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Cambridge University Press eBooks

Continuous-time impurity solvers

James E. Gubernatis, Naoki Kawashima, Philipp Werner

Published Jan 31, 2016• 1 citation
Publisher

Abstract

This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems. Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers. Quantum impurity models A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.

DOI
10.1017/cbo9780511902581.009

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This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems. Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers. Quantum impurity models A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.
Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of ``correlated electron'' materials as auxiliary problems whose solution gives the ``dynamical mean-field'' approximation to the self-energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. Derivations and descriptions of the algorithms are presented in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied, and outline prospects for future applications.
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.
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.
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
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.
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.