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Journal of Mathematical Physics

Self-Consistent Approximations in Many-Body Systems

M. Revzen, Lynn E. H. Trainor

Published Sep 1, 1969• 0 citations
Publisher

Abstract

A new stationary property of the grand canonical potential is presented. This property is used to define a class of self-consistent approximations which is particularly appropriate for the so-called ladder approximation. The conditions which guarantee self-consistency are only sufficient. A comparison is made with the already-known stationary properties related to Hartree-Fock and random-phase approximations. Remarks are made on the possibility of a collective excitation associated with the formalism.

DOI
10.1063/1.1665030

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Physical Review C•Dec 1, 1973•1 citation
A new stationary property of the grand canonical potential is presented. This property is used to define a class of self-consistent approximations which is particularly appropriate for the so-called ladder approximation. The conditions which guarantee self-consistency are only sufficient. A comparison is made with the already-known stationary properties related to Hartree-Fock and random-phase approximations. Remarks are made on the possibility of a collective excitation associated with the formalism.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
Some model Hamiltonians are proposed for quantum-mechanical many-body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature-domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self-energy corrections. The latter are expected to yield damping of single-particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree-Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree-Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.
The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Pock (HP) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.