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arXiv:0901.3213 · Physical Review C

Sign of the overlap of Hartree-Fock-Bogoliubov wave functions

Luis Miguel Robledo

Published Feb 20, 2009• 134 citations
arXivHTMLPDF

Abstract

The problem of how to compute accurately and efficiently the sign of the overlap between two general Hartree-Fock-Bogoliubov (HFB) wave functions is addressed. The results obtained can easily be extrapolated to the evaluation of the sign of the trace of a density operator exponential of one body operators.
DOI
10.1103/physrevc.79.021302

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Several technical aspects concerning the evaluation of the overlap between two mean-field wave functions of the Hartree-Fock-Bogoliubov type are discussed. The limit when several orbitals become fully occupied is derived as well as the formula to reduce the dimensionality of the problem when exactly empty orbitals are present. The formalism is also extended to deal with different bases for each of the wave functions. Several practical results concerning the evaluation of pfaffians, as well as the canonical decomposition of norm overlaps, are also discussed in the Appendices.
The nuclear many-body approach beyond the mean-field approx- imation demands overlap calculations of different many-body states. Norm overlaps between two different Hartree-Fock-Bogoliubov states can be calcu- lated by means of the Onishi formula. However, the formula leaves the sign of the norm overlap undetermined. Several approaches have been proposed by Hara-Hayashi-Ring, Neergard-Wust, and Robledo. In the present paper, the Neergard-Wust formula is examined whether it is applicable to practical numer- ical calculations, although the formula was dismissed by many nuclear theoreti- cians so far for unknown reasons.
In this Letter, we present a new expression for the overlaps of wave functions in Hartree-Fock-Bogoliubov based theories. Starting from the Pfaffian formula by Bertsch et al. [1], an exact and computationally stable formula for overlaps is derived. We illustrate the convenience of this new formulation with a numerical application in the context of the particle-number projection method. This new formula allows for substantially increased precision and versatility in chemical, atomic, and nuclear physics applications, particularly for methods dealing with superfluidity, symmetry restoration, and uses of nonorthogonal many-body basis states.
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)
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.