Self-learning Monte Carlo method: Continuous-time algorithm

The recently introduced self-learning Monte Carlo method is a general-purpose numerical method that speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. We implement this method in the framework of a continuous-time Monte Carlo method with an auxiliary field in quantum impurity models. We introduce and train a diagram generating function (DGF) to model the probability distribution of auxiliary field configurations in continuous imaginary time, at all orders of diagrammatic expansion. By using DGF to propose global moves in configuration space, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.

Figure 1

Schematic figure for the Markov chains in the original and self-learning continuous-time Monte Carlo methods to obtain an uncorrelated configuration. $n$ denotes the average expansion order that determines the size of the matrix $N_{\sigma }\left(\{s_i,{\tau }_i\}\right)$, and further determines the complexity of the simulation. See the last section of this Rapid Communication for a detailed discussion.

Figure 2

Effective interactions for different $U$ with $\beta =10$, $V=1$, and $K=1$.

Figure 3

For $5\times {10}^4$ independent configurations on the Markov chain of the original CT-AUX, the histogram is the distribution of the difference $ln{W}_i-ln{W}_i^{\mathrm{eff}}$. The upper-left and upper-right insets are distributions of $ln{W}_i$ and $ln{W}_i^{\mathrm{eff}}$, respectively. Here, $U=5$, $V=1$, $\beta =10$, and $K=1$.

Figure 4

Left: $U$ dependence of the autocorrelation time of the original CT-AUX with $\beta =10$. Right: $\beta$ dependence of the autocorrelation time with $U=2$. The other related parameters are $V=1$ and $K=1$. In both figures, the unit time is a local update (inserting/removing a auxiliary spin) in the original CT-AUX method.

Figure 5

Autocorrelation function of the auxiliary-spin magnetization for a system with $\beta =10$, $V=1$, and $K=1$. The unit time is defined in the main text. Inset: $U$ dependence of the autocorrelation time in the self-learning CT-AUX. We set the number of local updates on DGF to be $M_{\mathrm{eff}}=2\times {10}^3$ ($U=1,2,3,4$) and $M_{\mathrm{eff}}=5\times {10}^4$ ($U=5,6$).