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arXiv:2303.11199 · Physical review. B./Physical review. B

Tensor train continuous time solver for quantum impurity models

André Erpenbeck, Wei‐Ting Lin, Thomas Blommel, Lei Zhang, Sergei Iskakov, Liam Bernheimer, Y. Núñez-Fernández, Guy Cohen, Olivier Parcollet, Xavier Waintal, Emanuel Gull

Published Jun 26, 2023• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.107.245135

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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.
The investigation of quantum impurity models plays a crucial role in condensed matter physics because of their wide-ranging applications, such as embedding theories and transport problems. Traditional methods often fall short in producing accurate results for multi-orbital systems with complex interactions and off-diagonal hybridizations. Recently, tensor-train-based integration and summation techniques have shown promise as effective alternatives. In this study, we use tensor train methods to tackle quantum impurity problems formulated within the imaginary-time inchworm hybridization expansion framework. We identify key challenges in the inchworm expansion itself and its interplay with tensor-train-based methods. We demonstrate the accuracy and versatility of our approach by solving general quantum impurity problems. Our results suggest that tensor-train decomposition schemes offer a viable path toward accurate and efficient multi-orbital impurity solvers.
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 present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.
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 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.
Dynamical mean-field theory (DMFT) is one of the most widely-used methods to treat accurately electron correlation effects in ab-initio real material calculations. Many modern large-scale implementations of DMFT in electronic structure codes involve solving a quantum impurity model with a continuous-time quantum Monte Carlo (CT-QMC) solver [Rubtsov et al., Phys. Rev. B 72, 035122 (2005); Werner et al., Phys. Rev. Lett. 97, 076405 (2006); Werner and Millis, Phys. Rev. B 74, 155107 (2006); Gull et al., Rev. Mod. Phys. 83, 349 (2011)]. The main advantage of CT-QMC is that, unlike standard quantum Monte Carlo approaches, it is able to generate the local Green's functions $G(\ensuremath{\tau})$ of the correlated system on an arbitrarily fine imaginary time $\ensuremath{\tau}$ grid, and is free of any systematic errors. In this work, we extend a hybrid QMC solver proposed by Khatami et al. [Phys. Rev. E 81, 056703 (2010)] and Rost et al. [Phys. Rev. E 87, 053305 (2013)] to a multiorbital context. This has the advantage of enabling impurity solver QMC calculations to scale linearly with inverse temperature $\ensuremath{\beta}$, and permit its application to ddd- and fff-band materials. In addition, we present a Green's-function processing scheme which generates accurate quasicontinuous imaginary time solutions of the impurity problem which overcome errors inherent to standard QMC approaches. This solver and processing scheme are incorporated into a full DFT+DMFT calculation using the castep DFT code [Clark et al., Z. Kristallogr. 220, 567 (2005)]. Benchmark calculations for SrVO3{\mathrm{SrVO}}_{3}SrVO3​ properties are presented. The computational efficiency of this method is also demonstrated.