Fig. 7: The diagram and results of the imaginary-time BRA method.

a illustrates the schematic of imaginary-time BRA process. The O operators are depicted by a gradient of colors, with one instance inserted and fixed at the imaginary time τ = 0, and the other moving within the time axis. If successfully moved, it corresponds physically to a transition from the imaginary time point τ to a new time point \({\tau }^{{\prime} }\). The time difference is denoted by \(\Delta \tau={\tau }^{{\prime} }-\tau\). b The simulation results for the XXZ chain with L = 20 using imaginary-time BRA method are presented. The main plot displays the weight ratios with standard error for a fixed inverse temperature β = 40 and varying Δ = 0.1, 0.5, and 1.0. The small inset shows the weight ratios with error bars for fixed Δ = 0.1 and varying β = 2L, 3L, 4L, 5L. These results have not yet been multiplied by the reference values of \(\langle {S}_{1}^{x}(0){S}_{2}^{x}(0)\rangle\). It can be observed that when β is sufficiently large, the furthest correlation \(\langle {S}_{1}^{x}(\tau=\beta /2){S}_{2}^{x}(0)\rangle\) decays to nearly zero.