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arXiv:2501.00385

Pfaffian and minor formulas in number-conserved pairing theory

G. J. Fu

Published Dec 31, 2024• 0 citations
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Abstract

In this work, I present closed-form formulas for the norm and many-body density matrices between wave functions with exact particle numbers in pairing theory. The expressions, written as sums of Pfaffians and minors, apply to both even and odd particle-number systems and to pair condensates as well as broken-pair configurations. This formalism facilitates calculations using the generator coordinate method and symmetry restoration techniques, such as angular momentum projection. I apply these formulas to $^{70}$Ge and $^{69}$Ge using the angular-momentum-projected number-conserved BCS theory. The calculations describe low-lying collective bands and provide an interpretation of their intrinsic structures.

DOI
10.48550/arxiv.2501.00385

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In this work, I present closed-form formulas for the norm and many-body density matrices between wave functions with exact particle numbers in pairing theory. The expressions, written as sums of Pfaffians and minors, apply to both even and odd particle-number systems and to pair condensates as well as broken-pair configurations. This formalism facilitates calculations using the generator coordinate method and symmetry restoration techniques, such as angular momentum projection. I apply these formulas to 70^{70}70Ge and 69^{69}69Ge using the angular-momentum-projected number-conserved BCS theory. The calculations describe low-lying collective bands and provide an interpretation of their intrinsic structures.
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 Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.
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.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.