The development of numerical methods capable of simulating realistic materials with strongly correlated electrons, with controllable errors, is a central challenge in quantum many-body physics. Here we describe a framework for a general multiscale method based on embedding a self-energy of a strongly correlated subsystem into a self-energy generated by a method able to treat large weakly correlated systems approximately. As an example, we present the embedding of an exact diagonalization self-energy into a self-energy generated from self-consistent second-order perturbation theory. Using a quantum impurity model, generated from a cluster dynamical mean field approximation to the two-dimensional Hubbard model, as a benchmark, we illustrate that our method allows us to obtain accurate results at a fraction of the cost of typical Monte Carlo calculations. We test the method in multiple regimes of interaction strengths and dopings of the model. The general embedding framework we present avoids difficulties such as double counting corrections, frequency-dependent interactions, or vertex functions. As it is solely formulated at the level of the single-particle Green's function, it provides a promising route for the simulation of realistic materials that are currently difficult to study with other methods.
The development of numerical methods capable of simulating realistic materials with strongly correlated electrons, with controllable errors, is a central challenge in quantum many-body physics. Here we describe a framework for a general multiscale method based on embedding a self-energy of a strongly correlated subsystem into a self-energy generated by a method able to treat large weakly correlated systems approximately. As an example, we present the embedding of an exact diagonalization self-energy into a self-energy generated from self-consistent second-order perturbation theory. Using a quantum impurity model, generated from a cluster dynamical mean field approximation to the two-dimensional Hubbard model, as a benchmark, we illustrate that our method allows us to obtain accurate results at a fraction of the cost of typical Monte Carlo calculations. We test the method in multiple regimes of interaction strengths and dopings of the model. The general embedding framework we present avoids difficulties such as double counting corrections, frequency-dependent interactions, or vertex functions. As it is solely formulated at the level of the single-particle Green's function, it provides a promising route for the simulation of realistic materials that are currently difficult to study with other methods.
Abstract The degrees of freedom that confer to strongly correlated systems their many intriguing properties also render them fairly intractable through typical perturbative treatments. For this reason, the mechanisms responsible for their technologically promising properties remain mostly elusive. Computational approaches have played a major role in efforts to fill this void. In particular, dynamical mean field theory and its cluster extension, the dynamical cluster approximation have allowed significant progress. However, despite all the insightful results of these embedding schemes, computational constraints, such as the minus sign problem in quantum Monte Carlo (QMC), and the exponential growth of the Hilbert space in exact diagonalization (ED) methods, still limit the length scale within which correlations can be treated exactly in the formalism. A recent advance aiming to overcome these difficulties is the development of multiscale many body approaches whereby this challenge is addressed by introducing an intermediate length scale between the short length scale where correlations are treated exactly using a cluster solver such QMC or ED, and the long length scale where correlations are treated in a mean field manner. At this intermediate length scale correlations can be treated perturbatively. This is the essence of multiscale many-body methods. We will review various implementations of these multiscale many-body approaches, the results they have produced, and the outstanding challenges that should be addressed for further advances.
Generalized quantum impurity models—which feature a few localized and strongly correlated degrees of freedom coupled to itinerant conduction electrons—describe diverse physical systems, from magnetic moments in metals to nanoelectronics quantum devices such as quantum dots or single-molecule transistors. Correlated materials can also be understood as self-consistent impurity models through dynamical mean-field theory. Accurate simulation of such models is challenging, especially at low temperatures, due to many-body effects from electronic interactions, resulting in strong renormalization. In particular, the interplay between local impurity complexity and Kondo physics is highly nontrivial. A common approach, which we further develop in this work, is to consider instead a simpler effective impurity model that still captures the low-energy physics of interest. The mapping from a bare to an effective model is typically done perturbatively, but even this can be difficult for complex systems, and the resulting effective model parameters can nevertheless be quite inaccurate. Here we develop a nonperturbative, unsupervised machine learning approach to systematically obtain low-energy effective impurity-type models, based on the renormalization-group framework. The method is shown to be general and flexible, as well as accurate and systematically improvable. We benchmark the method against exact results for the Anderson impurity model, and we provide an outlook for more complex models beyond the reach of existing methods. Published by the American Physical Society 2024
A numerically implementable multi-scale many-body approach to strongly correlated electron systems is introduced. An extension to quantum cluster methods, it approximates correlations on any given length-scale commensurate with the strength of the correlations on the respective scale. Short length-scales are treated explicitly, long ones are addressed at a dynamical mean-field level and intermediate length-regime correlations are assumed to be weak and are approximated diagrammatically. To illustrate and test this method, we apply it to the one-dimensional Hubbard model. The resulting multi-scale self-energy provides a very good quantitative agreement with substantially more numerically expensive, explicit quantum Monte Carlo calculations.
Understanding emergent many-body phenomena in correlated materials remains one of the grandest challenges in solid-state quantum chemistry calculations. The individuality of electrons diminishes as correlation increases, which leads to the emergence of new features in the system. Seeking to address such problems accurately requires the aid of non-perturbative many-body techniques as well as the corresponding high-quality basis sets. In periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, and an elemental basis set independent of the physical properties of materials may lead to significant simulation inaccuracies. In fact, with the rapid progress of quantum chemistry methods in condensed-phase simulations, the need for a library of reliable Gaussian basis sets explicitly designed for periodic calculations has become urgent. In the meantime, the development of parameter-free, systematic and reliable quantum chemistry methods for simultaneously treating weak and strong correlations in periodic systems has never stopped. Their ability to treat realistic solid materials needs to be tested, and the obtained results can serve as benchmarks for empirical parameter-guided solid-state calculations. In this thesis, we present numerical studies of various solid-state systems, emphasizing the design and optimization of Gaussian basis sets and assessing the band structure properties of the periodic solid systems. The work begins by introducing the theoretical framework of the electronic structure theory and presenting the fundamentals of a fully self-consistent GW approximation and the self-energy embedding theory (SEET) in Chapters 2 and 3. It continues with a description of the Gaussian basis set optimization scheme we devised for solid-state quantum chemistry calculations. The scheme we present is designed to avoid a lack of material specificity within a given basis set by simultaneously minimizing the total energy of the system and optimizing the band energies when compared to the reference plane wave calculation while accounting for the overlap matrix condition number. We compare the quality of the Gaussian basis sets generated via our method against the existing basis sets. And the optimization scheme is tested to yield improved results. We finally present a quantitative study of electronic properties of realistic materials with the treatment of Green's function based weakly correlated self-consistent GW approximation and the strongly correlated self-energy embedding theory. In the transition metal oxide BiVO3, we found the inaccurate illustration of orbitals including t_{2g} in transition metal Vanadium by DFT. Our calculation shows a systematic trend of DFT failure in illustrating both insulating and metallic solutions, demonstrating the dangers of using DFT when the experimental data is scarce.
We apply the Green's function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems arising for strongly correlated solids within the self-energy embedding theory (SEET) framework. We describe the details of our GFCC solver implementation, investigate its performance, and highlight potential advantages and problems on examples of impurities created during the self-consistent SEET for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, our data also shows that when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems. These instabilities appear especially severe when the impurity size gets larger and multiple degenerate orbitals with strong correlations are present. We conclude that, to fully check the reliability of GFCCSD results and use them in fully ab initio calculations in the absence of experiments, a verification from a GFCC solver with higher order excitations is necessary.
Self-consistent quantum embedding forms a route to bridge local descriptions of electronic structure methods and the electronic structure of very large systems and condensed phases. Existing examples of quantum embedding methods include the density functional embedding of Cortona, Wesolowski and Warshel, based on the single-particle density and Carter, and the dynamical mean-field theory, based on the single-particle Greens function. In this talk, however, I will describe a new formalism for quantum embedding that is based on the single-particle density matrix, and which is based on a self-consistent approximation to the exact entanglement embedding of a subsystem in an environment. Unlike density functional embedding, this formalism can practically describe strongly correlated systems, and unlike dynamical mean-field theory, it is easily combined with traditional ground-state electronic structure methods, such as quantum chemical wavefunction methods and projector and variational Monte Carlo approaches. I will demonstrate the performance of the method on the phase diagram of the 1D and 2D Hubbard models and discuss applications to ab-initio systems.