Phonon Excitation in Quantum Lattice Model

Abstract

From the calculation of macroscopic sound velocity based upon the free energy derived by Matsubara and Matsuda within the molecular field approximation, it is shown that the ordinary spin waves of the Quantum Lattice Model (QLM) or the XY model do not directly correspond to the phonon excitation of liquid ⁴He. Instead of the spin waves, a new excitation with the usual macroscopic sound velocity is derived from the motion of the number operator, where the motion is taken into account to the second order in the time derivative within the effective-field approximation. The present calculation implies the possibility that the similar excitation exists in the one- or two-dimensional XY model which does not exhibit a long-range order.