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physica status solidi (b)

Reference System Approach in the Electron Liquid Theory I. General Relations

M. V. Vavrukh, Taras Krokhmalskii

Published Dec 1, 1991• 11 citations
Publisher

Abstract

Abstract A new formalism of the statistical description of the electron liquid is proposed. The renormalized statistical perturbation theory is formulated in terms of n‐particle correlation functions of some simpler model system, called the reference system. The functional representations of the partition function, free energy, ground state energy, and two‐particle correlation function are obtained for quantum systems on the basis of the method of collective variables. The approach under consideration describes, in particular, the strongly coupled electron liquid, it is general enough and can be generalized to the case of superconducting electron systems, systems in external fields, and also Bose systems.

DOI
10.1002/pssb.2221680213

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Progress of Theoretical Physics•Jan 1, 1960•1 citation
Abstract A new formalism of the statistical description of the electron liquid is proposed. The renormalized statistical perturbation theory is formulated in terms of n‐particle correlation functions of some simpler model system, called the reference system. The functional representations of the partition function, free energy, ground state energy, and two‐particle correlation function are obtained for quantum systems on the basis of the method of collective variables. The approach under consideration describes, in particular, the strongly coupled electron liquid, it is general enough and can be generalized to the case of superconducting electron systems, systems in external fields, and also Bose systems.
Abstract The diagrammatic strong‐coupling perturbation theory (SCPT) for correlated electron systems is developed for intersite Coulomb interaction and for a nonorthogonal basis set. The construction is based on iterations of exact closed equations for many‐electron Green functions (GFs) for Hubbard operators in terms of functional derivatives with respect to external sources. The graphs, which do not contain the contributions from the fluctuations of the local population numbers of the ion states, play a special role: a one‐to‐one correspondence is found between the subset of such graphs for the many‐electron GFs and the complete set of Feynman graphs of weak‐coupling perturbation theory (WCPT) for single‐electron GFs. This fact is used for formulation of the approximation of renormalized Fermions (ARF) in which the many‐electron quasi‐particles behave analogously to normal Fermions. Then, by analyzing: (a) Sham's equation, which connects the self‐energy and the exchange–correlation potential in density functional theory (DFT); and (b) the Galitskii and Migdal expressions for the total energy, written within WCPT and within ARF SCPT, a way we suggest a method to improve the description of the systems with correlated electrons within the local density approximation (LDA) to DFT. The formulation, in terms of renormalized Fermions LDA (RF LDA), is obtained by introducing the spectral weights of the many‐electron GFs into the definitions of the charge density, the overlap matrices, effective mixing and hopping matrix elements, into existing electronic structure codes, whereas the weights themselves have to be found from an additional set of equations. Compared with LDA+U and self‐interaction correction (SIC) methods, RF LDA has the advantage of taking into account the transfer of spectral weights, and, when formulated in terms of GFs, also allows for consideration of excitations and nonzero temperature. Going beyond the ARF SCPT, as well as RF LDA, and taking into account the fluctuations of ion population numbers would require writing completely new codes for ab initio calculations. The application of RF LDA for ab initio band structure calculations for rare earth metals is presented in part II of this study (this issue). © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also 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
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
Many theoretical methods for N -electron systems use a reference state whose electronic density in general differs from that of the associated eigenstate. In contrast, the density functional theory of Kohn and Sham postulates a model reference system for which these two densities are identical. An alternative postulate, requiring the reference state to have maximum projection on the true wave function (the Brueckner−Brenig construction), is shown here to provide a theory in which the ground-state energy is a functional of the reference-state density. This revised formalism makes it possible to identify elements of the theory with specific formulas and concepts of formal many-body theory. In particular, the one-electron wave functions are in one-to-one correspondence with physical quasiparticles of a generalized Fermi-liquid theory, and the one-electron energies are Landau quasiparticle energies.
The method of auxiliary variables is generalized so as to be applicable to the systems having strong interparticle interactions or obeying Fermi-Dirac statistics. The ground state energy and the excitation energy specrum are calculated to the zeroth approximation. It is shown that Feynman's relation for liquid helium and Tomonaga's expression for a one-dimensional Fermi gas can be derived as the special cases of the present theory. The theory is applied also to the classical statistical mechanics.