Efficient sampling using macrocanonical Monte Carlo and density of states mapping

In the context of Monte Carlo sampling for lattice models, the complexity of the energy landscape often leads to Markov chains being trapped in local optima, thereby increasing the correlation between samples and reducing sampling efficiency. This study proposes a Monte Carlo algorithm that effectively addresses the irregularities of the energy landscape through the introduction of the estimated density of states. This algorithm enhances the accuracy in the study of phase transitions and is not model specific. Although our algorithm is primarily demonstrated on the two-dimensional square lattice model, the method is also applicable to a broader range of lattice and higher-dimensional models. Furthermore, the study develops a method for estimating the density of states of large systems based on that of smaller systems, enabling high-precision density of states estimation within specific energy intervals in large systems without additional sampling. For regions of lower precision, a reweighting strategy is employed to adjust the density of states to enhance the precision further. This algorithm is not only significant within the field of lattice model sampling but may also inspire applications of the Monte Carlo method in other domains.