From measurements to inferences of physical quantities in numerical simulations

We propose a change of style for numerical estimations of physical quantities from measurements to inferences. We estimate the most probable quantities for all the parameter region simultaneously by using the raw data cooperatively. Estimations with higher precisions are made possible. We can obtain a physical quantity as a continuous function, which is processed to obtain another quantity. We applied the method to the Heisenberg spin-glass model in three dimensions. A dynamic correlation-length scaling analysis suggests that the spin-glass and the chiral-glass transitions occur at the same temperature with a common exponent $\nu$. The value is consistent with the experimental results. We explained a spin-chirality separation problem by a size-crossover effect.

Figure 1

Temperature dependencies of energy($E$), magnetization ($M$), the specific heat ($C$), and the effective critical exponent ${\beta }_{\mathrm{eff}}$ in the two-dimensional Ising ferromagnetic model. Error bars are smaller than the symbols and line widths.

Figure 2

Correlation function data for selected time steps for $L=39$ (circles) and $L=159$ (lines). Arrows depict crossover distances between short-range correlations and long-range correlations ($r_{\mathrm{c}}=9$ for SG and $r_{\mathrm{c}}=3$ for CG). Inset: Scaled correlation functions for data with $T=0.200,L=159$ depicted with the same line color.

Figure 3

The correlation length data estimated by the inference and that by the measurements (the second-moment method). Error bars for the inference data are smaller than linewidths. Inset: Temperature dependencies of the effective exponent ${\eta }_{\mathrm{eff}}$. The critical exponent $\eta$ obtained by the scaling analysis is also plotted.

Figure 4

A scaling plot of (a) the SG transition and (b) the CG transition. Data of 42 sets of temperature ranging from 0.170 to 0.220 are depicted with different color lines. Insets depict the same scaling plot using the correlation length data obtained by the measurements.