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Progress of Theoretical Physics

Singularities of a Mean Field Green's Function

BERTRAND G. GIRAUD

Published Dec 1, 1992• 2 citations
Publisher

Abstract

A mean field approximation allows microscopic calculations of an N-body propagator G(z) = (z − H)-1. We investigate singularities of the resulting approximate Green's function, Gmf(z), by comparing them to the binding energies calculated by the Hartree-Fock approximation. We show that each Hartree-Fock energy induces a singularity of Gmf.

DOI
10.1143/ptp.88.1147

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A mean field approximation allows microscopic calculations of an N-body propagator G(z) = (z − H)-1. We investigate singularities of the resulting approximate Green's function, Gmf(z), by comparing them to the binding energies calculated by the Hartree-Fock approximation. We show that each Hartree-Fock energy induces a singularity of Gmf.
The "phase transitions" predicted within finite microscopic systems by Hartree-Fock-Bogoliubov (HFB) and aspects of the use of the associated broken-syrometry bases in the random-phase approximation (RPA) are considered using soluble models-Evidence is presented in support of the conjecture that the success of these techniques lies in the fact that they mimic singularities in the dependence on interaction strengths of the exact solution.This conjecture provides a natural explanation for why such methods fail close to a point where a phase transition occurs and indicates possible directions for improvement.Phase diagrams at both zero and finite temperature are determined, and simple analytic expressions for the way in which critical scrcn;the scale with particle number are found.It is shown that the ''phase transitions" predicted at finite temperature are relevant.A connection between the singularities referred to above and real phase transitions found in the thermodynamic limit is discussed.It is found that only stable bases can be used in an RPA calculation.This is in particular true for those RPA modes which are not associated with the onset of instability of the basis; these modes do not describe any excited state when tht. basis is unstable.Outside transitional regions certain undesirable features of HFB are unearthed, notably that the HFB ground state energy is not necessarily an upper bound to the exact ground state energy.The effectiveness in this regime of the Hartree-Fock Seniority approximation as a substitute to projection methods is ■ -■■.ated.
The excitation of a many-body system by a one-body perturbation is considered. The stationary phase approximation to a functional-integral representation of the many-body evolution operator shows that the optimal mean field for describing the final expectation values of one- and two-body operators is that given by the time-dependent Hartree-Fock method. The forced Lipkin model is considered as a test of the mean-field approximation.NUCLEAR REACTIONS Mean-field approximation, inclusive observables. Time-dependent Hartree-Fock.
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.