Pairing correlations and particle-number projection methods

A systematic study of the pairing correlations derived from various particle-number projection methods is performed in an exactly soluble cranked-deformed shell model Hamiltonian. It is shown that most of the approximate particle-number projection methods including the method of Lipkin and Nogami, which is used quite extensively in nuclear structure studies, break down in the weak pairing limit. The results obtained from the recently formulated number-projected Hartree-Fock-Bogoliubov ~PHFB! equations, on the other hand, are in complete agreement with the exact solutions of the model Hamiltonian. The pairing energy calculated from the PHFB method is shown to be finite in all the studied limits. Pairing correlations play a central role in describing the properties of atomic nuclei @1,2#. The bulk of the nuclear phenomena, for instance, the odd-even mass differences and moments of inertia of deformed nuclei can be described by considering nucleons to be in a superfluid paired phase. The recent discovery of halo nuclei have also elucidated the im- portance of pairing correlations in the formation of such structures @3-5#. In the traditional nuclear structure models, the pairing correlations are treated in Bardeen-Cooper- Schrieffer ~BCS! or Hartree-Fock-Bogoliubov ~HFB! ap- proximations @2#. These approaches use a simple ansatz of a product state of quasiparticles for the ground-state wave function. The product ansatz leads to a set of nonlinear equations in the BCS theory or to the diagonalization of a matrix in the case of HFB approach. There exist now very fast numerical methods to find accurate solutions of these problems. The major advantage in the mean-field models is that the number of nonlinear equations or the dimensionality of the matrix to be diagonalized is quite small as compared to the exact analysis, which is impossible to apply in a realistic configuration space of medium and heavy mass nuclei. Nevertheless, the product ansatz of the mean-field ap- proach breaks the symmetries that the orginal many-body Hamiltonian obeys. For instance, in the case of BCS theory, the product state violates the gauge symmetery associated with the particle number. The BCS product state of quasipar- ticles does not have a well defined particle number. The most undesirable feature associated with broken symmetry is that it leads to sharp phase transitions. These phase transitions may be reasonable for a system with a very large number of particles, for example, a metallic superconductor. But for a finite mesoscopic system these phase transitions are washed out by fluctuations and are not observed in the experimental data or in the exact study of toy models. The sharp transition is an artifact of the simple ansatz of the product state, which does not contain fluctuations. The symmetry restoration in the mean-field studies is a major challenge to the nuclear structure models. Although the projection methods that restore the broken symmetries have been known for more than 30 years, but have been studied only in simple model cases due to the numerical difficulties. In most of the realistic studies, approximate pro- jection methods have been employed @6#. Only recently, the gradient method in the projected energy surface has been employed in realistic calculations with Gogny forces @7# . I t has been an unsolved problem whether simple HFB-like equations can be obtained with the projected energy func- tional. This would allow the usage of fast numerical algo- rithms for diagonalization of matrices as is done in the ordi- nary HFB theory. This problem has been recently addressed and it has been shown that the variation of an arbitrary en- ergy functional, which is completely expressed in terms of the HFB density matrix r and the pairing tensor k, results in the HFB like equation @8#. It is noted that the projected en- ergy functional can also be expressed in terms of the HFB density matrices r and k, and therefore one obtains HFB- like equations with modified expressions for the pairing po- tential and the Hartree-Fock field. In the present work, we shall limit ourselves to the resto- ration of the gauge symmetry associated with the particle- number projection, which is important in the discussion of the pairing correlations. Several approximate particle- number projection methods have been developed and the most popular of these is the method of Lipkin and Nogami @9-12#. The purpose of the present work is to perform a systematic study of these various approximate projection methods and to compare them with the recently developed number-projected HFB ~PHFB! approach. It is shown that most of the approximate methods including the Lipkin- Nogami method break down in the weak pairing limit as has been shown in earlier studies @13#. The PHFB method, on the other hand, gives finite pair correlations in all the cases stud- ied. The paper is organized as follows: a brief survey of the the particle-number projection methods is given in the next section and in the Appendix. The model Hamitonian used in the present work is expressed in Sec. III, the numerical re-