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International Journal of Modern Physics B

Electron correlation effects beyond the random phase approximation

J. D. Fan, Y. M. Malozovsky

Published Apr 7, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979216420066

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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.
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.
This chapter examines different problems that apply the field theory in condensed matter physics. It considers bosonic as well as fermionic systems and applies the techniques of the quantum field theory. The chapter provides a model describing electrons in a metal known as the electron gas model that considers a system of electrons interacting with each other via the instantaneous Coulomb force e2r. The electron gas model provides a framework from which introduces standard many-body approaches such as the Gell-Mann-Bruckner theory or random phase approximation and show how they are formulated within diagrammatic or functional integral techniques. The chapter also considers a quantum three-dimensional coulomb gas and apply the field-theoretic techniques to systems of interacting electrons. The terminology of the order parameter for the field ϕ is to indicate the treatment of the Coulomb interaction within a mean-field theory with the inclusion of Gaussian fluctuations around the mean-field value of the order parameter.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
Strong electronic correlations are the root of many fascinating phenomena of condensed matter physics. However, they also make calculations notoriously difficult. This thesis is concerned with the development and improvement of computational methods for strongly correlated electron systems. After a brief recapitulation of the Green’s function formalism,which is the theoretical basis of most such methods, we discuss the analytical continuation from imaginary to real frequencies. Especially the maximum entropy method is explained in detail and an implementation there of is presented.The remainder of this thesis deals with methods to compute Green’s functions on thei maginary frequency axis. First, the Anderson impurity model (AIM) is considered. It can be solved numerically – a task for which several methods exist. We focus on continuous-time quantum Monte Carlo with worm sampling, through which various kinds of correlation functions can be calculated. On this basis we derive and implement symmetric improved estimators that allow for the computation of the self-energy and vertex functions where the level of stochastic noise is largely reduced with respect to conventional estimators.As a next step the dynamical mean-field theory (DMFT) is introduced as a method to capture local correlations in the Hubbard model (HM) by self-consistently mapping it onto the AIM. We study the convergence behavior of this iterative procedure and the improvement of convergence due to mixing. Then, we compare electron scattering rates from DMFT in the weak-coupling regime of the square-lattice HM to results from Boltzmann scattering theory.While local correlations account for many interesting phenomena, such as the Mott-Hubbard metal-insulator transition, they are not sufficient to correctly describe, e.g., the physics of phase transitions in two- or three-dimensional systems. It is thus the final step in this thesis to also include nonlocal correlations by means of the dynamical vertex approximation (DΓA). For this we develop a new self-consistency scheme, which is then applied to several systems: We study the square-lattice HM, where we compare to benchmark methods in the weak-coupling regime, but also present results for intermediate and strong coupling. Furthermore, a two-band model with doping and a three-band Wannier model for strontium vanadate are studied. As an application of our new self-consistent DΓA, we study the effects of geometrical frustration in the kagome lattice, where we also compare our results to determinant quantum Monte Carlo.
In this second work in a series of papers the coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2{S}_{2}S2​ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.
The coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of correlations in a many-electron system. In this first work in a series of papers we concentrate on exact results and restrict ourselves to various ring approximations applicable essentially to the high-density regime. It is shown that the well-known random-phase approximation (RPA) can be formulated as a nonlinear three-dimensional integral equation for the four-point function S2{S}_{2}S2​ which provides a measure of the two-particle---hole pair component in the true ground-state wave function. Exact analytic solutions are presented for both S2{S}_{2}S2​ and the three-point particle-hole vertex function in the RPA, and their properties are discussed. The Tamm-Dancoff approximation to the ring summation is also formulated, and the analagous exact solutions are presented for the first time. It is demonstrated that both solutions are unique, and a comparison of the two approximations is presented in terms of the correlation energy.