Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.23160

Quantics tensor cross interpolation for high-order strong-coupling expansions

Kanghyeon Kim, Lei Geng, Philipp Werner, Aaram J. Kim

Published Aug 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Real-time impurity solvers enable the study of transport phenomena and the description of nonequilibrium lattice systems within the framework of dynamical mean-field theory (DMFT). They also provide direct access to the spectral functions of both equilibrium and nonequilibrium systems. A widely used approach is the self-consistent strong-coupling expansion, whose lowest-order implementation corresponds to the non-crossing approximation. Higher-order implementations, however, are computationally demanding because the number of diagram topologies grows factorially with expansion order, while the evaluation of self-energies and Green's functions requires increasingly high-dimensional integrations. Here, we demonstrate that the latter challenge can be mitigated by employing quantics tensor cross interpolation in a variable-separated framework. Compared with the previously used scale-separated approach, the new scheme yields substantially lower bond dimensions and capacitates self-consistent steady-state DMFT calculations up to fourth order. We illustrate its performance with representative results for both equilibrium and photo-doped systems. In addition, we analyze the convergence of the strong-coupling expansion in the challenging noninteracting limit by computing diagrams up to sixth order. At this order, the onset of the asymptotic regime of the strong-coupling expansion becomes apparent, which allows the application of extrapolation techniques.

DOI
10.48550/arxiv.2608.23160

Related papers

Accelerating a Strong-Coupling Non-Equilibrium Steady-State Impurity Solver using (Quantics) Tensor Trains

Bastian Schindler, Martin Eckstein

Including higher order diagrammatic corrections to the strong-coupling expansion is mainly limited by the evaluation of high-dimensional, time-ordered integrals. In this work we present and compare four different parametrizations of the integrands in order to obtain a low-rank (quantics) tensor-train representation using tensor cross interpolation. Particular emphasis is placed on a quantics time-difference formulation in which the required retarded convolutions are performed directly in quantics tensor-train form. Using controlled Gaussian benchmarks, we analyze the accuracy, bond dimensions, and computational scaling of the different approaches. We then validate the most promising formulations in self-consistent equilibrium and nonequilibrium DMFT calculations and demonstrate calculations up to the third order in the strong-coupling expansion. Finally, we extend the solver to impurity models with retarded density-density interactions and apply it within nonequilibrium extended DMFT. Our results show that tensor cross interpolation substantially reduces the cost of evaluating higher-order diagrams and provides a controlled, systematically improvable framework for nonequilibrium quantum impurity calculations.

arXiv:2608.13146•Aug 13, 2026•0 citations

Strong coupling impurity solver based on quantics tensor cross interpolation

Aaram J. Kim, Philipp Werner

Numerical methods capable of handling nonequilibrium impurity models are essential for the study of transport problems and the solution of the nonequilibrium dynamical mean-field theory (DMFT) equations. In the strong correlation regime, the self-consistently resummed hybridization expansion is an appealing strategy, which however has been employed so far mainly in the lowest-order noncrossing approximation. At higher orders, standard implementations become numerically costly, but a significant speed-up can be achieved by evaluating multidimensional integrals in an approximate factorized form. Here we develop a one-crossing approximation solver based on the recently introduced quantics tensor cross interpolation, and demonstrate its accuracy and efficiency with applications to the Anderson impurity model and nonequilibrium steady-state DMFT calculations for the Hubbard model.

arXiv:2411.19026•Physical review. B./Physical review. B•Mar 10, 2025•15 citations

Third-order strong-coupling impurity solver for real-frequency dynamical mean field theory: Accurate spectral functions for antiferromagnetic and photodoped states

Lei Geng, Aaram J. Kim, Philipp Werner

Strong-coupling expansions for impurity models up to third order are implemented here on the real-time axis using a quantics tensor cross interpolation framework. This approach provides an accurate impurity solver that can describe both equilibrium systems and nonequilibrium steady states in the intermediate to strong-interaction regime.

arXiv:2507.20385•Physical review. B./Physical review. B•Nov 8, 2025•3 citations

Weak-coupling tensor cross interpolation impurity solver for nonequilibrium dynamical mean-field theory

Shuta Matsuura, Hiroshi Shinaoka, Philipp Werner +1 more

Simulating nonequilibrium quantum many-body systems remains a major challenge due to the exponential growth of the computational complexity with real time. Here we implement a nonequilibrium impurity solver based on the weak-coupling expansion and the tensor cross interpolation (TCI), and apply it to nonequilibrium dynamical mean-field theory (DMFT). The method approximates the integrands of the high-dimensional integrals arising in the weak-coupling expansion in a tensor-train form, enabling efficient evaluations without stochastic sampling and thereby mitigating the sign problem affecting continuous-time quantum Monte Carlo (CT-QMC) methods. Benchmark calculations for an exactly solvable nonequilibrium impurity model agree well with the exact results and reveal a low-rank structure of the integrands. When applied to interaction-quench problems in the half-filled Hubbard model, the method reproduces fast thermalization at a critical interaction strength with accuracy comparable to CT-QMC. Away from half filling, where the sign problem becomes even more severe, the present approach remains well controlled, revealing a crossover instead of a sharply defined fast thermalization point in the 3/4-filled case. The solver can also be applied to steady-state DMFT problems, yielding accurate spectral functions in the metallic regime without analytic continuation.

arXiv:2607.00702•Physical review. B./Physical review. B•Oct 14, 2026•0 citations

Solving quantum impurity models in the non-equilibrium steady state with tensor trains

Martin Eckstein

We discuss the evaluation of the integrals for intermediate-order diagrams in the self-consistent strong-coupling expansion on the Keldysh contour using Tensor Cross Interpolation (TCI). TCI is used to factorize the nested parts of the integrand, allowing the integral to be computed as a recursion of convolution integrals, which are efficiently evaluated using the Fast Fourier Transform. The evaluation of diagrams where all vertices lie on one branch of the Keldysh contour resembles the structure of the imaginary-time formalism. For diagrams with time arguments on both contour branches, we find that it can be advantageous to parametrize the integrals in terms of physical time arguments with an additional sum over Keldysh indices. We benchmark the solution in relevant test cases, including the single impurity Anderson model and an exactly solvable electron-boson model. While the bond dimension increases with diagram order, the TCI-based integration efficiently handles low-order diagrams, making it a promising approach to go beyond the non-crossing approximation in steady-state non-equilibrium dynamical mean-field theory simulations.

arXiv:2410.19707•Oct 25, 2024•1 citation

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

Markus Frankenbach, Marc K. Ritter, Mathias Pelz +3 more

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 nonperturbatively across a wide range of interactions and temperatures. It gives access to a local, nonperturbative 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 in its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of 2 × 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 ≲ 2 % . Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, nonperturbative mpNRG vertex.

arXiv:2506.13359•Physical Review Research•Aug 27, 2025•1 citation
Real-time impurity solvers enable the study of transport phenomena and the description of nonequilibrium lattice systems within the framework of dynamical mean-field theory (DMFT). They also provide direct access to the spectral functions of both equilibrium and nonequilibrium systems. A widely used approach is the self-consistent strong-coupling expansion, whose lowest-order implementation corresponds to the non-crossing approximation. Higher-order implementations, however, are computationally demanding because the number of diagram topologies grows factorially with expansion order, while the evaluation of self-energies and Green's functions requires increasingly high-dimensional integrations. Here, we demonstrate that the latter challenge can be mitigated by employing quantics tensor cross interpolation in a variable-separated framework. Compared with the previously used scale-separated approach, the new scheme yields substantially lower bond dimensions and capacitates self-consistent steady-state DMFT calculations up to fourth order. We illustrate its performance with representative results for both equilibrium and photo-doped systems. In addition, we analyze the convergence of the strong-coupling expansion in the challenging noninteracting limit by computing diagrams up to sixth order. At this order, the onset of the asymptotic regime of the strong-coupling expansion becomes apparent, which allows the application of extrapolation techniques.
Including higher order diagrammatic corrections to the strong-coupling expansion is mainly limited by the evaluation of high-dimensional, time-ordered integrals. In this work we present and compare four different parametrizations of the integrands in order to obtain a low-rank (quantics) tensor-train representation using tensor cross interpolation. Particular emphasis is placed on a quantics time-difference formulation in which the required retarded convolutions are performed directly in quantics tensor-train form. Using controlled Gaussian benchmarks, we analyze the accuracy, bond dimensions, and computational scaling of the different approaches. We then validate the most promising formulations in self-consistent equilibrium and nonequilibrium DMFT calculations and demonstrate calculations up to the third order in the strong-coupling expansion. Finally, we extend the solver to impurity models with retarded density-density interactions and apply it within nonequilibrium extended DMFT. Our results show that tensor cross interpolation substantially reduces the cost of evaluating higher-order diagrams and provides a controlled, systematically improvable framework for nonequilibrium quantum impurity calculations.
Numerical methods capable of handling nonequilibrium impurity models are essential for the study of transport problems and the solution of the nonequilibrium dynamical mean-field theory (DMFT) equations. In the strong correlation regime, the self-consistently resummed hybridization expansion is an appealing strategy, which however has been employed so far mainly in the lowest-order noncrossing approximation. At higher orders, standard implementations become numerically costly, but a significant speed-up can be achieved by evaluating multidimensional integrals in an approximate factorized form. Here we develop a one-crossing approximation solver based on the recently introduced quantics tensor cross interpolation, and demonstrate its accuracy and efficiency with applications to the Anderson impurity model and nonequilibrium steady-state DMFT calculations for the Hubbard model.
Simulating nonequilibrium quantum many-body systems remains a major challenge due to the exponential growth of the computational complexity with real time. Here we implement a nonequilibrium impurity solver based on the weak-coupling expansion and the tensor cross interpolation (TCI), and apply it to nonequilibrium dynamical mean-field theory (DMFT). The method approximates the integrands of the high-dimensional integrals arising in the weak-coupling expansion in a tensor-train form, enabling efficient evaluations without stochastic sampling and thereby mitigating the sign problem affecting continuous-time quantum Monte Carlo (CT-QMC) methods. Benchmark calculations for an exactly solvable nonequilibrium impurity model agree well with the exact results and reveal a low-rank structure of the integrands. When applied to interaction-quench problems in the half-filled Hubbard model, the method reproduces fast thermalization at a critical interaction strength with accuracy comparable to CT-QMC. Away from half filling, where the sign problem becomes even more severe, the present approach remains well controlled, revealing a crossover instead of a sharply defined fast thermalization point in the 3/4-filled case. The solver can also be applied to steady-state DMFT problems, yielding accurate spectral functions in the metallic regime without analytic continuation.
We discuss the evaluation of the integrals for intermediate-order diagrams in the self-consistent strong-coupling expansion on the Keldysh contour using Tensor Cross Interpolation (TCI). TCI is used to factorize the nested parts of the integrand, allowing the integral to be computed as a recursion of convolution integrals, which are efficiently evaluated using the Fast Fourier Transform. The evaluation of diagrams where all vertices lie on one branch of the Keldysh contour resembles the structure of the imaginary-time formalism. For diagrams with time arguments on both contour branches, we find that it can be advantageous to parametrize the integrals in terms of physical time arguments with an additional sum over Keldysh indices. We benchmark the solution in relevant test cases, including the single impurity Anderson model and an exactly solvable electron-boson model. While the bond dimension increases with diagram order, the TCI-based integration efficiently handles low-order diagrams, making it a promising approach to go beyond the non-crossing approximation in steady-state non-equilibrium dynamical mean-field theory simulations.
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 nonperturbatively across a wide range of interactions and temperatures. It gives access to a local, nonperturbative 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 in its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of 2 × 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 ≲ 2 % . Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, nonperturbative mpNRG vertex.