Neural network impurity solver for real-frequency dynamical mean-field theory

TL;DR

This work addresses the need for fast, accurate real-frequency DMFT impurity solvers by introducing a transformer-based neural network that maps the hybridization function $oldsymbol{Δ}(ω)$ and impurity parameters $oldsymbol{p}=[U, ε_d, α]$ to the spectral function $ ilde{A}(ω)$. The model is trained on high-quality complex-time impurity data and regularized with a derivative loss $ig\\|rac{∂}{∂α} ilde{A}^{ ext{pred}}-rac{∂}{∂α} ilde{A}^{ ext{data}}igold|$, enabling stable extrapolation to real frequency via $A(ω) ≈ A(ω, α) - α ∂_α A(ω, α)$. With a cross-attention fusion of parameter and hybridization embeddings and a Softplus output to enforce positivity, the solver achieves quantitative accuracy across metallic, correlated, and insulating regimes in the half-filled and doped Hubbard models on the Bethe lattice, and remains stable within the DMFT self-consistency loop. The results suggest practical real-frequency DMFT calculations and potential extensions to real-space DMFT and multi-band impurities, enabling rapid, high-accuracy simulations of strongly correlated materials.

Abstract

We introduce a neural network impurity solver for real-frequency DMFT that employs a multihead cross-attention mechanism to map hybridization functions to spectral functions, conditioned on impurity parameters. Trained on high-quality MPS data from complex contour time evolution and incorporating derivative constraints with respect to the complex-time angle, our model achieves smooth generalization to the real-frequency axis. Benchmarking on the single-band Hubbard model for the Bethe lattice demonstrates quantitative accuracy across metallic, strongly correlated, and insulating regimes.

Figures (9)

• Figure 1: Schematic illustration of the neural network impurity solver. (a) The impurity solver takes as input the hybridization function $\Delta(\omega)$ together with the physical parameters $p=[U,\epsilon_d,\alpha]$, and produces the corresponding spectral function $\tilde{A}(\omega)$. (b) Neural network architecture of the solver. The hybridization function and the physical parameters are processed separately through FAN layers and feedforward networks, respectively. Positional encodings are added to these representations, which are then passed through a transformer block, followed by fitting layers and a softplus projection to generate the predicted spectral function $\tilde{A}(\omega)$. (c) Internal structure of a transformer block. The hybridization function and parameter representations are first normalized (Layer norm) and then coupled through multihead attention. The attention output is followed by a second Layer norm and a feedforward network, each equipped with residual connections.
• Figure 2: Spectral functions for different interaction strengths $U$. Panels (a)--(c) show results obtained without the $\alpha$-derivative constraint, while panels (d)--(f) correspond to results with the constraint. Thick transparent lines are reference results from the MPS real-time solver, and dashed lines denote predictions from the neural network solver. Insets show zoomed regions around $\omega=0$.
• Figure 3: Self-energy for the same interaction strengths $U$ as in Fig. \ref{['fig:dmftnn_half_spectral']}, showing results without( Panels (a)-(c) ), and with( Panels (d)-(e) ) the $\alpha$-derivative constraint. Thick lines represent real-time MPS reference results, and dashed lines show neural network predictions. Insets highlight the low-frequency region near $\omega=0$, and the red dotted line indicates a Fermi liquid behavior of $-\text{Im}\Sigma(\omega)\sim\omega^2$.
• Figure 4: Quasiparticle weight predicted by the neural network solver compared to the reference.
• Figure 5: Spectral functions for hole-doped cases with $\langle \hat{n}_\sigma\rangle\sim 0.4$. Panels (a)--(c) show results for $U/D = 1.35$, and panels (d)--(f) correspond to $U/D = 1.80$. Thick lines represent reference results from the real-time MPS solver, and dashed lines denote predictions from the neural network solver with $\gamma=1$.