In this paper, both the local region and sub-Monte Carlo steps were introduced to study the non-equilibrium steady state of two-dimensional Ising model. As the size of local region can vary on a large scale, it is very important to determine the effective local region size. In our studies, a series of local regions with different sizes were taken into consideration, and the results show that, when the temperature of a local region is far away from the critical temperature of the system, no matter the local region size is large or small, the results obtained by our calculations are almost the same; however, when the temperature of the local region is near to the critical point, the difference between the calculated results is very large if the local region size is very small, but this difference will decrease quickly when the local region size is enlarged, and the calculation results will tend to be the same when the local region size is larger than a certain value. Furthermore, the numerical results show that the effective local region size is almost unaffected by the temperature gradient of the system. It is clear that the local region and expanded Monte Carlo method provide us an efficient numerical method to study the non-equilibrium steady state.
New quantum Monte Carlo study of quantum critical phenomena with Trotter-number-dependent finite-size scaling and non-equilibrium relaxation
