Symmetry-enforced self-learning Monte Carlo method applied to the Holstein model

The 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 Rapid Communication, 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 a 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 an evaluation of the critical point for the charge density wave transition at half filling with high precision. We argue that our work opens a research area of quantum Monte Carlo, providing a general procedure to deal with ergodicity in situations involving Hamiltonians with multiple, distinct low-energy states.

Figure 1

The symmetric functions used to construct the phonon potential have minima at $\pm \left|\alpha \right|$ with $\alpha =-\frac{g}{{\mathrm{\Omega }}^2}$. The blue line is $\frac{1}{4}{(X-\alpha )}^4-\frac{{\alpha }^2}{2}{(X-\alpha )}^2$ and the red line is $\frac{1}{6}{(X-\alpha )}^6-\frac{{\alpha }^2}{4}{(X-\alpha )}^4$.

Figure 2

(a) Comparison of autocorrelation time of the CDW structure factor vs $L$ for DQMC and SLMC. Simulations were done at the critical point $T_c$ for the CDW transition. SLMC greatly suppresses the autocorrelation time, with dynamical exponent $z\sim 2.9$, while for DQMC, $z\sim 5.1$. (b) Comparison of CPU time to obtain one statistically independent configuration (${\tau }_L$ sweeps) between DQMC and SLMC. Power-law fitting gives $\sim L^{11}$ for DQMC and $\sim L^7$ for SLMC. Including the prefactor, SLMC provides a $\times 50$ speedup for $L=12$ and $\times 300$ speedup for $L=20$.

Figure 3

(a) Finite-size scaling analysis showing $S_{\text{CDW}}$ $L^{-7/4}$ vs $T/t$. The critical point $T_c=0.244\left(3\right)$ is determined from the crossing of finite-size data. (b) Data collapse of $S_{\text{CDW}}$ $L^{-7/4}$ vs $L^{1/\nu }$ $(T-T_c)/T_c$ with $\nu =1$. Note the quality of the collapse is better than that in Ref. [34], due to the larger $L$ obtained with SLMC.