Compressing local vertex functions from the multipoint numerical renormalization group using quantics tensor cross interpolation

TL;DR

This work tackles the compression of local two-particle vertices computed by mpNRG for the SIAM by representing imaginary- and real-frequency vertices as quantics tensor trains via QTCI. By combining partial spectral functions with Matsubara and Keldysh kernels and employing the symmetric improved estimator, the authors obtain compact representations of the core and full four-point vertex with bond dimensions on the order of a few hundred across a wide parameter range. Imaginary-frequency vertices reach accuracies around $0.002$ with moderate ranks ($\chi \lesssim 120$), while real-frequency vertices achieve about $0.02$ accuracy with $\chi \lesssim 170$, enabling high-resolution calculations beyond previous capabilities. This establishes a path toward tensor-train–based diagrammatic methods in correlated lattice models starting from non-perturbative mpNRG data and suggests feasibility for QTCI-enabled DMFT–parquet calculations with momentum dependence and real-frequency dynamics.

Abstract

The multipoint numerical renormalization group (mpNRG) is a powerful impurity solver that provides accurate spectral data useful for computing local, dynamic correlation functions in imaginary or real frequencies non-perturbatively across a wide range of interactions and temperatures. It gives access to a local, non-perturbative four-point vertex in imaginary and real frequencies, which can be used as input for subsequent computations such as diagrammatic extensions of dynamical mean--field theory. However, computing and manipulating the real-frequency four-point vertex on large, dense grids quickly becomes numerically challenging when the density and/or the extent of the frequency grid is increased. In this paper, we compute four-point vertices in a strongly compressed quantics tensor train format using quantics tensor cross interpolation, starting from discrete partial spectral functions obtained from mpNRG. This enables evaluations of the vertex on frequency grids with resolutions far beyond the reach of previous implementations. We benchmark this approach on the four-point vertex of the single-impurity Anderson model across a wide range of physical parameters, both in its full form and its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of $2\cdot 10^{-3}$ with maximum bond dimensions not exceeding 120. The more complex full real-frequency vertex requires maximum bond dimensions not exceeding 170 for an accuracy of $\lesssim 2\%$. Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, non-perturbative mpNRG vertex.

Figures (12)

• Figure 1: Maximum error of Matsubara (circles) and Keldysh (diamonds) core vertex evaluations with SVD truncations as described in the main text. The error is measured relative to the maximum of the respective vertex function. By $\tau$ we denote the target TCI tolerance, and use a cutoff of $S_{\text{cut}}=10^{-2}\tau$ for Matsubara. For Keldysh, we choose an SVD cutoff of $10^{-6}$ times the largest singular value, and find that this yields results that are sufficiently accurate for a tolerance of $\tau=10^{-3}$. We show maximum errors over 64000 sampling points for the Matsubara core vertex and $2\cdot10^6$ sampling points for the more complicated Keldysh core vertex. All errors are well below the respective target tolerance $\tau$.
• Figure 2: Singular values $S_i$ of regular Matsubara kernels (Eq. \ref{['eq:MatsubaraKernel']}, solid line) and broadened, fully retarded Keldysh kernels (Eq. \ref{['eq:keldysh1Dkerneldef']}, dashed line). We show kernels at inverse temperatures $\beta\in\{20,200,2000\}$ and interaction $u=0.5$. The frequency grids are bosonic with $2^{12}$ points, with the Keldysh grid ranging from $-0.65$ to $\omega_{\max}=0.65$.
• Figure 3: QTCI-compression of the Matsubara core vertex $\Gamma_{\mathrm{core}}(\omega,\nu,\nu')$ in the $p$-channel at $\beta=2000$, with $R=8$ and tolerance $\tau= 10^{-3}$. Heatmaps show the $\log_{10}$ absolute value of $\Gamma_{\mathrm{core}}^{\uparrow\uparrow}$ in (a,b) and $\Gamma_{\mathrm{core}}^{\uparrow\downarrow}$ in (d,e) on the slice $\omega=0$. $n_\nu$ and $n_{\nu'}$ enumerate the fermionic Matsubara frequencies $\nu,\nu'$. Left column: Reference data $\Gamma_{\mathrm{core}}^{\mathrm{ref}}$. Center column: QTCI representation $\Gamma_{\mathrm{core}}^{\mathrm{QTCI}}$. Right column: Normalized error $\varepsilon_{\boldsymbol{\sigma}}[\Gamma_{\mathrm{core}}]\lesssim1.58\cdot10^{-3}$ defined in Eq. \ref{['eq:tciconvergence']}. We reproduce key features of the vertex on a large frequency box with a comparatively low QTT rank of $\chi=107$ and $\chi=106$, respectively.
• Figure 4: (a) Rank and (b) RAM usage of the interleaved QTT representation of the Matsubara core vertex $\Gamma_{\mathrm{core}}^{\uparrow\uparrow}$ in the $p$-channel vs. frequency grid size for different tolerances. The grid has $2^R$ points in each frequency argument. For the target tolerance of $\tau=10^{-3}$, ranks saturate at $\chi \approx 100$. Dotted worst-case lines in (a) and (b) indicate the maximum rank of a $3R$-leg QTT (hence the even-odd alternation in the worst case of (a)) and the RAM requirements of dense grids with $2^{3R}$ points, respectively.
• Figure 5: Rank of the Matsubara core vertex $\Gamma_{\mathrm{core}}^{\uparrow\uparrow}$ and full vertex $\Gamma^{\uparrow\uparrow}$ in the $p$-channel versus (a) TCI tolerance $\tau$, (b) inverse temperature $\beta$ and (c) interaction strength $u$.