Exact Particle Number Projection Research Articles

Particle-number projection in the finite-temperature mean-field approximation

Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical deformed nucleus $^{162}$Dy, a typical spherical nucleus $^{148}$Sm, and a transitional nucleus $^{150}$Sm in which the pairing condensate is deformed. We compare the results of this projection with results from the saddle-point approximation and exact shell model Monte Carlo calculations. We find that the approximate canonical HF entropy in the particle-number projection decreases monotonically to zero in the limit when the temperature goes to zero. However, in a nucleus with a strong pairing condensate, the approximate canonical HFB entropy in the particle-number projection decreases monotonically to a negative value, reflecting the violation of particle-number conservation. Computationally, the exact particle-number projection is more efficient than calculating the derivatives required in the saddle-point approximation.

Variation after particle-number projection for the Hartree-Fock-Bogoliubov method with the Skyrme energy density functional

Variation after particle-number restoration is incorporated for the first time into the Hartree-Fock-Bogoliubov (HFB) framework employing the Skyrme energy density functional with zero-range pairing. The resulting projected HFB equations can be expressed in terms of the local gauge-angle-dependent densities. Results of projected calculations are compared with those obtained within the Lipkin-Nogami method in the standard version and with the Lipkin-Nogami method followed by exact particle-number projection.

QUANTUM NUMBER PROJECTIONS OF NUCLEAR DEFORMED STATES

We combine the exact particle-number projection method with the method of Peierls-Yoccoz in order to build the simultaneous eigen-functions of the particle number and the angular momentum operators. In the axial symmetry case, the general expression of the system energy resulting from this double projection is derived. In order to overcome the complexity of the method, the calculations are performed within the Gaussian overlap approximation. It turns out that, on the one hand, the double projection introduces a non–negligible correction of the energy of the system, and on the other hand, this correction is sensitive to the deformation. Future calculations have to therefore include an evaluation of the equilibrium deformation.

Large-scale HFB calculations for deformed nuclei with the exact particle number projection

Recent theoretical advances in the large-scale HFBTHO calculations of nuclear ground-state properties are presented with the emphasis on the exact particle number projection. The applicability of the widely used Lipkin-Nogami procedure is discussed together with the analysis of the particle number projection after variation.

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis. The program HFBTHO (v1.66p)

We describe the program HFBTHO for axially deformed configurational Hartree–Fock–Bogolyubov calculations with Skyrme-forces and zero-range pairing interaction using Harmonic-Oscillator and/or Transformed Harmonic-Oscillator states. The particle-number symmetry is approximately restored using the Lipkin–Nogami prescription, followed by an exact particle number projection after the variation. The program can be used in a variety of applications, including systematic studies of wide ranges of nuclei, both spherical and axially deformed, extending all the way out to nucleon drip lines. Program summary Title of the program: HFBTHO (v1.66p) Catalogue number: ADUI Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUI Licensing provisions: none Computers on which the program has been tested: Pentium-III, Pentium-IV, AMD-Athlon, IBM Power 3, IBM Power 4, Intel Xeon Operating systems: LINUX, Windows Programming language used: FORTRAN-95 Memory required to execute with typical data: 59 MB when using N sh = 20 No. of bits in a word: 64 No. of processors used: 1 Has the code been vectorized?: No No. of bytes in distributed program, including test data, etc.: 195 285 No. of lines in distributed program: 12 058 Distribution format: tar.gz Nature of physical problem: The solution of self-consistent mean-field equations for weakly bound paired nuclei requires a correct description of the asymptotic properties of nuclear quasiparticle wave functions. In the present implementation, this is achieved by using the single-particle wave functions of the Transformed Harmonic Oscillator, which allows for an accurate description of deformation effects and pairing correlations in nuclei arbitrarily close to the particle drip lines. Method of solution: The program uses the axially Transformed Harmonic Oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogolyubov Hamiltonian based on the Skyrme-forces and zero-range pairing interaction until a self-consistent solution is found. Restrictions on the complexity of the problem: Axial-, time-reversal-, and space-inversion symmetries are assumed. Only quasiparticle vacua of even–even nuclei can be calculated. Typical running time: 4 s per iteration on an Intel Xeon 2.8 GHz processor when using N sh = 20 Unusual features of the program: none

Ground State Correlations Associated with Number Symmetry Breaking

The ability of an exact particle-number projection to incorporate correlations in the ground state is investigated. The method is compared to the Lipkin-Nogami prescription and to the exact Richardson solution.

Systematic study of deformed nuclei at the drip lines and beyond

An improved prescription for choosing a transformed harmonic-oscillator (THO) basis for use in configuration-space Hartree-Fock-Bogoliubov (HFB) calculations is presented. The new $\text{HFB}+\text{THO}$ framework that follows accurately reproduces the results of coordinate-space HFB calculations for spherical nuclei, including those that are weakly bound. Furthermore, it is fully automated, facilitating its use in systematic investigations of large sets of nuclei throughout the periodic table. As a first application, we have carried out calculations using the Skyrme force SLy4 and volume pairing, with exact particle-number projection following application of the Lipkin-Nogami prescription. Calculations were performed for all even-even nuclei from the proton drip line to the neutron drip line having proton numbers $Z=2,4,\dots{},108$ and neutron numbers $N=2,4,\dots{},188$. We focus on nuclei near the neutron drip line and find that there exist numerous particle-bound even-even nuclei (i.e., nuclei with negative Fermi energies) that have at the same time negative two-neutron separation energies. This phenomenon, which was earlier noted for light nuclei, is attributed to bound shape isomers beyond the drip line.

Restoring of broken symmetries in the generator-coordinate method

Corrections to the potential and kinetic part of the collective hamiltonian due to particle-number projection have been derived within the generator-coordinate plus generalized gaussian-overlap (GCM + GOA) approach. The collective Schrödinger equation was constructed from the single-particle plus pairing hamiltonian using BCS generating functions. This approximation, connected with a rotation in gauge space, turns out to yield results very close to those of an exact particle-number projection. No corrections to the mass parameters were found. The GCM approach to the particle-number projection is simply generalized to restoring any broken symmetry, such as e.g. angular-momentum projection.