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arXiv:2610.28542

Acceleration of CTQMC via Denoising Diffusion Models with Auxiliary Spectral Functions

Rishi Rao, Li Zhu

Published Oct 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Continuous-time Quantum Monte Carlo (CTQMC) impurity solvers are a numerically exact, but expensive, option for solving the Dynamical Mean Field Theory (DMFT) equations, with statistical noise from undersampling presenting a major challenge to low temperature studies. This noise is especially egregious within the sampled self-energy, which can destabilize the self-consistency loop upon insertion into the Dyson equation. In this work, we introduce a neural denoiser for fermionic imaginary time Green's functions based on the diffusion model architecture trained on synthetic data. The network predicts positive and normalized auxiliary spectral functions consistent with the noisy imaginary axis CTQMC data, which is then refined via autodifferentiation to match the measured self-energy, its first two high-frequency moments, and impurity occupation similar to the Constrained Residual Minimization (CRM) method. This leads to Green's functions and self-energies that are causal by design. Through this denoising framework we are able to reduce the required CTQMC measurement steps by a factor of 10-100. We benchmark our method with a half-filled Hubbard model on a Bethe lattice, a two-patch Dynamical Cluster Approximation (DCA) of the two-dimensional Hubbard model, and the SrVO$_3$ system. We reproduce the Mott transition upon increasing Coulomb interaction for the half-filled Hubbard model with only 5000 CTQMC measurements, the pseudogap transition for the DCA two-dimensional Hubbard model with only 5$\times$10$^5$ measurements, and match the spectral function of SrVO$_3$ with 50 times fewer measurements than the reference. Since we generate auxiliary spectral functions, we recover the approximate spectral functions of the systems for free without performing analytic continuation.

DOI
10.48550/arxiv.2610.28542

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Continuous-time Quantum Monte Carlo (CTQMC) impurity solvers are a numerically exact, but expensive, option for solving the Dynamical Mean Field Theory (DMFT) equations, with statistical noise from undersampling presenting a major challenge to low temperature studies. This noise is especially egregious within the sampled self-energy, which can destabilize the self-consistency loop upon insertion into the Dyson equation. In this work, we introduce a neural denoiser for fermionic imaginary time Green's functions based on the diffusion model architecture trained on synthetic data. The network predicts positive and normalized auxiliary spectral functions consistent with the noisy imaginary axis CTQMC data, which is then refined via autodifferentiation to match the measured self-energy, its first two high-frequency moments, and impurity occupation similar to the Constrained Residual Minimization (CRM) method. This leads to Green's functions and self-energies that are causal by design. Through this denoising framework we are able to reduce the required CTQMC measurement steps by a factor of 10-100. We benchmark our method with a half-filled Hubbard model on a Bethe lattice, a two-patch Dynamical Cluster Approximation (DCA) of the two-dimensional Hubbard model, and the SrVO3_33​ system. We reproduce the Mott transition upon increasing Coulomb interaction for the half-filled Hubbard model with only 5000 CTQMC measurements, the pseudogap transition for the DCA two-dimensional Hubbard model with only 5×\times×105^55 measurements, and match the spectral function of SrVO3_33​ with 50 times fewer measurements than the reference. Since we generate auxiliary spectral functions, we recover the approximate spectral functions of the systems for free without performing analytic continuation.
We present an algorithm for solving the self-consistency equations of the dynamical mean-field theory (DMFT) with high precision and efficiency at low temperatures. In each DMFT iteration, the impurity problem is mapped to an auxiliary Hamiltonian, for which the Green function is computed by combining determinantal quantum Monte Carlo (BSS-QMC) calculations with a multigrid extrapolation procedure. The method is numerically exact, i.e., yields results which are free of significant Trotter errors, but retains the BSS advantage, compared to direct QMC impurity solvers, of linear (instead of cubic) scaling with the inverse temperature. The new algorithm is applied to the half-filled Hubbard model close to the Mott transition; detailed comparisons with exact diagonalization, Hirsch-Fye QMC, and continuous-time QMC are provided.
We develop a method for calculating the self-energy of a quantum impurity coupled to a continuous bath by stochastically generating a distribution of finite Anderson models that are solved by exact diagonalization, using the noninteracting local spectral function as a probability distribution for the sampling. The method enables calculation of the full analytic self-energy and single-particle Green's function in the complex frequency plane, without analytic continuation, and can be used for both finite and zero temperature at arbitrary fillings. Results are in good agreement with imaginary frequency data from continuous-time quantum Monte Carlo calculations for the single-impurity Anderson model and the two-orbital Hubbard model within dynamical mean-field theory (DMFT) as well as real frequency data for self-energy of the single-band Hubbard model within DMFT using the numerical renormalization group. The method should be applicable to a wide range of quantum impurity models and particularly useful when high-precision real frequency results are sought.
An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a dynamical mean field theory (DMFT) approach for the description of strongly correlated systems. The inversion of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix with k2{k}^{2}k2 operations given by the diagram expansion order kkk in the CTQMC fast update and the multiplication of the $k\ifmmode\times\else\texttimes\fi{}k$ matrix, and the noninteracting properties with $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{m\ensuremath{-}1}$ operations to measure the mmm-point correlators, are computationally time consuming. Here we propose the CTQMC method in combination with a machine learning technique, which eliminates the $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}$ and $k\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{{n}_{\text{max}}}^{3}$ operations for the two-point impurity Green's functions ${G}_{\ensuremath{\sigma}}(i{\ensuremath{\omega}}_{n})$ and four-point vertices ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$, respectively. This method not only predicts the accurate physical properties at low temperature, but also dramatically decreases the computational times of ${\ensuremath{\chi}}_{\ensuremath{\sigma},\overline{\ensuremath{\sigma}}}(i{\ensuremath{\omega}}_{n{}_{1}},i{\ensuremath{\omega}}_{n{}_{2}},i{\ensuremath{\omega}}_{n{}_{3}},i{\ensuremath{\omega}}_{n{}_{4}})$ for the nonlocal extension of DMFT approximation.
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 SrVO3_33​ and SrMnO3_33​. The learned solvers achieve orders-of-magnitude speedup and can initialize controlled calculations, dramatically accelerating DMFT while preserving accuracy.
The acceleration of exact continuous time quantum Monte Carlo (CTQMC) approaches in multi-site or multi-orbital systems is extremely interesting work, because these approaches are very time-consuming in terms of numerical computation and might account for the nature of exotic behaviors such as high-temperature superconductivity and Mott insulator behavior observed in the strongly correlated materials. We extend the recently developed interaction-expansion CTQMC method in combination with a machine learning (CTQMC+ML) approach for the single-site and single-orbital systems to multi-site and multi-orbital ones. This method can be applied to explore the nonlocal correlation effects in lattice models and to study the electronic structure of real materials via an ab-initio density functional theory plus dynamical mean field theory approach. We find that our CTQMC+ML method for multi-site (and multi-orbital) systems accurately predicts the impurity Green’s function with less computational time than the CTQMC approaches, as in the case of the single-site and single-orbital version.
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger UUU than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.