We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision.The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm.It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction.Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates 1=N 2 or faster, where N is the number of function evaluations.The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations.Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.
We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision.The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm.It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction.Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates 1=N 2 or faster, where N is the number of function evaluations.The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations.Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.
The simulation of strongly correlated quantum impurity models is a significant challenge in modern condensed matter physics that has multiple important applications. Thus far, the most successful methods for approaching this challenge involve Monte Carlo techniques that accurately and reliably sample perturbative expansions to any order. However, the cost of obtaining high precision through these methods is high. Recently, tensor train decomposition techniques have been developed as an alternative to Monte Carlo integration. In this study, we apply these techniques to the single-impurity Anderson model at equilibrium by calculating the systematic expansion in power of the hybridization of the impurity with the bath. We demonstrate the performance of the method in a paradigmatic application, examining the first-order phase transition on the infinite-dimensional Bethe lattice, which can be mapped to an impurity model through dynamical mean field theory. Our results indicate that using tensor train decomposition schemes allows the calculation of finite-temperature Green's functions and thermodynamic observables with unprecedented accuracy. The methodology holds promise for future applications to frustrated multiorbital systems, using a combination of partially summed series with other techniques pioneered in diagrammatic and continuous time quantum Monte Carlo.
We apply the tensor cross interpolation (TCI) algorithm to solve equilibrium quantum impurity problems with high precision based on the weak-coupling expansion. The TCI algorithm, a kind of active learning method, factorizes high-dimensional integrals that appear in the perturbative expansion into a product of low-dimensional ones, enabling us to evaluate higher-order terms efficiently. This method is free from the sign problem which quantum Monte Carlo methods sometimes suffer from, and allows one to directly calculate the free energy. We benchmark the TCI impurity solver on an exactly solvable impurity model and find good agreement with the exact solutions. We also incorporate the TCI impurity solver into the dynamical mean-field theory to solve the Hubbard model and show that the metal-to-Mott insulator transition is correctly described with comparable accuracy to the Monte Carlo methods. Behind the effectiveness of the TCI approach for quantum impurity problems lies the fact that the integrands in the weak-coupling expansion naturally have a low-rank structure in the tensor-train representation.
The physics of N-body quantum systems reveals the emergence of numerous unexpected collective phenomena, such as the 0.7 anomaly, the Kondo effect, and high critical temperature superconductivity. From a theoretical perspective, describing these systems is challenging due to the exponential growth of the state space as the number of particles increases. For instance, the quantitative understanding of the Kondo effect required the development of new concepts and mathematical tools, including the (numerical) renormalization group. This pioneering work, which earned Wilson the Nobel Prize in 1978, laid the foundation for many subsequent advances.In this thesis, we focus on the physics of quantum dots, described by the Anderson model. This model exhibits rich phenomenology, both in the charge domain (Coulomb blockade) and the spin domain (Kondo effect). While this model is well understood in equilibrium, we propose an innovative technique to study its properties in the non-equilibrium regime—a historically challenging area to address. Our approach enables exact calculations with controlled errors.This method builds on the "Diagrammatic Monte Carlo" technique, which systematically computes a large number of Feynman diagrams. Traditionally, this technique stochastically samples the space of diagrams. In our work, we replaced this approach with a machine learning method based on tensor networks and the cross-interpolation algorithm. The latter identifies and leverages the specific structure of the diagram space, greatly accelerating the calculations. A key advantage of this method is its insensitivity to the sign problem that limits stochastic approaches.Using this new technique, we were able to compute diagrams up to much higher orders than previously possible (<30), enabling us to explore the physics of non-equilibrium quantum dots with unprecedented precision.
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.
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.
Feynman diagrams are an essential tool for simulating strongly correlated electron systems. However, stochastic quantum Monte Carlo sampling suffers from the sign problem, particularly when solving a multiorbital quantum impurity model. Recently, two approaches have been proposed for efficient numerical treatment of Feynman diagrams: tensor cross interpolation (TCI) to replace stochastic sampling and the quantics tensor train (QTT) representation for compressing space-time dependence. One of the remaining challenges is the nontrivial task of identifying low-rank structures in weak-coupling Feynman diagrams for multiorbital electron-phonon systems. In particular, the traditional TCI algorithm faces an ergodicity problem, which prevents it from fully exploring the multiorbital space. To address this, we incorporate a new algorithm called global search, which resolves this issue. By combining this approach with QTT, we uncover low-rank structures and achieve efficient numerical integration with exponential resolution in time and faster-than-power-law convergence of error relative to computational cost. Additionally, our approach does not require the division of discontinuous regions necessary in nonquantics TCI.