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Physical Review C

Fock-space diagonalization of the state-dependent pairing Hamiltonian with the Woods-Saxon mean field

HERVÉ MOLIQUE, J. Dudek

Published Oct 1, 1997• 95 citations
Publisher

Abstract

A particle-number conserving approach is presented to solve the nuclear mean-field plus pairing Hamiltonian problem with a realistic deformed Woods-Saxon single-particle potential. The method is designed for the state-dependent monopole pairing Hamiltonian $H{\^{}}_{\mathrm{pair}}={\ensuremath{\sum}}_{\ensuremath{\alpha}\ensuremath{\beta}}{G}_{\ensuremath{\alpha}\ensuremath{\beta}}{c}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}{c}_{\ensuremath{\beta}}$ with an arbitrary set of matrix elements ${G}_{\ensuremath{\alpha}\ensuremath{\beta}}.$ Symmetries of the Hamiltonians on the many-body level are discussed using the language of $\mathcal{P}$ symmetry introduced earlier in the literature and are employed to diagonalize the problem; the only essential approximation used is a many-body (Fock-space) basis cutoff. An optimal basis construction is discussed and the stability of the final result with respect to the basis cutoff is illustrated in details. Extensions of the concept of $\mathcal{P}$ symmetry are introduced and their consequences for an optimal many-body basis cutoff construction are exploited. An algorithm is constructed allowing to solve the pairing problems in the many-body spaces corresponding to $p\ensuremath{\sim}40$ particles on $n\ensuremath{\sim}80$ levels and for several dozens of lowest lying states with precision $\ensuremath{\sim}$(1--2) % within seconds of the CPU time on a CRAY computer. Among applications, the presence of the low-lying seniority $s=0$ solutions, that are usually poorly described in terms of the standard approximations (BCS, HFB), is discussed and demonstrated to play a role in the interpretation of the spectra of rotating nuclei.

DOI
10.1103/physrevc.56.1795

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A particle-number conserving approach is presented to solve the nuclear mean-field plus pairing Hamiltonian problem with a realistic deformed Woods-Saxon single-particle potential. The method is designed for the state-dependent monopole pairing Hamiltonian Hˆpair=\ensuremath∑\ensuremathα\ensuremathβG\ensuremathα\ensuremathβc\ensuremathα\ifmmode†\else\textdagger\fic\ensuremathα\ifmmode\elseˉ\textasciimacron\fi\ifmmode†\else\textdagger\fic\ensuremathβ\ifmmode\elseˉ\textasciimacron\fic\ensuremathβH{\^{}}_{\mathrm{pair}}={\ensuremath{\sum}}_{\ensuremath{\alpha}\ensuremath{\beta}}{G}_{\ensuremath{\alpha}\ensuremath{\beta}}{c}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}{c}_{\ensuremath{\beta}}Hˆpair​=\ensuremath∑\ensuremathα\ensuremathβ​G\ensuremathα\ensuremathβ​c\ensuremathα\ifmmode†\else\textdagger\fi​c\ensuremathα\ifmmode\elseˉ​\textasciimacron\fi\ifmmode†\else\textdagger\fi​c\ensuremathβ\ifmmode\elseˉ​\textasciimacron\fi​c\ensuremathβ​ with an arbitrary set of matrix elements G\ensuremathα\ensuremathβ.{G}_{\ensuremath{\alpha}\ensuremath{\beta}}.G\ensuremathα\ensuremathβ​. Symmetries of the Hamiltonians on the many-body level are discussed using the language of P\mathcal{P}P symmetry introduced earlier in the literature and are employed to diagonalize the problem; the only essential approximation used is a many-body (Fock-space) basis cutoff. An optimal basis construction is discussed and the stability of the final result with respect to the basis cutoff is illustrated in details. Extensions of the concept of P\mathcal{P}P symmetry are introduced and their consequences for an optimal many-body basis cutoff construction are exploited. An algorithm is constructed allowing to solve the pairing problems in the many-body spaces corresponding to p\ensuremath∼40p\ensuremath{\sim}40p\ensuremath∼40 particles on n\ensuremath∼80n\ensuremath{\sim}80n\ensuremath∼80 levels and for several dozens of lowest lying states with precision \ensuremath∼\ensuremath{\sim}\ensuremath∼(1--2) % within seconds of the CPU time on a CRAY computer. Among applications, the presence of the low-lying seniority s=0s=0s=0 solutions, that are usually poorly described in terms of the standard approximations (BCS, HFB), is discussed and demonstrated to play a role in the interpretation of the spectra of rotating nuclei.
A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a kkk-particle Hilbert subspace can be mapped onto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Feng Pan, V. G. Gueorguiev, and J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004)] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis shows that the standard pairing model with the first step approximation displays similar pair structures of the first few exact low-lying states of the model, which, therefore, provides a link between the two models.
Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei. In their best current level of implementation, their accuracy is of the order of a few percent error on the ground-state correlation energy. Recently implemented variants of these methods are operating a breakthrough in the description of medium-mass open-shell nuclei at a polynomial computational cost while putting state-of-the-art models of internucleon interactions to the test.Purpose: As progress in the design of internucleon interactions is made, and as questions one wishes to answer are refined in connection with increasingly available experimental data, further efforts must be made to tailor many-body methods that can reach an even higher precision for an even larger number of observable quantum states or nuclei. The objective of the present work is to contribute to such a quest by designing and testing a new many-body scheme.Methods: We formulate a truncated configuration-interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total NNN-body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two- and four-quasiparticle excitations. Furthermore, the extent by which the underlying BCS state breaks U(1)\text{U}(1)U(1) symmetry is optimized in the presence of the projected two- and four-quasiparticle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multireference energy density functional calculations. The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem.Results: By construction, the method reproduces exact results for N=2N=2N=2 and N=4N=4N=4. For N=(8,16,20)N=(8,\phantom{\rule{0.16em}{0ex}}16,\phantom{\rule{0.16em}{0ex}}20)N=(8,16,20), the error in the ground-state correlation energy is less than (0.006%, 0.1%, 0.15%) across the entire range of internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to the low-lying spectroscopy, which it does with high accuracy.Conclusions: The numerical cost of the newly designed variational method is polynomial (N6{N}^{6}N6) in system size. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate, and versatile ab initio description of mid-mass open-shell nuclei in the future.
The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted in the present work.
The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrodinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrodinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-orderedk-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.
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.
Combining several techniques, we propose an efficient and numerically reliable method to perform the quantum number projection and configuration mixing for most general mean-field states, i.e., the Hartree-Fock-Bogoliubov (HFB) type product states without symmetry restrictions. As for the example of calculations, we show the results of the simultaneous parity, number and angular-momentum projection from HFB type states generated from the cranked Woods-Saxon mean-field with a very large basis that is composed of Nmax = 20 spherical harmonic oscillator shells.