We studied classical spins on a square lattice, for L= 16, 32 and 64, with periodic boundary conditions. The simulation was a combination of Monte Carlo and spin-dynamics methods applied to Hamiltonian (2.1). We used the Metropolis Monte Carlo method to produce initial spin configurations (IC) at a given temperature , but close to . Then, each initial configuration was evolved in time using the equations of motion for the spins. We accumulated statistics of the fluctuating number of free vortices during the time evolution to determine the free vortex lifetime. A vortex in a given unit cell is considered free if there are no vortices or antivortices in any of the 8 surrounding unit cells.
In the Monte Carlo, we used the first Monte Carlo steps (MCS) for equilibration, writing data after each 500 or 1000 MCS. Individual spins were updated by adding increments in arbitrary directions, and then renormalizing to unit lengths. We generated between 25 and 100 IC at each temperature, so that the relative error in is 1 to 3 percent.
The time evolution simulation solves numerically the Landau---Lifshitz spin equations of motion, using a fourth order Runge---Kutta scheme.
The free vortex lifetime was determined during the time evolution simulation. The number of free vortices was counted at each time step and the times of its decrements were recorded. If is the time between the and decrements, is the number of free vortices (plus antivortices) in the system before the decrement, and is the change of , then from this event the estimate of the vortex lifetime from the time interval is
The denominator introduces a weighting factor that reduces the importance of fluctuations involving a change in of more than one. This formula gives correct results if , where dt is the integration time step. Also, this condition assures that most of the fluctuations and the processes which occur will change by one or two. [ for vortex pair creation/annihilation, when a vortex changes from bound to free or vice-versa.] The choice of the time step dt depends on T and L. Increasing either of these increases the average number of vortices, and diminishes the time scale over which their numbers fluctuate, and requires a decrease of dt. The smallest dt we used was for a system at temperature . In order to check the reliability of the simulations with small time steps we compared the components of the spins(, ) with precision at equal times for runs with and out to a final time , finding no differences.
We also considered the free-vortex number-number time correlation function, as another way to obtain the time scale of the vortex number fluctuations, and as a check of the lifetime measurement. The definition is
where is the instantaneous deviation in from its time-independent average, .
Under the phenomenological assumption of linear response, if the number of free vortices deviates slightly from the equilibrium number at a given temperature, then the rate at which the system relaxes back to equilibrium is proportional to the deviation from equilibrium. If it is valid, it leads to a relaxation time ;
For the simulations of the correlation function we used 40 initial configurations and a time step for all temperatures. Each IC was integrated in time up to . Each point is from approximately 300 measurements since we constrained the argument of in the interval (). The successive measurements for a given IC and time t in Eq. (3.2) were taken with a shift of 1.0 time unit in order to minimize the correlation of the data. The final data points were obtained using a weighted average over the initial configurations.