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Overlap formula between the HFB vacua for the symmetry restoration

Zao-Chun Gao, Qing-Li Hu, Y. S. Chen

Published Jun 13, 2013• 0 citations
OpenAlex

Abstract

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 $V^{-1}$ 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 ($N_\text{mesh}=16$) is enough to perform the particle number projection even for the heavy nuclear system.

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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.
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.
The restoration of particle number and angular momentum symmetry using projection operators has been theoretically investigated for mean-field Hartree-Fock-Bogoliubov (HFB) states. A computer code for the projection of particle number was then implemented. To do so efficiently and avoid the sign ambiguity of the Onishi formula, the computation of overlap between quasi-particle vaccums using Pfaffians was investigated. Different algorithms for the the Pfaffian from~\cite{GBBertschRobledo},\cite{Wimmer}, was tested for performance and accuracy. For the same reason different truncations of the model space was also investigated. The particle number projector was then implemented in \textit{HOSPHE} \cite{Carlsson} for calculations of ground states pertaining to an effective Quadrupole plus Pairing Hamiltonian calculated using the SLy4 parametrization of the Skyrme interaction.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, we extend the Pfaffian formula for the many-body traces of HFB density operators introduced by L. M. Robledo in Ref. [1] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. We validate our method through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, I extend the Pfaffian formula for the many-body traces of HFB density operators introduced by Robledo [L. M. Robledo, Phys. Rev. C. 79, 021302(R) (2009)] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. The method is validated through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
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.