Improvement to the Projected BCS Approximation

We consider the structure of the number projected BCS wave function in the particle-hole basis, and use it to study several approximate treatments of pairing. The analysis is carried out for the exactly solvable Richardson model involving a pure pairing hamiltonian acting in a space of equally spaced doubly degenerate levels at half filling. 1. Inroduction Pairing correlations are ubiquitous in strongly-correlated systems, ranging from condensed matter to quantum optics to cold atomic gases to atomic nuclei. A traditional starting point for the description of these correlations is through the use of the Bardeen Cooper Schrieffer (BCS) approximation [1], whereby the correlations are described by means of a coherent state of collective pairs that breaks the conservation of particle number. This method is especially useful in the description of systems with a very large number of interacting particles where the fluctuations in the particle number is negligible. For systems with a fairly small number of particles, e.g. atomic nuclei or superconducting grains, it is important to restore particle number, through the use of the number projected BCS (PHBCS) approximation [2]. Pairing correlations can be treated exactly using the Richardson method when the hamiltonian is of a pure BCS form [3]. Likewise, more general pairing Hamiltonians are exactly solvable if they can be expressed as a linear combination of the set of integrals that define the Richardson-Gaudin models [4]. This exact solvability has enabled the test of approximate methods of treating pairing for a wide variety of systems, like small superconducting grains [5, 6] or realistic atomic nuclei [7, 8, 9, 10, 11]. Such tests have illustrated that for a large enough number of active orbits over which the pairing acts, even the PBCS approximation misses important pairing correlations, making its use in large scale energy density functional treatments of finite nuclei suspect. This has led to a multitude of efforts to develop improved approximate treatments of pairing correlations. This includes, e.g., the use of RPA methods [8] and the use of coupled cluster methods [11]. In this work we study the accuracy of the PBCS approximations and propose an alternative method based on a generalization of the PBCS wave function for an improved approximate treatment of pairing correlations. The method starts with the PBCS approximation, which is then expanded in terms of particle-hole excitations around the Hartree-Fock Fermi surface. In the PBCS approximation, each term in the series expansion is defined by the expansion coefficients of a single collective pair and furthermore the contribution of each term is prescribed. XXXVIII Symposium on Nuclear Physics (Cocoyoc 2015) IOP Publishing Journal of Physics: Conference Series 639 (2015) 012009 doi:10.1088/1742-6596/639/1/012009 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 In our improved method, we diagonalize the Hamiltonian in the space of collective particlehole pair excitations, thereby permitting the contribution of each term in the full series to be modified and indeed optimized. We gradually increase the number of terms included in the series expansion until convergence is achieved. We refer to this new approximation, for the sake of terminology, as the Particle-Hole BCS (PHBCS) approximation [6]. The structure of the paper is as follows. In section II, we describe the PHBCS approximation and detail its differences relative to the other approximations against which we will compare it. In Section III, we describe the model that we use to carry out comparative tests of the various approximations and then in Section IV we describe the results of this comparison and draw some conclusions. In Section V we summarize the main results of the work and outline some issues for future consideration. 2. The Particle-Hole BCS approximation Consider a set of N particle pairs moving in a space of Ω doubly-degenerate single-particle states i, ī and denote the single-particle creation and annihilation operators associated with these states as ci , c † ī and ci, cī, respectively. Furthermore, denote the operators that create and annihilate a pair of particles in doubly-degenerate time-reversed states as P † i = c † ic † ī , (1) Pi = [ P † i ]† = cīci , (2) which satisfy the commutation [ Pi, P † j ] = δij (1−Ni) [ Ni, P † j ] = 2δijP † j (3) where Ni = c † ici + c † ī cī . The traditional PBCS state can be expressed as a condensate of M = N2 collective pairs, viz. |PBCS⟩ = 1 √ Z1,Ω,M [ Γ†(x) ]M | 0⟩ , Γ†(x) = Ω ∑