An impurity model describes an atom or molecule embedded in some host or bath, with which it can exchange electrons. This exchange of electrons allows the impurity to make transitions between different quantum states, and leads to a non-trivial dynamics. Therefore, despite the zero dimensional nature (which makes impurity problems computationally much more tractable than fermionic lattice models), their numerical simulation remains a challenging task. Methods such as exact diagonalization or numerical RG, which explicitly treat a finite number of bath states, work well for single orbital models. However, because the number of bath states must be increased proportional to the number of orbitals, the computational effort grows exponentially with system size, and requires severe truncations of the bath already for two orbitals. Monte Carlo methods have the advantage that the bath is integrated out and thus the (infinite) size of the bath Hilbert space does not affect the simulation. While restricted to finite temperature, Monte Carlo methods are thus the method of choice for the solution of large multi-orbital or cluster impurity problems. Over the last few years, significant progress has been made (both in terms of efficiency and flexibility) with the development of diagrammatic Monte Carlo techniques. These notes provide an overview over two recently developed, complementary methods: (i) the weakcoupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and (ii) the strong-coupling approach, which can handle impurity
An impurity model describes an atom or molecule embedded in some host or bath, with which it can exchange electrons. This exchange of electrons allows the impurity to make transitions between different quantum states, and leads to a non-trivial dynamics. Therefore, despite the zero dimensional nature (which makes impurity problems computationally much more tractable than fermionic lattice models), their numerical simulation remains a challenging task. Methods such as exact diagonalization or numerical RG, which explicitly treat a finite number of bath states, work well for single orbital models. However, because the number of bath states must be increased proportional to the number of orbitals, the computational effort grows exponentially with system size, and requires severe truncations of the bath already for two orbitals. Monte Carlo methods have the advantage that the bath is integrated out and thus the (infinite) size of the bath Hilbert space does not affect the simulation. While restricted to finite temperature, Monte Carlo methods are thus the method of choice for the solution of large multi-orbital or cluster impurity problems. Over the last few years, significant progress has been made (both in terms of efficiency and flexibility) with the development of diagrammatic Monte Carlo techniques. These notes provide an overview over two recently developed, complementary methods: (i) the weakcoupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and (ii) the strong-coupling approach, which can handle impurity
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.
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 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, which describe the coupling between interacting orbitals and a non-interacting bath, play a central role in the physics of strongly correlated electron systems. Solving a quantum impurity model in general requires the use of non-perturbative numerical methods. Hamiltonian-based approaches, which rely on an explicit bath discretization, are typically limited to a small number of bath sites or small entanglement, and diagrammatic methods suffer from sign problems, slow convergence, or diagram truncation approximations. Here we show that these two classes of methods can be combined: augmenting diagrammatic methods with a small auxiliary bath can reduce the residual problem to a regime where low-order perturbation theory is highly accurate and rapidly converging. In a simple benchmark, the precision of the hybrid approach surpasses bold-line calculations by several orders of magnitude; for a strongly interacting two-orbital model with a severe sign problem, convergence is achieved at three orders of magnitude lower computational cost than competing methods; and convergence to the unknown exact result is rapidly accelerated in a difficult realistic problem. Our results establish a practical route to high-precision quantum impurity solutions in correlated quantum systems.
Theoretical studies of electron and energy transport in nanostructures usually involve nonequilibrium quantum impurity models, which feature a locally interacting quantum system (impurity) coupled to noninteracting bosonic and/or electronic baths each at its own equilibrium. In these models, the interplay of strong correlation, nonequilibrium effects and dissipation leads to rich and complex phenomena. As a nonequilibrium strongly correlated many-body problem, these models, except at some rare cases, have no exact analytical solution, and hence their treatment heavily relies upon numerical techniques. The goal of this thesis is to develop numerically accurate and relatively low-cost techniques for treatment of strongly correlated open nonequilibirum quantum systems. Such impurity solvers are in high demand both as standalone methods for simulation of response in nanoscale junctions and as part of divide-and-conquer schemes such as dynamical mean field theory (DMFT).For nonequilibirum open quantum systems, two characteristic energy scales governing the physics are intra-system interactions and strength of the system-baths couplings. Thus, systems strongly coupled to their baths with weak intra-system interactions or weakly coupled systems with strong local interactions can be treated within developed perturbation theory approaches. My research focuses on development of nonequilibirum dual techniques, which are capable to accurately treat situations where perturbative techniques are inapplicable due to absence of a small parameter in the system and which reduce to standard perturbative expansions in the limits of weak coupling or weak intra-system interaction. \tDual techniques rely on ability to solve exactly a simplified reference system. The latter includes exactly all intra-system interactions and accounts approximately for system-bath couplings. Dual techniques (described below) allow to account for deviation from the correct hybridization function in a organized and controlled manner of superperturbative expansion. Here, I use Markovian Lindbald quantum master equation (QME) as a solver for the reference system. In the thesis I discuss possibility of mapping between true non-Markov dynamics and simpler Markov Lindblad QME (mapping between physical and reference systems), numerical methods for the mapping and for solution of the resulting Lindblad QME, and application of the mapping in novel dual formulations: auxiliary-QME dual fermion and dual boson approaches. \tFirst I analytically prove that the continuous fermionic baths that the impurity is coupled to can be mapped onto a set of dissipative auxiliary modes under Lindblad QME. Based on this mapping, complicated non-Markovian dynamics of the open fermionic system can be treated by a much simpler Markovian Lindblad QME. However, nice mapping generally requires many auxiliary modes, and numerically only a small number of modes can be exactly diagonalized due to exponential increase of the Hilbert space. To perturbatively treat the difference between the true bath and the small-size auxiliary one, we introduce dual-fermion method. It can directly target the steady state and avoid long-time propagation. Besides current-voltage relation, it can also provide spectral functions, indicating itself as a potential impurity solver for DMFT. Furthermore, we generalize dual-fermion method to consider additionally the effects of bosonic environment; hence it is called dual-boson method. It can be used for studying electron-phonon interaction and light-matter interaction in nanojunctions. By comparing with numerically exact results we argue that both methods are very accurate and relatively cheap. \tFinally, matrix product state representation of the Lindblad QME enables us to treat a large number of auxiliary modes, which provide an exponentially improved mapping. Adding counting field to the auxiliary QME, we can compute the generating function of time-dependent full-counting statistics of electron transport, which yields not only electronic current but also higher-order cumulants (e.g., zero-frequency noise).
Two of the primary sources of error in the Cluster dynamical mean field theory (CDMFT) technique arise from the use of finite size clusters and finite size baths, which makes the development of impurity solvers that can treat larger systems an essential goal. In this work we introduce an impurity solver based on the recently developed dynamical variational Monte Carlo (dVMC) method. Variational Monte Carlo possesses a favorable scaling as a function of system size, which enables the treatment of systems beyond the reach of current exact diagonalization solvers. To benchmark the technique, we perform a systematic set of CDMFT calculations on the single-band one-dimensional Hubbard model. We compare to results obtained with an exact diagonalization solver for small clusters, and against the exact solution in the thermodynamic limit obtained by Lieb and Wu for larger clusters. The development of improved impurity solvers will help extend the reach of quantum cluster methods, which can be applied to a wide range of strongly correlated electron systems, promising new insights into their emergent behavior.