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.
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.
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.
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).
This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
We present a quantum impurity solver based on a pseudoparticle framework, which combines diagrammatic resummations for a three-point vertex with diagrammatic Monte Carlo sampling of a four-point vertex. This recently proposed approach (A. J. Kim et al., arXiv:2112.15549) is generalized here to fermionic impurity problems and we discuss the technical details of the implementation, including the time-stepping approach, the Monte Carlo updates, and the routines for checking the two-particle irreducibility of the four-point vertex. We also explain how the vertex information can be efficiently stored using a Dubiner basis representation. The convergence properties of the algorithm are demonstrated with applications to exactly solvable impurity models and dynamical mean field theory simulations of the single-orbital Hubbard model. It is furthermore shown that the algorithm can handle a two-orbital problem with off-diagonal hybridizations, which would cause a severe sign problem in standard hybridization-expansion Monte Carlo simulations. Since the vertex-based algorithm successfully handles sign-oscillating integrals in equilibrium and samples only connected diagrams, it may be a promising approach for real-time simulations.
We extend a previously proposed rotation and truncation scheme to optimize quantum Anderson impurity calculations with exact diagonalization [Y. Lu, M. H\"oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B 90, 085102 (2014)] to density-matrix renormalization group (DMRG) calculations. The method reduces the solution of a full impurity problem with virtually unlimited bath sites to that of a small subsystem based on a natural impurity orbital basis set. The later is solved by DMRG in combination with a restricted-active-space truncation scheme. The method allows one to compute Green's functions directly on the real frequency or time axis. We critically test the convergence of the truncation scheme using a one-band Hubbard model solved in the dynamical mean-field theory. The projection is exact in the limit of both infinitely large and small Coulomb interactions. For all parameter ranges, the accuracy of the projected solution converges exponentially to the exact solution with increasing subsystem size.
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.