Ensemble-free configurational temperature for spin systems

An estimator for the dynamical temperature in an arbitrary ensemble is derived in the framework of the conjugate variables theorem. We prove directly that its average indeed gives the inverse temperature and that it is independent of the ensemble. We test this estimator numerically by a simulation of the two-dimensional XY model in the canonical ensemble. As this model is critical in the whole region of temperatures below the Berezinski-Kosterlitz-Thouless critical temperature T_{BKT}, we use a generalization of Wolff's unicluster algorithm. The numerical results allow us to confirm the robustness of the analytical expression for the microscopic estimator of the temperature. This microscopic estimator has also the advantage that it gives a direct measure of the thermalization process and can be used to compute absolute errors associated with statistical fluctuations. In consequence, this estimator allows for a direct, absolute, and stringent test of the ergodicity of the underlying Markov process, which encodes the algorithm used in a numerical simulation.