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EPJ Web of Conferences

New Pfaffian formulae for an overlap of multiple quasiparticle states

Makito Oi, T. Mizusaki

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The Pfaffian formulae recently derived by the authors are presented. They can be widely applied to evaluations of a variety of quantities, necessary in nuclear many-body theories, such as the generator coordinate method (GCM). As an example, an application to the energy expectation is presented.
DOI
10.1051/epjconf/20146602075

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•Jun 13, 2013•0 citations
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.
We derive an expression that allows for the unambiguous evaluation of the overlap between two arbitrary quasiparticle vacua, including its sign. Our expression is based on the Pfaffian of a skew-symmetric matrix, extending the overlap formula recently proposed by Robledo [Phys. Rev. C 79, 021302(R) (2009)] to the most general case of quasiparticle vacua, including the one of the overlap between two different blocked nnn-quasiparticle states for either even or odd systems. The powerfulness of the method is illustrated for a few typical matrix elements that appear in realistic angular-momentum-restored generator coordinate method calculations when breaking time-reversal invariance and using the full model space of occupied single-particle states.
Background: State-of-the-art multi-reference energy density functional calculations require the computation of norm overlaps between different Bogoliubov quasiparticle many-body states. It is only recently that the efficient and unambiguous calculation of such norm kernels has become available under the form of Pfaffians [L. M. Robledo, Phys. Rev. C 79, 021302 (2009)]. Recently developed particle-number-restored Bogoliubov coupled-cluster (PNR-BCC) and particle-number-restored Bogoliubov many-body perturbation (PNR-BMBPT) ab initio theories [T. Duguet and A. Signoracci, J. Phys. G 44, 015103 (2017)] make use of generalized norm kernels incorporating explicit many-body correlations. In PNR-BCC and PNR-BMBPT, the Bogoliubov states involved in the norm kernels differ specifically via a global gauge rotation.Purpose: The goal of this work is threefold. We wish (i) to propose and implement an alternative to the Pfaffian method to compute unambiguously the norm overlap between arbitrary Bogoliubov quasiparticle states, (ii) to extend the first point to explicitly correlated norm kernels, and (iii) to scrutinize the analytical content of the correlated norm kernels employed in PNR-BMBPT. Point (i) constitutes the purpose of the present paper while points (ii) and (iii) are addressed in a forthcoming paper.Methods: We generalize the method used in another work [T. Duguet and A. Signoracci, J. Phys. G 44, 015103 (2017)] in such a way that it is applicable to kernels involving arbitrary pairs of Bogoliubov states. The formalism is presently explicated in detail in the case of the uncorrelated overlap between arbitrary Bogoliubov states. The power of the method is numerically illustrated and benchmarked against known results on the basis of toy models of increasing complexity.Results: The norm overlap between arbitrary Bogoliubov product states is obtained under a closed-form expression allowing its computation without any phase ambiguity. The formula is physically intuitive, accurate, and versatile. It equally applies to norm overlaps between Bogoliubov states of even or odd number parity. Numerical applications illustrate these features and provide a transparent representation of the content of the norm overlaps.Conclusions: The complex norm overlap between arbitrary Bogoliubov states is computed, without any phase ambiguity, via elementary linear algebra operations. The method can be used in any configuration mixing of orthogonal and non-orthogonal product states. Furthermore, the closed-form expression extends naturally to correlated overlaps at play in PNR-BCC and PNR-BMBPT. As such, the straight overlap between Bogoliubov states is the zero-order reduction of more involved norm kernels to be studied in a forthcoming paper.
We find that the overlap between alpha-like four-body condensation wave function and m-scheme basis state can be represented by the hyper-Pfaffian, which is a natural extension of the Pfaffian for thepair-condensation wave function. This overlap is useful for nuclear structure calculations with the variational Monte Carlo method. As the hyper-Pfaffian has, however, only a few mathematical relations, it is quite difficult to use for numerical computations. We also find that a specific case of hyper-Pfaffian can be expressed by the sum of standard Pfaffians.
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.