We present extensive Monte Carlo spin-dynamics simulations of the classical $\mathrm{XY}$ model in three dimensions on a simple cubic lattice with periodic boundary conditions. A recently developed efficient integration algorithm for the equations of motion is used, which allows a substantial improvement of statistics and large integration times. We find spin-wave peaks in a wide range around the critical point and spin diffusion for all temperatures. At the critical point we find evidence for a violation of dynamic scaling in the sense that independent components of the dynamic structure factor $S(\mathbf{q},\omega )$ require different dynamic exponents in order to obtain scaling. Below the critical point we investigate the dispersion relation of the spin waves and the linewidths of $S(\mathbf{q},\omega )$ and find agreement with mode coupling theory. Apart from strong spin-wave peaks we observe additional peaks in $S(\mathbf{q},\omega )$ which can be attributed to two-spin-wave interactions. The overall line shapes are also discussed and compared to mode coupling predictions. Finally, we present results for the transport coefficient $D(q,\omega )$ of the out-of-plane magnetization component at the critical point, which is related to the thermal conductivity of ${}^4$He near the superfluid-normal transition.

• Received 8 December 1998

DOI:https://doi.org/10.1103/PhysRevB.60.3375