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Physics Procedia

Continuous-time quantum impurity solvers

Emanuel Gull, Philipp Werner, Andrew J. Millis, Matthias Troyer

Published Jan 1, 2010• 2 citations
Publisher

Abstract

We present a brief overview of two continuous–time quantum Monte Carlo impurity solvers–a diagrammatic expansion of the partition function in the interaction and in the impurity-bath hybridization and mention a recently developed continuous-time auxiliary field method. We show that continuous-time methods deliver substantial gains in computational effeciency over previous QMC discrete-time algorithms.

DOI
10.1016/j.phpro.2010.09.025

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We present a brief overview of two continuous–time quantum Monte Carlo impurity solvers–a diagrammatic expansion of the partition function in the interaction and in the impurity-bath hybridization and mention a recently developed continuous-time auxiliary field method. We show that continuous-time methods deliver substantial gains in computational effeciency over previous QMC discrete-time algorithms.
Impurity solvers play an essential role in the numerical investigation of strongly correlated electrons systems within the ``dynamical mean field'' approximation. Recently, a new class of continuous-time solvers has been developed based on a diagrammatic expansion of the partition function in either the interactions or the impurity-bath hybridization. We investigate the performance of these two complementary approaches and compare them to the well-established Hirsch-Fye method. The results show that the continuous-time methods, and, in particular, the version which expands in the hybridization, provide substantial gains in computational efficiency.
We extend the recently developed real-time diagrammatic Monte Carlo method, in its hybridization expansion formulation, to the full Kadanoff-Baym-Keldysh contour. This allows us to study real-time dynamics in correlated impurity models starting from an arbitrary, even interacting, initial density matrix. As a proof of concept, we apply the algorithm to study the nonequilibrium dynamics after a local quantum quench in the Anderson impurity model. Being a completely general approach to real-time dynamics in quantum impurity models, it can be used as a solver for nonequilibrium dynamical mean field theory.
Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization $\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\rightarrow}0$) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates and of appropriate $\ensuremath{\Delta}\ensuremath{\tau}$ values is included.
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.