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Progress of Theoretical Physics

Stability Condition for the Hartree-Fock-Bogoliubov Ground State under Constraints

R. Okamoto, K. Suzuki

Published Aug 1, 1983• 1 citation
Publisher

Abstract

A general stability condition is discussed for the Hartree-Fock-Bogoliubov variational problem with some constraints. It is proved that the criterion for stability can be expressed in the same form as that given for the case without constraint.
DOI
10.1143/ptp.70.606

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A necessary condition for the stability of the Hartree-Fock solution in many-body problem is presented. If this condition is not satisfied, then the solution becomes no longer stable as was discussed by Overhauser in connection with the one-dimensional spin wave model. A variational method is proposed in this case to construct a stable solution which has definitely a lower energy than the Hartree-Fock value. The workability of our method is tested in some realistic examples like the B.C.S.-Bogolyubov theory of superconductivity. Then the method is applied to field theory which seems to be inconsistent in view of the presence of “ghost states”. A canonical transformation leads to a new vacuum state with lower energy, but the high momentum part of the coupling is not damped.
Hartree-Fock-Bogoliubov (HFB) theory is generalized by eliminating the restriction to quasiparticle vacuum states. However, the fundamental HFB transformation to quasiparticles is retained. The theory is formulated in terms of the quasiparticle density matrix RRR, but, in contrast to conventional HFB, there is no restriction to vacuum densities R2=R{R}^{2}=RR2=R.
This paper describes a general group theoretical analysis of the temperature–Hartree-Fock–Bogoliubov (HFB) equation and its solution. The action of the symmetry group G0 of the system on the HFB Hamiltonian, the HFB density matrix, and the HFB Green function are defined. It is shown that the HFB equation and its solution is classified by a subgroup G of G0, which is the invariance group of the HFB Hamiltonian, the HFB density matrix, and the HFB Green function corresponding the solution. General expression of the instability of a solution and its decomposition into R-rep (single-valued irreducible representation over the real number field) components of the invariance group of the solution are obtained. The self-consistent field (SCF) condition is decomposed into R-rep components of G0.