Self-Learning Monte Carlo study of Holstein Model

We design a self-learning Monte Carlo algorithm for the study of Holstein model at half filling, which is notorious for its slow Monte Carlo dynamics in a wide temperature range across the charge-density-wave phase transition. Self-learning Monte Carlo method extracts an effective bosonic Hamiltonian that captures the low-energy physics of the problem originated from the coupling between electrons and phonons. The update of the effective Hamiltonian is more efficient and greatly reduces the auto-correlation among the phonon configurations. Because of its considerable speedup over traditional Determinant quantum Monte Carlo method, we are able to simulate Holstein model with much larger system sizes and study the scaling behaviors near the critical point and acquire the critical exponents with high accuracy.