Symmetry Enforced Self-Learning Monte Carlo Method Applied to the Holstein Model

Self-learning Monte Carlo method (SLMC), using a trained effective model to guide Monte Carlo sampling processes, is a powerful general-purpose numerical method recently introduced to speed up simulations in (quantum) many-body systems. In this work, we further improve the efficiency of SLMC by enforcing physical symmetries on the effective model. We demonstrate its effectiveness in the Holstein Hamiltonian, one of the most fundamental many-body descriptions of electron-phonon coupling. Simulations of the Holstein model are notoriously difficult due to the combination of the typical cubic scaling of fermionic Monte Carlo and the presence of extremely long autocorrelation times. Our method addresses both bottlenecks. This enables simulations on large lattices in the most difficult parameter regions, and evaluation of the critical point for the charge density wave transition at half-filling with high precision. We argue that our work opens a new research area of quantum Monte Carlo (QMC), providing a general procedure to deal with ergodicity in situations involving Hamiltonians with multiple, distinct low energy states.