Neural network as low-cost surrogates for impurity solvers in quantum embedding methods

Abstract

A promising application of ML is in creating low-cost surrogate models to replace computational bottlenecks in quantum many-body simulations. Here, we explore whether a NN can be trained in the low-data regime, with one to two orders of magnitude fewer training examples than previous works, as an efficient substitute for the impurity solver in DMFT simulations of correlated electron models. We show that the NN solver achieves accuracy comparable to popular CTQMC impurity solvers when interpolating between samples within the training set. While the NN's performance decreases notably when extrapolating to lower temperatures outside the training distribution, its output still provides an excellent initial guess for input to more accurate CTQMC impurity solvers, thus accelerating the time to solution up to a factor of five. We discuss our results in the context of rapid phase-space exploration.

Figures (7)

• Figure 1: (a) Left: The DMFT self-consistency loop. The computationally expensive part is the many-body calculation used to solve the impurity problem. Our goal is to replace this solver with an efficient ML architecture. Right: The NN used to replace the impurity solver in the DMFT loop, consists of $4$ fully-connected dense layers of neurons with GELU activations. It accepts the Weiss mean-field Green's function $G_0(\tau)$, with the inverse temperature $\beta$ and the Hubbard interaction strength $U$ as input features, and predicts the impurity site's self-energy $\Sigma(\tau)$. (b) Illustration of the data generation by the single shot impurity CTQMC solver for a fixed $(U_{i},\beta_{i})$ grid for $10$ synthetic Weiss fields $G_{0}^{n}$ generated by finite pole expansion and the corresponding self-energies $\Sigma^{(n)}(\tau)$, both in terms of their Legendre coefficients $G_{0, i}^{l=0,2,\dots,58}$ and $\Sigma_{i}^{l=0,2,\dots,58}$, respectively, forming the features and target pairs for training of the NN $(\hat{{X}}_{32}, \hat{{Y}}_{30})$.
• Figure 2: Imaginary-time Green's functions $G(\tau)$ for (a),(b) a single DMFT iteration and (c),(d) fully converged solutions. Results are shown for parameters in the metallic ($U/t = 2.50$, $\beta t = 6$, left column) and insulating ($U/t = 9.0$, $\beta t= 30$, right column) regions of the phase diagram. We obtain excellent agreement between the CTHYB QMC solver and the NN solver; the NN solver reproduces both $G(\tau)$ and $G_{0}(\tau)$ for the fully converged solutions, with an RMSE of $\mathrm{10}^{-3}$ at convergence. Both solvers achieve convergence in a comparable number of DMFT iterations; the QMC solver, however, required a runtime of $\sim 2.1\times10^{3}~\mathrm{s}$ compared to the NN solver, which converged in $\sim 1.6\times10^{-1}~\mathrm{s}$, both timed on a single Intel Xeon Gold $6248R$ core ($3.00~\text{GHz}$) on the ISAAC cluster ISAAC.
• Figure 3: A comparison of the convergence of the DMFT loop using either the QMC (CTHYB) solver or its low-cost surrogate NN. Both solvers were run for $U/t = 5$ and $\beta t = 20$ and converged to root-mean-square error (RMSE) $\epsilon \le \epsilon_\mathrm{tol} = 0.001$ in 17 iterations.
• Figure 4: The evolution of (a) the low-frequency slope in the imaginary part of the self-energy $\Sigma(i\omega_n)$ and (b) double occupancy on the impurity site as a function of $U/t$ at a fixed $\beta t = 25$. Results are shown for the converged solutions obtained using a QMC (blue) and NN (red) impurity solver. Two curves are shown in each case. One is obtained starting with an insulating solution and sweeping downward in $U/t$ to estimate $U_{\mathrm{c}1}$ ($\square$) while the other starts with a metallic solution and sweeps upward in the interaction to obtain $U_{\mathrm{c}2}$ ($\bigcirc$).
• Figure 5: Double occupancy $D = \langle n_\uparrow n_\downarrow \rangle$ phase diagram tracked by the NN across uniform $10^{4}$ phase points in the $(U/t,\, T/t)$ grid by sweeping upwards in $U/t$ for (a) metallic and downwards for the (b) insulating branch. Circles and triangles denote the QMC spinodal points $U_{\mathrm{c}2}$ and $U_{\mathrm{c}1}$, respectively, extracted from the sign change in the low-frequency self-energy slope (see Fig. \ref{['fig:hysteresis']}). The hollow white squares represent the NN training grid. The qualitatively accurate results of the NN for the phase diagram and coexistence region signify the interpolation and extrapolation capabilities of the NN impurity solver.