Electron correlation is a measure of the errors that are inherent in the Hartree-Fock theory or orbital models. When the electron density is high, correlation is weak and the traditional electronic theory works well. However, at a low density of electrons correlation effects become strong and the traditional theory fails to describe the electron system correctly. Therefore, the electron correlation plays a radical role in such materials as high-temperature superconductors and heavy fermions, etc. To date, there is no agreement on how to deal with higher-order terms (correlation energy) in the series of electron’s ground state energy although a method that is termed diagrammatic iteration approach (DIA) was developed more than one decade ago by the authors of this article. That is why no consensus on the origin and mechanism of superconductivity has been engaged in superconductivity community. From the viewpoint of methodology, the DIA is indeed an approach to higher-order terms from the lower-order ones, i.e. it is a new method to show how to go beyond the random phase approximation (RPA) step by step by iteration. Here, we are logically presenting it to the community of modern physics with more analyses and hope to attract more attention to it and promote its applications.
Electron correlation is a measure of the errors that are inherent in the Hartree-Fock theory or orbital models. When the electron density is high, correlation is weak and the traditional electronic theory works well. However, at a low density of electrons correlation effects become strong and the traditional theory fails to describe the electron system correctly. Therefore, the electron correlation plays a radical role in such materials as high-temperature superconductors and heavy fermions, etc. To date, there is no agreement on how to deal with higher-order terms (correlation energy) in the series of electron’s ground state energy although a method that is termed diagrammatic iteration approach (DIA) was developed more than one decade ago by the authors of this article. That is why no consensus on the origin and mechanism of superconductivity has been engaged in superconductivity community. From the viewpoint of methodology, the DIA is indeed an approach to higher-order terms from the lower-order ones, i.e. it is a new method to show how to go beyond the random phase approximation (RPA) step by step by iteration. Here, we are logically presenting it to the community of modern physics with more analyses and hope to attract more attention to it and promote its applications.
The methods that have been used to deal with a many-particle system can be basically sorted into three types: Hamiltonian, field theory and phenomenological method. The first two methods are more popular. Traditionally, the Hamiltonian method has been widely adopted in the conventional electronic theory for metals, alloys and semiconductors. Basically, the mean-field approximation (MFA) that has been working well for a weakly coupled system like a metal is employed to simplify a Hamiltonian corresponding to a particular electron system. However, for a strongly coupled many-particle system like a cuprate superconductor MFA should in principle not apply. Therefore, the field theory on the basis of Green’s function and the Feynman diagrams must be invoked. In this method, one is however more familiar with the random phase approximation (RPA) that gives rise to the same results as MFA because of being short of the information for higher-order terms of interaction. For a strongly coupled electron system, it is obvious that one has to deal with higher-order terms of a pair interaction to get a correct solution. Any ignorance of the higher-order terms implies that the more sophisticated information contained in those terms is discarded. However, to date one has not reached a consensus on how to deal with the higher-order terms beyond RPA. We preset here a method that is termed the diagrammatic iteration approach (DIA) and able to derive higher-order terms of the interaction from the information of lower-order ones on the basis of Feynman diagram, with which one is able to go beyond RPA step by step. It is in principle possible that all of higher-order terms can be obtained, and then sorted to groups of diagrams. It turns out that each of the groups can be replaced by an equivalent one, forming a diagrammatic Dyson-equation-like relation. The diagrammatic solution is eventually “translated” to a four-dimensional integral equation. The method can be applied to a layered 2D system that is a model system of cuprate superconductors and others such as atomic, nuclear, heavy-fermion systems, etc.
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in that it does not have an effective mean-field reference wave function. The ground-state wave functions from all Hamiltonians in the model have the same one-electron reduced density matrix (1-RDM); consequently, one-electron theories such as the Hartree–Fock and density functional theories are inapplicable. In contrast, the ground-state two-electron reduced density matrix (2-RDM) has a one-to-one mapping to the ground-state wave function. For a range of lattices including linear, ladder, and square topologies, we variationally compute the 2-RDM subject to constraints, known as N -representability conditions, that are necessary for the 2-RDM to represent an N -electron ensemble density matrix. We find that for all model Hamiltonians the 2-RDM is accurately computed as long as the D, Q, and G N -representability conditions are supplemented with the T 2 condition. Energies, orbital and geminal occupation numbers, and correlation functions are computed. The model has applications to superconductivity as well as more general pairing phenomena in electronic systems. Effective methods are needed to treat strong electron correlation arising in the study of materials including the study of high-temperature superconductivity and phase transitions.
Abstract Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.
Accurately describing many-body effects in multi-orbital systems remains a major challenge in theoretical condensed matter physics. At present, there is a significant methodological gap between the numerical tools used in ab initio computational materials science and those developed to study strong electronic correlations. The former can treat realistic, large-scale systems but typically neglect many-body effects, while the latter focus on simplified models with only a few degrees of freedom, as only such models can be solved accurately in the presence of strong interactions. The purpose of this thesis is to bridge these two approaches and establish a systematic theoretical framework for realistic correlated electronic materials. This involves a full-cycle methodology that begins with constructing ab initio interacting models from density-functional theory, solving them using dynamical mean-field theory to capture local correlations, and extending beyond to incorporate non-local collective electronic fluctuations. To this end, we introduce the "dual" approach to strong correlations, which includes the dual fermion, dual boson, and dual triply irreducible local expansion methods. The central idea of the dual theories is to shift the reference point of the conventional Feynman diagrammatic expansion from a non-interacting electronic system to an interacting but exactly solvable one. Integrating out this reference system recasts the expansion in an effective dual space, where all diagrammatic building blocks are renormalized by the corresponding impurity quantities. This procedure transforms a non-perturbative expansion in the original variables into a perturbative expansion in terms of dual fermionic and bosonic fields, exact in both the weak- and strong-coupling limits. In this thesis, we collect and systematize the major developments of dual techniques achieved to date.
It is important to understand the phase diagram of electronic states in the CuO2 plane to clarify the mechanism of high-temperature superconductivity. We investigate the ground state of electronic models with strong correlation by employing the optimization variational Monte Carlo method. We consider the two-dimensional Hubbard model as well as the three-band [Formula: see text]–[Formula: see text] model. We use the improved wave function that takes account of inter-site electron correlation to go beyond the Gutzwiller wave function. The ground state energy is lowered considerably, which now gives the best estimate of the ground state energy for the two-dimensional Hubbard model. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We investigate the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion [Formula: see text], the band parameter [Formula: see text] and the electron density [Formula: see text] for the Hubbard model. We show phase diagrams that include superconducting and antiferromagnetic phases. We expect that high-temperature superconductivity occurs near the boundary between antiferromagnetic phase and superconducting one. Since the three-band [Formula: see text]–[Formula: see text] model contains many-band parameters, high-temperature superconductivity may be more likely to occur in the [Formula: see text]–[Formula: see text] model than in single-band models.