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arXiv:1903.11636 · Physical review. B./Physical review. B

Quantum Monte Carlo algorithm for out-of-equilibrium Green's functions at long times

Corentin Bertrand, Olivier Parcollet, Antoine Maillard, Xavier Waintal

Published Sep 12, 2019• 44 citations
arXivHTMLPDF

Abstract

We present a quantum Monte Carlo algorithm for computing the perturbative expansion in power of the coupling constant $U$ of the out-of-equilibrium Green's functions of interacting Hamiltonians of fermions. The algorithm extends the one presented in Profumo et al. [Phys. Rev. B 91, 245154 (2015)], and inherits its main property: it can reach the infinite time (steady state) limit since the computational cost to compute order ${U}^{n}$ is uniform versus time; the computing time increases as ${2}^{n}$. The algorithm is based on the Schwinger-Keldysh formalism and can be used for both equilibrium and out-of-equilibrium calculations. It is stable at both small and long real times including in the stationary regime, because of its automatic cancellation of the disconnected Feynman diagrams. We apply this technique to the Anderson quantum impurity model in the quantum dot geometry to obtain the Green's function and self-energy expansion up to order ${U}^{10}$ at very low temperature. We benchmark our results at weak and intermediate coupling with high precision numerical renormalization-group computations as well as analytical results.

DOI
10.1103/physrevb.100.125129

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We present a simple, general purpose, quantum Monte Carlo algorithm for out-of-equilibrium interacting nanoelectronic systems. It allows one to systematically compute the expansion of any physical observable (such as current or density) in powers of the electron-electron interaction coupling constant UUU. It is based on the out-of-equilibrium Keldysh Green's function formalism in real-time and corresponds to evaluating all the Feynman diagrams to a given order Un{U}^{n}Un (up to n=15n=15n=15 in the present work). A key idea is to explicitly sum over the Keldysh indices in order to enforce the unitarity of the time evolution. The coefficients of the expansion can easily be obtained for long-time, stationary regimes, even at zero temperature. We then illustrate our approach with an application to the Anderson model, an archetype interacting mesoscopic system. We recover various results of the literature such as the spin susceptibility or the ``Kondo ridge'' in the current-voltage characteristics. In this case, we found the Monte Carlo free of the sign problem even at zero temperature, in the stationary regime and in absence of a particle-hole symmetry. The main limitation of the method is the lack of convergence of the expansion in UUU for large UUU, i.e., a mathematical property of the model rather than a limitation of the Monte Carlo algorithm. Standard extrapolation methods of divergent series can be used to evaluate the series in the strong correlation regime.
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This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
We express the recently introduced real-time diagrammatic quantum Monte Carlo [Phys. Rev. B 91, 245154 (2015)] in the Larkin-Ovchinnikov basis in Keldysh space. Based on a perturbation expansion in the local interaction UUU, the special form of the interaction vertex allows us to write diagrammatic rules in which vacuum Feynman diagrams directly vanish: This reproduces the main property of the previous algorithm, without the cost of the exponential sum over Keldysh indices. In an importance sampling procedure, this implies that only interaction times in the vicinity of the measurement time contribute, and such an algorithm can directly address the long-time limit needed in the study of steady states in out-of-equilibrium systems. We then implement and discuss different variants of Monte Carlo algorithms in the Larkin-Ovchinnikov basis. A sign problem reappears, showing that the cancellation of vacuum diagrams has no direct impact on it.
Quantum impurity models are paradigmatic models of interacting quantum matter, as well as key computational primitives for modern electronic-structure methods. They describe a small subsystem of interacting fermions coupled to a large, noninteracting bath. We perform a comprehensive study of the computational complexity of simulating impurity models, delineating the boundary between classical and quantum tractability for this class of problems. Our main finding is that static properties of quantum impurity models can be calculated efficiently on a classical computer. Specifically, we give classical algorithms that (1) estimate the ground-state energy to additive precision δδδ in time poly(n,δ−1)\mathrm{poly}(n,δ^{-1})poly(n,δ−1), and (2) estimate the partition function at inverse temperature βββ to relative precision δδδ in time poly(n,β,δ−1)\mathrm{poly}(n,β,δ^{-1})poly(n,β,δ−1), where nnn is the system size. These results improve the previous best-known complexity for ground-state energy estimation from quasipolynomial to polynomial time, while establishing for the first time rigorous polynomial-time guarantees for simulating impurity models in thermal equilibrium. On the other hand, we find that simulating dynamical properties of impurity models is hard for classical computers but easy on a quantum computer. As a canonical example, we show that computing their nonequilibrium Green's functions captures the full power of quantum computation, even at finite temperature. Taken together, our results rule out superpolynomial quantum speedups for computing static properties, but provide an avenue for quantum advantage in simulating impurity physics out of equilibrium.
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