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arXiv:2011.05522 · Quantum Science and Technology

Beyond quantum cluster theories: multiscale approaches for strongly correlated systems

Herbert F. Fotso, Ka-Ming Tam, Juana Moreno

Published Apr 14, 2022• 6 citations

Abstract

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.

DOI: 10.1088/2058-9565/ac676b

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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.

Many technologically important materials display complex behavior due to strong electronic correlations. These include magnets, magnetoresistive materials, lanthanides and actinides and high temperature superconductors. Some of these materials display complex phase diagrams with multiple competing ground states, spin and charge ordering and complex excitations like those associated with spin and charge separation. An unbiased treatment of these phenomena requires a theory able to treat the correlations over all relevant length scales. In part, due to the minus sign problem, numerically exact methods employed to study these systems scale exponentially. Thus, despite great recent progress such as the development of the dynamical mean field approximation and its cluster extensions, there is as yet no unbiased study of these complex phenomena. The Multi-Scale Many-Body (MSMB) formalism and algorithms address these problems by separating the correlations into different length scales and treating each with an appropriate approximation: strong correlations at short length scales are treated by numerically exact Quantum Monte Carlo (QMC) simulations, those at intermediate length scales are treated by Feynman diagrammatic methods, while the weak correlations at long length scales are treated by a (dynamical) mean field approximation with a self-consistently determined effective medium.

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.

We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size

{L}_{c}

embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as

\mathcal{O}(1/{L}_{c})

, so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order

\mathcal{O}(1/{L}_{c}^{3})

with third-order and three-body corrections down by an additional factor of

\mathcal{O}(1/{L}_{c})

.

This article reviews quantum cluster theories, a set of approximations for infinite lattice models which treat correlations within the cluster explicitly, and correlations at longer length scales either perturbatively or within a mean-field approximation. These methods become exact when the cluster size diverges, and most recover the corresponding mean-field approximation when the cluster size becomes 1. Although quantum cluster theories were originally developed to treat disordered systems, they have more recently been applied to the study of ordered and disordered correlated systems, which will be the focus of this review. After a brief historical review, the authors provide detailed derivations of three cluster formalisms: the cluster perturbation theory, the dynamical cluster approximation, and the cellular dynamical mean-field theory. They compare their advantages and review their applications to common models of correlated electron systems.

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.

Quantum many-body calculation methods, including exact diagonalization, quantum Monte Carlo, density matrix and tensor renormalization, and dynamic mean field theory, as well as emerging methods harnessing the power of artificial intelligence and quantum computing, can be used to accurately and efficiently calculate the physical properties of correlated quantum many-body systems. The quantum particles in the interacting, non-perturbative many-body system are highly entangled, and the mean field theory often lacks sufficient accuracy or may even break down for certain problems. It is thus very necessary to develop new methods for studying the exotic states and emergent phenomena in correlated systems, such as high-temperature superconductivity and frustrated quantum magnetism. Recently, the rapid progress in the field of many-body calculation has shown the characteristics of interdisciplinary cross, and there is increasingly significant interplay and integration with machine learning, materials science, quantum simulation and computation, among other fields. This paper outlines the brief history of many-body calculations, major approaches and their current status, as well as main challenges in this field.