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Condensed Matter Physics

DIAGRAM TECHNIQUE FOR THE HUBBARD MODEL. LADDER DIAGRAM SUMMATION

V. A. Moskalenko, L.Z. Kon

Published Jan 1, 1998• 12 citations
Publisher

Abstract

A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations.Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation.The hopping matrix elements are considered as a perturbation.One-particle Matsubara-Green function of the model has been investigated and Dyson equation has been obtained.New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants.They become zero when Coulomb interaction is zero.The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system.The integral equations, which sum such diagrams, have been established for two different channels.The coherent potential approximation has been used to simplify and solve these equations.On this basis a metal-dielectric phase transition has been investigated.

DOI
10.5488/cmp.1.1.23

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A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations.Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation.The hopping matrix elements are considered as a perturbation.One-particle Matsubara-Green function of the model has been investigated and Dyson equation has been obtained.New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants.They become zero when Coulomb interaction is zero.The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system.The integral equations, which sum such diagrams, have been established for two different channels.The coherent potential approximation has been used to simplify and solve these equations.On this basis a metal-dielectric phase transition has been investigated.
A new diagram technique, previously proposed for Hubbard Model, was used for the Periodic Anderson Model (PAM).The Hamiltonian of localized electrons H 0 f and of free conduction electrons H 0 c are considered as a main part of the Hamiltonian of PAM and the hybridization term is treated as a perturbation.H 0 f contains on-site Coulomb interaction and is diagonalized in Hubbard operators.This diagram technique is used for Matsubara Green functions and contains apart from the Wick product of free Green functions new contributions equal to on-site cumulants depending of the Coulomb interaction.If such cumulants are ignored we obtaine the so called Hubbard I approximation for PAM with 3 branches of renormalized energy spectrum of quasiparticles.They are investigated in detail and the chemical potential of the system is determined
In a large variety of systems, like the high‐temperature superconductors, the dominant parameter of the system is the on‐site Coulomb repulsion that becomes comparable with the kinetic energy in the vicinity of the metal‐insulator transition. This implies that perturbation expansions must include this interaction in the zeroth order Hamiltonian and treat the hopping term as a perturbation. Thus, one gets a situation radically different from the usual one in which the particles are initially free and needs to develop a nonstandard perturbation theory based on a cumulant expansion. In this paper, we revisit the general formalism of the cumulant expansion approach for the single band and two‐band Hubbard model, being interested in introducing a systematic way to perform approximations which take into account cooperative phenomena. As an example we analyze the effect of the charge fluctuations on the single‐particle spectral properties.
We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
Abstract The theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003
The Hubbard-Holstein model after canonical Lang-Firsov transformation is investigated. Two peculiarities of the model are taken into account: a) the strong electron correlations and b) the fact that the hoppings of polarons between lattice sites are accompanied by an indefinite number of phonons. A new diagram technique based on the classification of irreducible many-particle Green functions or Kubo cumulants is developed. Phonon renormalization of electronic functions is considered. A theorem about connected vacuum diagrams is established, and Dyson equation for electron Green function is obtained.