Accelerated continuous time quantum Monte Carlo method with machine learning

An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a dynamical mean field theory (DMFT) approach for the description of strongly correlated systems. The inversion of the $k\times k$ matrix with $k^2$ operations given by the diagram expansion order $k$ in the CTQMC fast update and the multiplication of the $k\times k$ matrix, and the noninteracting properties with $k\times {\omega }_{n_{\text{max}}}^{m-1}$ operations to measure the $m$-point correlators, are computationally time consuming. Here we propose the CTQMC method in combination with a machine learning technique, which eliminates the $k\times {\omega }_{n_{\text{max}}}$ and $k\times {\omega }_{n_{\text{max}}}^3$ operations for the two-point impurity Green's functions $G_{\sigma }\left(i{\omega }_n\right)$ and four-point vertices ${\chi }_{\sigma ,\overline{\sigma }}(i{\omega }_{n{}_1},i{\omega }_{n{}_2},i{\omega }_{n{}_3},i{\omega }_{n{}_4})$, respectively. This method not only predicts the accurate physical properties at low temperature, but also dramatically decreases the computational times of ${\chi }_{\sigma ,\overline{\sigma }}(i{\omega }_{n{}_1},i{\omega }_{n{}_2},i{\omega }_{n{}_3},i{\omega }_{n{}_4})$ for the nonlocal extension of DMFT approximation.

Figure 1

Schematic architecture of a one-dimensional convolutional autoencoder. The input imaginary time ${\tau }_k$, which is obtained from a continuous time quantum Monte Carlo (CTQMC) approach, is an array of a $k$ size, where $k$ is a diagram expansion order. The deep layered structure with three convolution and deconvolution layers is employed for a machine learning (ML) process. After the final deconvolution layer, we add one fully connected layer to match the desired number of output nodes. The two-point impurity Green's functions $G_{\sigma }\left(i{\omega }_n\right)$ and four-point vertices ${\chi }_{\sigma ,\overline{\sigma }}(i{\omega }_{n{}_1},i{\omega }_{n{}_2},i{\omega }_{n{}_3},i{\omega }_{n{}_4})$ are output datasets.

Figure 2

(a) Imaginary part of the impurity Green's function $\text{Im}\left[G_{\sigma }\left(i{\omega }_n\right)\right]$ as a function of ${\omega }_n$ at half-filling for $U=2.0$ and 3.0. “Exact” means that the exact results are obtained from Eq. (4). “$\mathrm{CTQMC}+\mathrm{ML}$” is the continuous time quantum Monte Carlo in combination with the ML. The trainings of ML for $U=2.0$ and 3.0 are independently performed to obtain each $\text{Im}\left[G_{\sigma }\left(i{\omega }_n\right)\right]$. We use the conventional CTQMC measurements of Eq. (4) with $6\times {10}^5$ and $8\times {10}^4$ different combinations of a dataset for exact and $\mathrm{CTQMC}+\mathrm{ML}$ results, respectively. The bandwidth $W$ and the temperature $T$ are set to $W=2.0$ and $T=0.01$, respectively. The real parts of the impurity Green's functions are set to 0 at half-filling. (b) $\langle |\text{Im}\left[G_{\sigma }^{\text{Ex}}\left(i{\omega }_0\right)\right]-\text{Im}\left[G_{\sigma }^{\text{ML}}\left(i{\omega }_0\right)\right]|\rangle$ at first Matsubara frequency as numbers of the trained input dataset. Here $\langle |\text{Im}\left[G_{\sigma }^{\text{Ex}}\left(i{\omega }_0\right)\right]-\text{Im}\left[G_{\sigma }^{\text{ML}}\left(i{\omega }_0\right)\right]|\rangle$ mean the errors of each ML process, and they are decreasing with increasing the number of input dataset. Inset in (b): $\text{Im}\left[G_{\sigma }\left(i{\omega }_n\right)\right]$ as a function of ${\omega }_n$ for $0.2\times {10}^4$ and $8\times {10}^4$ different trained datasets at $U=2.0$.

Figure 3

(Left) Imaginary part of the impurity Green's function $\text{Im}\left[G_{\sigma }\left(i{\omega }_0\right)\right]$ at the first Matsubara frequency ${\omega }_0$ and (right) double occupancy $D$ as a function of $U$. The ML kernel is constructed on input and output data of both $U=2.0$ and 3.0, unlike the case with independent ML trainings for each $U=2.0$ and 3.0 in Fig. 2. $\text{Im}\left[G_{\sigma }\left(i{\omega }_0\right)\right]$ and $D$ between $U=2.0$ and 3.0 are predicted with the semisupervised learning which can predict intermediate circumstance of two limiting cases provided as the trained datasets.

Figure 4

(a) and (b) The real and the imaginary parts of $G_{\sigma }\left(i{\omega }_n\right)$ as a function of ${\omega }_n$, respectively, for $U=2.0$. The chemical potentials $\mu$ employed in Eq. (5) for dopings are 0.4 and 0.8.

Figure 5

(a) and (b) The real parts of four-point vertices ${\chi }_{\uparrow ,\downarrow }$ and ${\chi }_{\uparrow ,\uparrow }$ with $k\times {\omega }_{n_{\text{max}}}^3$ operations, respectively, for $U=2.0$ and $T=0.01$ at half-filling. Here ${\chi }_{\uparrow ,\downarrow }\left(i{\omega }_n^{\prime }\right)$ and ${\chi }_{\uparrow ,\uparrow }\left(i{\omega }_n^{\prime \prime }\right)$ mean ${\chi }_{\uparrow ,\downarrow }(i{\omega }_n^{\prime },i{\omega }_n^{\prime },i{\omega }_n^{\prime },i{\omega }_n^{\prime })$ and ${\chi }_{\uparrow ,\uparrow }(i{\omega }_{n=0},i{\omega }_{n=0},i{\omega }_n^{\prime \prime },i{\omega }_n^{\prime \prime })$, respectively.