Self-Learning Monte Carlo Method in Fermion Systems

Zi Yang Meng, Junwei Liu, Huitao Shen +2 more

James E. Gubernatis

S. D. Bloom, S. M. Grimes, Robert F. Hausman

Zi Yang Meng, Liang Fu, Junwei Liu +2 more

As a new general-purpose Monte Carlo method, self-learning Monte Carlo (SLMC) has been numerically demonstrated in boson systems. In this work, we propose a new type of update algorithm, dubbed cumulative update, which could be naturally integrated into SLMC. Based on many local updates in the self-learned effective model, cumulative update efficiently proposes a global move with a high acceptance probability in the original model. Cumulative update can reduce the conventional computational cost of a full sweep O(N4) to O(N3), and also effectively reduce the auto-correlation time to be 1. We numerically show its efficiency through the well-known double-exchange model. With cumulative update, the SLMC could be several hundreds of times faster than the conventional local update method. By employing the cumulative update, SLMC can be generally used in any sign-problem-free Fermion systems and maximize the performance of Monte Carlo simulation.

Junwei Liu, Huitao Shen, Yang Qi +2 more

We develop the self-learning Monte Carlo (SLMC) method, a general-purpose numerical method recently introduced to simulate many-body systems, for studying interacting fermion systems. Our method uses a highly efficient update algorithm, which we design and dub ``cumulative update'', to generate new candidate configurations in the Markov chain based on a self-learned bosonic effective model. From a general analysis and a numerical study of the double exchange model as an example, we find that the SLMC with cumulative update drastically reduces the computational cost of the simulation, while remaining statistically exact. Remarkably, its computational complexity is far less than the conventional algorithm with local updates.

Yasuhiro Hatsugai, Masatoshi Imada