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arXiv:cond-mat/0405408 · Physical Review Letters

Projective Quantum Monte Carlo Method for the Anderson Impurity Model and its Application to Dynamical Mean Field Theory

M. Feldbacher, Karsten Held, Fakher F. Assaad

Published Sep 23, 2004• 32 citations
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Abstract

We develop a projective quantum Monte Carlo algorithm of the Hirsch-Fye type for obtaining ground state properties of the Anderson impurity model. This method is employed to solve the self-consistency equations of dynamical mean field theory. It is shown that the approach converges rapidly to the ground state so that reliable zero-temperature results are obtained. As a first application, we study the Mott-Hubbard metal-insulator transition of the frustrated one-band Hubbard model, reconfirming the numerical renormalization group results.

DOI
10.1103/physrevlett.93.136405

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We develop a projective quantum Monte Carlo algorithm of the Hirsch-Fye type for obtaining ground state properties of the Anderson impurity model. This method is employed to solve the self-consistency equations of dynamical mean field theory. It is shown that the approach converges rapidly to the ground state so that reliable zero-temperature results are obtained. As a first application, we study the Mott-Hubbard metal-insulator transition of the frustrated one-band Hubbard model, reconfirming the numerical renormalization group results.
We present a new algorithm which allows for direct numerically exact solutions within dynamical mean-field theory (DMFT). It is based on the established Hirsch-Fye quantum Monte Carlo (HF-QMC) method. However, the DMFT impurity model is solved not at fixed imaginary-time discretization Delta_tau, but for a range of discretization grids; by extrapolation, unbiased Green functions are obtained in each DMFT iteration. In contrast to conventional HF-QMC, the multigrid algorithm converges to the exact DMFT fixed points. It extends the useful range of Delta_tau, is precise and reliable even in the immediate vicinity of phase transitions and is more efficient, also in comparison to continuous-time methods. Using this algorithm, we show that the spectral weight transfer at the Mott transition has been overestimated in a recent density matrix renormalization group study.
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.
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.
A new numerical method for the solution of the dynamical mean field theory's self-consistent equations is introduced. The method uses the density matrix renormalization group technique to solve the associated impurity problem. The new algorithm makes no a priori approximations and is only limited by the number of sites that can be considered. We obtain accurate estimates of the critical values of the metal-insulator transitions and provide evidence of substructure in the Hubbard bands of the correlated metal. With this algorithm, more complex models having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.
We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean-field theory (DMFT). We benchmark the solver in the two-band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization-group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the antiferromagnetic phase transition in the unfrustrated Bethe lattice. The Neel temperature obtained by the fast impurity solver agrees very well with the QMC results in the large Hubbard U limit. The method is a promising tool to be used in combination with the LDA+DMFT\mathrm{LDA}+\mathrm{DMFT}LDA+DMFT to study Mott insulators starting from first principles.
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.