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Physical Review

Slater Determinants and Quasiparticle Vacuum States Constructed from a Finite Number of Fermion States

A. M. Navon, A. Bose

Published Jan 20, 1969• 5 citations
Publisher

Abstract

A theoretical study of single-particle transformations, quasiparticle transformations, Slater determinants, and quasiparticle vacuum states is made for a given finite number of single-fermion states. The usual unitary single-particle or quasiparticle transformations are generalized. The new transformations are the most general transformations which transform Slater determinants into Slater determinants or quasiparticle vacuum states into quasiparticle vacuum states, and they contain as particular cases Thouless's nonunitary transformation. It is shown that several Brillouin theorems used in Hartree-Fock and Hartree-Bogolubov variation theories follow from the form of the continuous variations which can be performed in the space of Slater determinants and in the space of quasiparticle vacuum states.

DOI
10.1103/physrev.177.1514

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Physical Review A•Aug 1, 1985•171 citations
A theoretical study of single-particle transformations, quasiparticle transformations, Slater determinants, and quasiparticle vacuum states is made for a given finite number of single-fermion states. The usual unitary single-particle or quasiparticle transformations are generalized. The new transformations are the most general transformations which transform Slater determinants into Slater determinants or quasiparticle vacuum states into quasiparticle vacuum states, and they contain as particular cases Thouless's nonunitary transformation. It is shown that several Brillouin theorems used in Hartree-Fock and Hartree-Bogolubov variation theories follow from the form of the continuous variations which can be performed in the space of Slater determinants and in the space of quasiparticle vacuum states.
Mean-field methods such as Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based. The HF and HFB wave functions are built out of independent quasiparticles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become nonunitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a nonunitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wave functions. The action of projection operators on nonunitary-based HF states is also discussed and applied, in a variation-after-projection approach, to the one-dimensional Hubbard model with periodic boundary conditions.
A variational principle is formulated to determine the single-particle states, their pairing, and the occupation number distribution for a trial state vector of the Bardeen, Cooper, Schrieffer type. The equations which are derived generalize those of the Hartree-Fock method obtained with a Slater determinant trial wave function. It is pointed out that in a suitable representation the vacuum state of a general quasi-particle transformation has such a trial form which exhibits directly the pairing of single-particle states. Another variational principle determines the excitation energies.Two coupling cases are distinguished: the commutative case in which the self-consistent densities and energies are related to quantities which all commute, and the more general noncommutative case. The latter is of importance in critical-field phenomena. The equations for the commutative case can be written in a matrix form which retains its validity in the more general noncommutative case. The simple matrix commutator equations appear as direct generalizations of the density matrix form of the Hartree-Fock equations.The equations for small oscillations have an equally simple form. Their connection with a diagonal representation of the quasi-particle energies is exhibited in a way which remains valid in the general coupling case. The "unphysical" solutions are excluded by the supplementary condition. The contact with the Green's function approach is established. The generalized matrix form of the Green's function equations shows especially clearly the symmetry properties of the method.
A general theory is proposed for the study of the single-particle description of a nondegenerate finite-particle system in a truncated shell-model space. In this theory, it is established that one can introduce a single-paritcle representation satisfying the requirements that single-particle operators to be defined should obey the Fermion-commutation relation, and that the corresponding free vacuum should also be the lowest eigenstate of the Hamiltonian. It is proved that such a representation based on the true ground state can be derived by the unitary transformation of the usual Fermion operators similarly to the quasi-particle transformation method. The equation is given for the determination of the unitary operator proper to new representation. This approach is proved to be understood as a natural extension of the usual Hartree-Fock theory. The theory is applied to a normal system, and a successive linearization procedure is proposed in order to solve the equation for the unitary operator. The advantages of the theory are discussed in connection with the nuclear shell-model problems.
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.
The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.