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Continuous-time auxiliary field Monte Carlo for quantum impurity models

Emanuel Gull, Olivier Parcollet, Matthias Troyer

Published Nov 20, 2012• 122 citations
OpenAlex

Abstract

We present a continuous-time Monte Carlo method for quantum impurity models, which combines a weak-coupling expansion with an auxiliary-field decomposition. The method is considerably more efficient than Hirsch-Fye and free of time discretization errors, and is particularly useful as impurity solver in large cluster dynamical mean-field theory (DMFT) calculations.

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We present a continuous-time Monte Carlo method for quantum impurity models, which combines a weak-coupling expansion with an auxiliary-field decomposition. The method is considerably more efficient than Hirsch-Fye and free of time discretization errors, and is particularly useful as impurity solver in large cluster dynamical mean-field theory (DMFT) calculations.
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.
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.
Andrew J. Millis Department of Physics, Columbia University, 538 West, 120th Street, New York, NY 10027, U.S.A. An atom or molecule embedded in an infinite host material defines a quantum impurity model (QIM). Exchange with the host causes the impurity to make transitions between different quantum states, leading to a nontrivial dynamics which may be described as a quantum field theory in zero space and one time dimension. QIMs exhibit many phenomena of intrinsic theoretical interest including quantum phase transitions (QPTs). The development over the past 15 years of dynamical mean field theory (DMFT) shows that they also serve as an auxiliary problem from whose solution important information relating to the physics of interacting lattice models can be obtained. Numerical simulation of these models remains a challenging task but the development of diagrammatic Monte Carlo techniques over the last few years has led to significant improvements in efficiency and flexibility. Highly precise results on few-orbital models can be computed and large Hilbert-space multiorbital models are now within reach. This chapter provides an overview of recent developments in quantum Monte Carlo methods and their application to multi-orbital systems. It presents both the weak-coupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and the strong-coupling approach, which can handle QIMs with strong interactions.
We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with four- and eight-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different bandwidth within the single-site dynamical mean field theory. We compare the critical interaction Uc/t{U}_{c}/tUc​/t with those obtained from solving the dynamical mean field equations with the continuous-time and Hirsch-Fye quantum Monte Carlo approaches. In both test cases we observe reasonable values of the metal-insulator critical interaction strength Uc/t{U}_{c}/tUc​/t and the nature of Mott physics in the self-energy behavior. While some details of the spectral functions cannot be captured by the semiclassical approximation due to the freezing of dynamical fluctuations, the main features are reproduced by the approach.
The solution of an auxiliary quantum impurity system is the computationally expensive step in dynamical mean field theory simulations of lattice models and materials. In this review, we discuss Monte Carlo based impurity solvers, which are suitable for a wide range of applications. In particular, we present an efficient implementation of the hybridization expansion approach, which enables the simulation of multiorbital impurity problems with off-diagonal and complex hybridizations, and dynamically screened (retarded) density-density interactions. As a complementary approach, we discuss an impurity solver based on the determinant Monte Carlo method, which scales favorably with inverse temperature and hence provides access to the very low temperature regime. The usefulness of these state-of-the-art impurity solvers is demonstrated with applications to the downfolding problem, i.e., the systematic derivation of dynamically screened interactions for low-energy effective models, and to pyrochlore iridates, where the spin-orbit coupling leads to complex hybridization functions in a multi-orbital system. As a benchmark for cluster extensions of dynamical mean field theory, we also present results from lattice Monte Carlo simulations for the momentum dependence of the pseudo-gap in the half-filled two-dimensional Hubbard model.