Operator Lanczos Approach enabling Neural Quantum States as Real-Frequency Impurity Solvers

TL;DR

This work presents a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction, which has excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies.

Abstract

To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard-Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems.

Figures (4)

• Figure 1: Graphical representation of the SCOL decomposition Eq. \ref{['eq:scol_decomp']} of a three-orbital Hamiltonian with segment depth $l_s = 3$ and $N_s = L_c/l_s$.
• Figure 2: Single-impurity Anderson model benchmark for the NQS+SCOL impurity solver. Panel (a) shows the log--log spectral function $\pi \mathcal{A}(\omega)$ for three interaction strengths $U=0.8, 1.6, 3.2$ and for several commutator segment depths $l_s$, compared to NRG reference data (black line). The SCOL set for all $l_s$ has $N_s = L_c/l_s$ and $N_{ab}=2^{14}$ samples are used in Eq. \ref{['eq:ONestimation']}. The corresponding spectra on a linear frequency axis $\omega/U$ are shown in panel (b), highlighting the quasiparticle peak and the Hubbard side bands. In this panel we use $l_s=8$ for $U=0.8$ and $U=1.6$, and $l_s=7$ for $U=3.2$. The imaginary part of the self-energy is presented in panel (c), $-\mathfrak{Im}\,\Sigma(\omega)/U$ again compared to NRG. The bath is a semi-circular band of half-bandwidth $D=2$, a Wilson discretization parameter $\Lambda=3$ and a Wilson chain of length $L_c=35$.
• Figure 3: Panel (a) shows the spectral function $\pi \mathcal{A}(\omega)$ for the two-orbital (MO2) Hubbard--Kanamori model, obtained with NQS+SCOL and compared to NRG. The imaginary part of the MO2 self-energy, $-\mathfrak{Im}\,\Sigma(\omega)$, is shown in panel (b). All calculations for MO2 were done for $\Lambda=3$ with chain length $L_c=29$, segment depth $l_s=5$, $N_s = L_c/l_s$ and $N_{ab} = 2^{12}$. Panel (c) shows the spectral function $\pi \mathcal{A}(\omega)$ for the three-orbital Hubbard--Kanamori impurity (MO3) for segment depths $l_s = 2,3,4$, $N_s = L_c/(l_s+2)$ and $N_{ab} = 2^{12}$, compared to the NRG reference with $L_c = 31$.
• Figure 4: Schematic representation of the combinatorial phase map (CPM) that supplies the complex phase of the NQS wave function.