Self-learning Monte Carlo method and cumulative update in fermion systems

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.

Figure 1

The trained parameters $J_n$ for the effective model in Eq. (8) for $L=4$.

Figure 2

For ${10}^5$ independent spin configurations on the Markov chain of $H$, the blue histogram is the distribution of the energy difference $\mathcal{E}\left(S\right)$ of these configurations between the original model $H\left(S\right)$ and the effective model $H_{\mathrm{eff}}\left(S\right)$. The red curve is the fitted Gaussian distribution with $\sigma =0.012$. The upper-left and upper-right insets are distributions of $E\left(S\right)$ and $E_{\mathrm{eff}}\left(S\right)$, respectively.

Figure 3

The autocorrelation function $C\left(\tilde{\tau }\right)=\langle M\left(\tilde{\tau }\right)M(\tilde{\tau }+\mathrm{\Delta }\tilde{\tau })\rangle -{\langle M\left(\tilde{\tau }\right)\rangle }^2$ of the magnetization with $\tilde{\tau }$ (number of EDs) as the time unit for both the SLMC and the conventional method on the system with $L=4$. Here, for SLMC we choose $l_{\text{c}}=16\times 4^3$. The dotted curves are the fitted ${\tilde{\tau }}_0$ based on the equation $C\left(\tilde{\tau }\right)=C\left(0\right)exp(-\tilde{\tau }/{\tilde{\tau }}_0)$.

Figure 4

The autocorrelation time ${\tilde{\tau }}_0$ for both the SLMC and the conventional method on systems with $L=4,6,\text{and}8$. The red dashed curve, obtained through extrapolating the two data points with the well-known dynamical exponent $z\simeq 2$, shows the scaling behavior of the conventional method [38].