to one-dimensional quantum sine-Gorden model, and obtain the phase diagram and excitation spectrum of the system. The effect of quantum lattice fluctuation on the ground-state properties of the system is discussed in detail. The results show that a quantum phase transition from a gapless state to gapped dimerized phase occurs at nonzero value of the spin-phonon coupling. Moreover, our results indicate that the two-magnon excitation spectrum is more sensitive to the spin-phonon coupling than the one-magnon excitation spectrum.
Excitation spectrum and T=0 dynamics of the one-dimensional planar spin- ferromagnet