Monte carlo simulation for continuous spin model (e.g. XY or Heisenberg model)

Question

Unlike the Ising model, the XY model and the Heisenberg model have a continuous spectrum. So one need discretize them for a numerical simulation. But how to make sure the discretization procedure reliable?

For example, for the XY model with variables \(\theta_i\) if \(\theta_i\) is discretized by choosing a unit 2πN, how to make sure the N is large enough?

Comments

One usually checks how much the observables of interest change when simulating with different discretizations. The error of the finer discretization is usually a little less than the discrepancy between the results. If the discrepancy is deemed too large, one needs to refine the discretization. Repeat until satisfied, or until resources run out.