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Physical Review A

Self-consistent effective particle-hole interaction in the Lipkin model

G. Wagner, VICTOR A. KHODEL, Peter Schuck

Published Dec 1, 1995• 1 citation
Publisher

Abstract

Ground-state energies and frequencies of collective modes for the Lipkin model are evaluated within the framework of a microscopic functional approach using functional relations between the ground-state energy, the matrix of the effective interaction R^ between particles in the medium, and the linear response function of a system. A strict functional equation for R^ is derived. A special version of the well-known local approximation is elaborated to transform this equation into a system of integro-differential equations, which is solved numerically. Most of the numerical calculations are carried out in the region of moderate coupling constants, where the ground state possesses the same symmetry as the noninteracting system. The deviation of the calculated ground-state energies from the exact values is found not to exceed 0.1% in a wide region of the variation of the interaction strength, including the vicinity of the phase-transition point. This is better by approximately one order of magnitude than the results from the established many-body approaches.

DOI
10.1103/physreva.52.4523

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Ground-state energies and frequencies of collective modes for the Lipkin model are evaluated within the framework of a microscopic functional approach using functional relations between the ground-state energy, the matrix of the effective interaction R^ between particles in the medium, and the linear response function of a system. A strict functional equation for R^ is derived. A special version of the well-known local approximation is elaborated to transform this equation into a system of integro-differential equations, which is solved numerically. Most of the numerical calculations are carried out in the region of moderate coupling constants, where the ground state possesses the same symmetry as the noninteracting system. The deviation of the calculated ground-state energies from the exact values is found not to exceed 0.1% in a wide region of the variation of the interaction strength, including the vicinity of the phase-transition point. This is better by approximately one order of magnitude than the results from the established many-body approaches.
A new microscopic approach to the theory of dynamic correlation, ground state and collective excitations of many‐body systems is suggested. These dynamic and ground state properties are known as being interconnected and should be considered on the same base. A useful tool for this goal is a response function formalism. An exact functional equation for effective interaction and linear response function is obtained. The reduction method of the functional equation to a closed set of equations is given. In the simplest case the functional equation may be transformed into nonlinear integrodifferential equation. Few examples of many‐body systems are considered on the base of this equation. The developed method is applied to gas‐liquid phase transition, for evaluation of transport coefficients and collective excitation width.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
Earlier papers in this series have shown how the extended coupled-cluster method (ECCM) provides a widely applicable biorthogonal formulation of quantum many-body and quantum field theory. Rules for calculating matrix elements of arbitrary operators have been formulated, and an exact description of the excited states has been given in terms of a generalized random-phase approximation for the many-body, quasilocal c-number (classical), ECCM cluster amplitudes which completely and exactly parametrize the system. The present paper introduces a variational functional in order to treat the excited states differently within the ECCM framework, so as to allow extra flexibility in the choice of approximation schemes. The method is tested on the SU(2) quasispin model of Lipkin, Meshkov, and Glick [Nucl. Phys. 62, 188 (1965); 62, 199 (1965)], which contains the spontaneous breakdown of parity, and which models a shape transition in finite atomic nuclei. The method very well describes both the ground and excited states of the model Hamiltonian over a broad range of coupling parameters on both sides of the critical value corresponding to the transition.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
In an accompanying paper, we have described a new approach to the computation of ground-state energies and of correlated particle-hole excitations in terms of a phenomenological pseudo-Hamiltonian containing two- and more-body smooth effective potentials which sum the results of short-range correlations and of density fluctuation amplitudes which describe long-range correlations. In this paper we study the problem of relating the pseudopotentials to a (possibly singular) microscopic interaction with the aid of tools developed within the framework of the coupled cluster theory of K\"ummel and his collaborators, the results containing both familiar and unfamiliar elements. For example, the formulas derived, which depend on Bethe-Goldstone and Bethe-Faddeev amplitudes, include new definitions of particle-hole scattering matrix elements. An important consistency check is satisfied, in that the problem of defining the phenomenological potentials in terms of the microscopic ones must yield two separate but equivalent solutions, once within the framework of the theory of the ground-state energy and a second time within the framework of the theory of excitations. The entire package is studied with the aid of a modified version of coupled cluster theory, and shown to form a self-consistent entity. Among the desirable features of the formalism is that the large gap in the single-particle energy spectrum often utilized in existing formalisms is naturally absent from the current one.