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arXiv:2404.05410 · Physical review. B./Physical review. B

Solving equilibrium quantum impurity problems on the L-shaped Kadanoff-Baym contour

Ruofan Chen, Chu Guo

Published Oct 7, 2024• 11 citations
arXivHTMLPDF

Abstract

The path integral formalism is the building block of many powerful numerical methods for quantum impurity problems. However, existing fermionic path integral-based numerical calculations have only been performed in either the imaginary-time or the real-time axis, while the most generic scenario formulated on the L-shaped Kadanoff-Baym contour is left unexplored. In this work, we extended the recently developed Grassmann time-evolving matrix product operator (GTEMPO) method to solve quantum impurity problems directly on the Kadanoff-Baym contour. The resulting method is numerically exact, with only two sources of numerical errors, e.g., the time discretization error and the matrix product state bond truncation error. The accuracy of this method is numerically demonstrated against exact solutions in the noninteracting case and against existing calculations on the real- and imaginary-time axes for the single-orbital Anderson impurity model. We also show that the numerical errors of the method can be well suppressed as we refine the hyperparameters. Our method is a perfect benchmarking baseline for its alternatives which often employ less-controlled approximations, and can also be used as a real-time impurity solver in dynamical mean field theory.

DOI
10.1103/physrevb.110.165114

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The path integral formalism is the building block of many powerful numerical methods for quantum impurity problems. However, existing fermionic path integral-based numerical calculations have only been performed in either the imaginary-time or the real-time axis, while the most generic scenario formulated on the L-shaped Kadanoff-Baym contour is left unexplored. In this work, we extended the recently developed Grassmann time-evolving matrix product operator (GTEMPO) method to solve quantum impurity problems directly on the Kadanoff-Baym contour. The resulting method is numerically exact, with only two sources of numerical errors, e.g., the time discretization error and the matrix product state bond truncation error. The accuracy of this method is numerically demonstrated against exact solutions in the noninteracting case and against existing calculations on the real- and imaginary-time axes for the single-orbital Anderson impurity model. We also show that the numerical errors of the method can be well suppressed as we refine the hyperparameters. Our method is a perfect benchmarking baseline for its alternatives which often employ less-controlled approximations, and can also be used as a real-time impurity solver in dynamical mean field theory.
The Grassmann time-evolving matrix product operator (GTEMPO) method has proven to be an accurate and efficient numerical method for the real-time dynamics of quantum impurity problems. However, its application for imaginary-time calculations is much less competitive than well-established methods such as the continuous-time quantum Monte Carlo (CTQMC). In this work, we unleash the full power of GTEMPO for zero-temperature imaginary-time calculations: The multitime impurity state is time-translationally invariant with infinite boundary conditions; therefore, it can be represented as an infinite Grassmann matrix product state (GMPS) with a nontrivial unit cell in a single time step, instead of an open boundary GMPS spanning the whole imaginary-time axis. We devise a very efficient infinite GTEMPO algorithm targeted at zero-temperature equilibrium quantum impurity problems, which is known to be a hard regime for quantum Monte Carlo methods. To demonstrate the performance of our method, we benchmark it against exact solutions in the noninteracting limit and against CTQMC calculations in the Anderson impurity models with up to two orbitals, where we show that the required bond dimension of the infinite GMPS is much smaller than its finite-temperature counterpart.
An emergent numerical approach to solve quantum impurity problems is to encode the impurity path integral as a matrix product state (MPS). For time-dependent problems, the cost of this approach generally scales with the evolution time. Here, we consider a common nonequilibrium scenario where an impurity, initially in equilibrium with a thermal bath, is driven out of equilibrium by a sudden quench of the impurity Hamiltonian. Despite no time-translational invariance in the problem, we show that we could still make full use of the infinite MPS technique, resulting in a method whose cost is essentially independent of the evolution time. We demonstrate the effectiveness of this method in the integrable case against exact diagonalization and against existing calculations on the L-shaped Kadanoff-Baym contour in the general case. Our method could be a very competitive method for studying long-time nonequilibrium quantum dynamics and potentially used as an efficient impurity solver in the nonequilibrium dynamical mean field theory.
The Grassmann time-evolving matrix product operator (GTEMPO) method, which represents the Feynman-Vernon influence functional as a temporal matrix product state, has been shown to be a flexible and potentially scalable solution for fermionic quantum impurity problems. In this work, we extend GTEMPO to solve fermionic impurity problems in the Nambu formalism, in which the impurity is coupled to a superconducting bath. A key insight is that by employing the Bogoliubov transformation for the superconducting bath, one could obtain the analytic expression of the Feynman-Vernon influence functional in a similar form to the case of a normal bath, after which the core algorithms of GTEMPO can be straightforwardly adapted. We demonstrate the accuracy of our method by benchmarking it against exact diagonalization in several exactly solvable cases, and against the continuous-time quantum Monte Carlo method using converged dynamical mean field theory (DMFT) iterations on the imaginary contour in the non-integrable case. In all cases, we perform both imaginary- and real-time calculations to illustrate the flexibility of our method. These results illustrate that our method could be potentially useful as an impurity solver in DMFT as well as its non-equilibrium extension for fermionic impurity problems in the Nambu formalism.
We present an infinite Grassmann time-evolving matrix product operator method for quantum impurity problems, which directly works in the steady state. The method embraces the well-established infinite matrix product state algorithms with the recently developed GTEMPO method, and benefits from both sides: it obtains real-time Green's functions without sampling noises and bath discretization error, it is applicable for any temperature without the sign problem, and its computational cost is independent of the transient dynamics and does not scale with the number of baths. We benchmark the method on the finite-temperature equilibrium Green's function in the noninteracting limit against exact solutions and in the single-orbital Anderson impurity model against GTEMPO calculations. We also study the zero-temperature nonequilibrium steady state of an impurity coupled to two baths with a voltage bias, obtaining consistent particle currents with existing calculations. The method is ideal for studying steady-state quantum transport, and can be readily used as an efficient real-time impurity solver in the dynamical mean-field theory and its nonequilibrium extension.
Abstract Tensor-network-based methods are promising candidates to solve quantum impurity problems (QIP). They are free of sampling noises and the sign problem compared to state-of-the-art continuous-time quantum Monte Carlo methods. Recent progress made in tensor-network-based impurity solvers is to use the Feynman–Vernon influence functional to integrate out the bath analytically, retaining only the impurity dynamics and representing it compactly as a matrix product state. The recently proposed Grassmann time-evolving matrix product operator (GTEMPO) method is one of the representative methods in this direction. In this work, we systematically study the performance of GTEMPO in solving equilibrium QIPs at a finite temperature with a semicircular spectrum density of the bath. Our results show that its computational cost would generally increase as the temperature goes down and scale exponentially with the number of orbitals. In particular, the single-orbital Anderson impurity model can be efficiently solved with this method, for two orbitals we estimate that one could possibly reach inverse temperature β ≈ 20 if high-performance computing techniques are utilized, while beyond that only very high-temperature regimes can be reached in the current formalism. Our work paves the way to apply GTEMPO as an imaginary-time impurity solver.
Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.