Flowing States and Vortices in the Classical XY Model in an External Field

Abstract

Uniformly flowing states and vortices in the classical XY model in an external field are studied. This is done by using a continuum approximation and by paying attention to particular solutions to nonlinear partial differential equations for two angles 8 and cp of rotation of spins for which cp satisfies the Laplace equation. For these two states equations for 8 have forms similar to that in the classical Ising model in a transverse field. The uniformly flowing states are therefore described by kink-type excitations identical to those in the two-dimensional Ising model. Phonon modes associated with the uniformly flowing states are also studied, which are similar to Bogoliubov phonons. Vortex solutions and vortex formation energy are studied in close similarity to the case of liquid He'. By comparing the energies of these two states, an expression for critical velocity is obtained. By making correspondence to the case of liquid He', numerical values of the critical velocity and of the velocity of phonons around the uniformly flowing states are estimated. For the former the numerical value is in fair agreement with experimental data.