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 $T_{\text{BKT}}$, 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.

Figure 1

Measured value ${\beta }_M$ as a function of ${\beta }_I$. The inset shows a typical error bar, of order ${10}^{-4}$, which is not observable in the main plot.

Figure 2

Thermalization path of the dynamical temperature estimator $\widehat{\beta }$ for system sizes of $L=16$, 32, and 64 at $T=0.4$.

Figure 3

Autocorrelation function $C_O\left(t\right)$ for energy $E$, magnetization $M$, and inverse temperature $\beta$ as a function of ${log}_2\left(t\right)$ for temperature $T=0.2$ (top) and $T=0.7$ (bottom).

Figure 4

Binning analysis for energy $E$, magnetization $M$, and inverse temperature $\beta$ for $T=0.4$ (top) and $T=0.7$ (bottom).

Figure 5

Histograms of block averages of $\widehat{\beta }$ for $T=0.4$ (top) and $T=0.7$ (bottom) compared to the corresponding Gaussian distribution represented by the solid red line.