Neural-Network Quantum Embedding Solvers for Correlated Materials

Abstract

Quantum impurity solvers are the computational bottleneck of quantum embedding approaches to correlated materials, such as dynamical mean-field theory (DMFT). We show that neural networks trained on synthetic, material-agnostic data learn the impurity mapping from hybridization functions and local interactions to Green's functions with quantitative accuracy for both model systems and real materials, providing fast solvers for single- and multi-orbital models. Benchmarks against numerically controlled quantum Monte Carlo show that the method reproduces the Mott transition, multi-orbital phase diagrams of Hubbard-Kanamori models, and the electronic properties of SrVO$_3$ and SrMnO$_3$. The learned solvers achieve orders-of-magnitude speedup and can initialize controlled calculations, dramatically accelerating DMFT while preserving accuracy.

Figures (14)

• Figure 1: Comparison of the $G$Net solver (symbols) with CTSEG (lines) for the half-filled one orbital DMFT solution of the Hubbard model on the Bethe lattice. Left column: $U$-scan at $\beta=50$. Right column: $\beta$-scan at $U = 4.6$. (a,c) $G(\tau)$ (a) and Im$\Sigma(i\omega)$ (c) for $U \in \{1,7\}$. On panel (c), for $U = 1$, Im$\Sigma(i\omega)$ is multiplied by 10 for readability. (b,d) Same for $\beta = 10,70$. Inset (c) $-\mathrm{Im}G(i\omega_0)$ vs $U$ showing DMFT metal-insulator transition hysteresis, with grey (black) arrows indicating insulating (metallic) branches. Inset (d) $-\mathrm{Im}G(i\omega_0)$ vs $\beta$, close to the Mott transition.
• Figure 2: (a) Metal-insulator phase boundaries obtained using $G$Net for 2-orbital at $\beta=50$ (markers), compared with the CTHYB results from Ref. werner2007high. The colored squares denote the parameter values at which the Green's functions are directly compared with CTHYB in panels (b) and (c). Blue squares denote $J/U=0$, while orange/red squares denote $J/U=0.25$. Crosses in panels (b) and (c) denote the NN solutions, whereas solid lines denote the CTHYB results. The insets show the NN-predicted $G(\beta/2)$ along horizontal cuts in the phase diagram. The vertical dotted lines in the insets correspond to the position of the phase transition taken from Ref. werner2007high.
• Figure 3: DFT+DMFT results for SrVO$_3$ (left) and SrMnO$_3$ (right) at $\beta = 10$ using $G$Net (crosses) for $M=3$ orbitals at $\beta=10$, CTHYB (solid blue), and a converged computation with the $G$Net solver followed by one iteration with CTHYB (NN+1$\times$CTHYB, dashed red). For each material, we show $G(\tau)$ (top), Im$\Sigma(i\omega)$ (bottom), and Im$G(i\omega)$ (bottom, insets). Top insets: impurity occupancy $n(\mu)$ with the NN density-solver (crosses) and CTHYB (circles).
• Figure 4: Convergence of the impurity occupancy $n_{\mathrm{imp}}$ (a), $\mathrm{Im}\Sigma(i\omega_0)$ (b), $\mathrm{Im}G(i\omega_0)$ (inset), vs the iterations of the DMFT self-consistency loop, using CTHYB, starting from a DFT solution (circle, blue) and updating $\mu$ or a converged DMFT computation using $G$Net and density solvers (cross, red), for the case of SrVO$_3$. Here, $i\omega_0$ denotes the first Matsubara frequency.
• Figure 5: Upper panel: Test loss of the $1$-orbital NN solver as a function of number of epochs. Lower panels: Averaged distance as defined in Eg. (\ref{['eq:G_diff_tau']}), for epochs $50$ (left) and $3000$ (right). The line and crosses are this averaged distance, the blue shaded regions indicate the standard deviation of this measure over the test set.