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Journal of Mathematical Physics

Criteria for Slater Determinants and Quasiparticle Vacuum States

A. M. Navon

Published May 1, 1969• 4 citations
Publisher

Abstract

New criteria characterizing Slater determinants and quasiparticle vacuum states are obtained. These criteria are expressed as quadratic homogeneous equations in the coefficients of the development of the trial wavefunction in a basis of Slater determinants. A redefinition of quasiparticle vacuum states permits the introduction of quasiparticle transformations which are more general than the generalized Bogoliubov transformations.

DOI
10.1063/1.1664912

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arXiv:cond-mat/0503016•New Journal of Physics•Aug 26, 2005•28 citations
New criteria characterizing Slater determinants and quasiparticle vacuum states are obtained. These criteria are expressed as quadratic homogeneous equations in the coefficients of the development of the trial wavefunction in a basis of Slater determinants. A redefinition of quasiparticle vacuum states permits the introduction of quasiparticle transformations which are more general than the generalized Bogoliubov transformations.
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.
The principle of compensation of dangerous diagrams has previously been the only general criterion for determining the coefficients in the Bogoliubov canonical transformation to quasiparticles. In this paper it is shown that the criterion of maximum overlap of the true ground state with the BCS ground state gives the same condition for determining the coefficients. The criterion of diagonalizing the quadratic part of the reaction operator, which is a generalization of the Brillouin-Brueckner condition, is also equivalent to the above criteria. Finally these criteria are evaluated from the standpoint of "least labor for the most accuracy."
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.
A scattering approach to entanglement in mesoscopic conductors with independent fermionic quasi-particles is discussed. We focus on conductors in the tunnelling limit, where a redefinition of the quasi-particle vacuum transforms the wavefunction from a many-body product state of non-interacting particles to a state describing entangled two-particle excitations out of the new vacuum (Samuelsson, Sukhorukov and Büttiker 2003 Phys. Rev. Lett. 91 157002). The approach is illustrated with two examples: (i) a normal–superconducting system, where the transformation is made between Bogoliubov–de Gennes quasi-particles and Cooper pairs, and (ii) a normal system, where the transformation is made between electron quasi-particles and electron–hole pairs. This is compared to a scheme where an effective two-particle state is derived from the manybody scattering state by a reduced density matrix approach.