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Zeitschrift für Naturforschung A

Eine gruppentheoretische Methode zur Berechnung von Skalarprodukten zwischen HFB-Zuständen / A Group Theoretical Method for Calculating Overlaps of HFB States

Heinz Günter Becker

Published Jul 1, 1973• 2 citations
Publisher

Abstract

Overlaps of Hartree-Fock-Bogoljubov states occurring in modified HFB theories (e. g. projected HFB) are calculated using group theoretical methods. For this purpose the Bogoliubov-Valatin transformation is generalized. The resulting nonunitary transformation leaves the Fermi commutation relations invariant, but the characteristic property of the creation and annihilation operators, namely that they are hermitean conjugate to each other, is lost. A suitable factorization of the new transformation is derived, which also applies to the unitary Bogoliubov-Valatin transformation but the structure of which is completely different from the well-known Bloch-Messiah decomposition. Furthermore, the relation between various representations of the HFB state vector is clarified.

DOI
10.1515/zna-1973-0701

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Overlaps of Hartree- Fock- Bogoljubov states occurring in modified HFB theories (e.g., projected HFB) are calculated using group theoretical methods. For this purpose the BogoliubovValatin transformation is generalized. The resulting nonunitary transformation leaves the Fermi commutation relations invariant, but the characteristic property of the creation and annihilation operators, namely that they are hermitean conjugate to each other, is lost. A suitable factorization of the new transformation is derived, which also applies to the unitary Bogoliubov- Valatin transformation but the structure of which is completely different from the well known Bloch-- Messiah decomposition. Furthermore, the relation between various representations of the HFB state vector is clarified. (GE)

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Overlaps of Hartree-Fock-Bogoljubov states occurring in modified HFB theories (e. g. projected HFB) are calculated using group theoretical methods. For this purpose the Bogoliubov-Valatin transformation is generalized. The resulting nonunitary transformation leaves the Fermi commutation relations invariant, but the characteristic property of the creation and annihilation operators, namely that they are hermitean conjugate to each other, is lost. A suitable factorization of the new transformation is derived, which also applies to the unitary Bogoliubov-Valatin transformation but the structure of which is completely different from the well-known Bloch-Messiah decomposition. Furthermore, the relation between various representations of the HFB state vector is clarified.
Overlaps of Hartree- Fock- Bogoljubov states occurring in modified HFB theories (e.g., projected HFB) are calculated using group theoretical methods. For this purpose the BogoliubovValatin transformation is generalized. The resulting nonunitary transformation leaves the Fermi commutation relations invariant, but the characteristic property of the creation and annihilation operators, namely that they are hermitean conjugate to each other, is lost. A suitable factorization of the new transformation is derived, which also applies to the unitary Bogoliubov- Valatin transformation but the structure of which is completely different from the well known Bloch-- Messiah decomposition. Furthermore, the relation between various representations of the HFB state vector is clarified. (GE)
Abstract The structure of the Bogoljubov -Valatin transformation and the HFB state is investigated by group-theoretical methods. A direct proof of the Bloch-Messiah factorization theorem shows that the second factor which contains the BCS part of the BV transformation can be chosen always real. An unitary representation of the Bogoljubov-Valatin group in the Fock space is constructed which allows to write down various equivalent expressions for the HFB state vector. The relation between the "unitary form" and the usual form of | HFB〉 is clarified using the "quasispin" concept in connection with a theorem about the factorization of the D(3) Lie group.
Using the Pfaffian identity, the formula for the overlap between Hartree-Fock-Bogoliubov (HFB) vacua, recently proposed by Bertsch and Robledo [Phys. Rev. Lett. {\bf 108}, 042505(2012)] is deduced into a simpler form. This new formula does not explicitly depend on the Bloch-Messiah decomposition of the HFB transformation, but it requires the existence of the V−1V^{-1}V−1 matrix. It presents the significant convenience of application for the symmetry restoration in the wave functions of general Bogoliubov form. Numerical calculations show that this formula is able to reproduce the results of the Onishi formula with the same accuracy as well as having the correct sign. It is also shown that a small number of the integrating mesh points (Nmesh=16N_\text{mesh}=16Nmesh​=16) is enough to perform the particle number projection even for the heavy nuclear system.
A formula to calculate the norm overlap between Hartree–Fock–Bogoliubov (HFB) states with odd number parity (one quasi-particle excited states) is derived with the help of Grassmann numbers and Fermion coherent states. The final formula is expressed in terms of a product of the Pfaffian for a neighboring even–even system (the zero quasi-particle state), and an extra factor consisting of the Bogoliubov transformation matrix and the anti-symmetric matrix in Thoulessʼ HFB ansatz for the even–even system.
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.
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.