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arXiv:1904.11969 · Physical review. B./Physical review. B

Cancellation of vacuum diagrams and the long-time limit in out-of-equilibrium diagrammatic quantum Monte Carlo

Alice Moutenet, Priyanka Seth, Michel Ferrero, Olivier Parcollet

Published Aug 14, 2019• 17 citations
arXivHTMLPDF

Abstract

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 $U$, 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.

DOI
10.1103/physrevb.100.085125

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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.
We present a quantum Monte Carlo algorithm for computing the perturbative expansion in power of the coupling constant UUU 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 Un{U}^{n}Un is uniform versus time; the computing time increases as 2n{2}^{n}2n. 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 U10{U}^{10}U10 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.
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.
We present a simple trick that allows us to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional diagrammatic Monte Carlo algorithm, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increases only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model.
We develop a diagrammatic Monte Carlo method for the real-time dynamics of dissipative quantum impurity models. These are small open quantum systems with interaction and local Markovian dissipation, coupled to a large quantum bath. Our algorithm samples the hybridization expansion formulated on a single real-time contour, rather than on the double Keldysh one, as it naturally arises in the thermofield/vectorized representation of the Lindblad dynamics. We show that local Markovian dissipation generally helps the convergence of the diagrammatic Monte Carlo sampling by reducing the sign problem, thus allowing to reach longer timescales as compared to the conventional unitary case. We apply our method to an Anderson impurity model in the presence of local dephasing and discuss its effect on the charge and spin dynamics of the impurity.
A Monte Carlo sampling of diagrammatic corrections to the noncrossing approximation is shown to provide numerically exact estimates of the long-time dynamics and steady-state properties of nonequilibrium quantum impurity models. This ``bold'' expansion converges uniformly in time and significantly ameliorates the sign problem that has heretofore limited the power of real-time Monte Carlo approaches to strongly interacting real-time quantum problems. The approach enables the study of previously intractable problems ranging from generic long-time nonequilibrium transport characteristics in systems with large on-site repulsion to the direct description of spectral functions on the real frequency axis in dynamical mean field theory.
Nonlocal effective interactions are inherent to non-relativistic quantum many-body systems, but their systematic resummation poses a significant challenge known as the ``vertex problem" in many-body perturbation theory. We introduce a renormalization scheme based on a projection-based renormalization condition that selectively resums the most essential nonlocal contributions to the effective interaction vertex, avoiding the computational complexity of the full vertex function. This enables us to derive a renormalized Feynman diagrammatic series with large parameters canceled by counter-diagrams, efficiently generated using a perturbative expansion of the parquet equations and computed using a diagrammatic Monte Carlo algorithm. Applying our approach to a 3D Yukawa Fermi liquid, we demonstrate that the renormalized perturbation theory remains predictive even in the strongly correlated regime and uncover significant sign cancellations between different channels contributing to the scattering amplitude. Our work establishes a novel framework for investigating strong correlations in quantum many-body systems, offering a systematic approach to explore nonlocal theories for challenging systems like the electron liquid in material science.