We address the problem of two pairs of fermions living on an arbitrary number of single particle levels of a potential well (mean field) and interacting through a pairing force. The associated solutions of the Richardson’s equations are classified in terms of a number vl, which reduces to the seniority v in the limit of large values of the pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G → ∞ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values G + cr and G − cr of the coupling which can exist on a single particle level with no pair degeneracy. Notably G + cr and G − cr merge when the number of single particle levels goes to infinity, where they coincide with the Gcr (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of Gcr the system undergoes a transition from a mean field to a pairing dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes finding out that the former, for G> G − cr, come close to the BCS predictions, whereas the latter display a divergence at Gcr, signaling the onset of a long range off-diagonal order in the system. PACS numbers: 21.60.-n, 21.30 Fe, 24.10.Cn 1 1
The lowest seniority-zero eigenstates of an exactly solvable multilevel pairing Hamiltonian for a finite Fermi system are examined at different pairing regimes. After briefly reviewing the form of the eigenstates in the Richardson formalism, we discuss a different representation of these states in terms of the collective pairs resulting from the diagonalization of the Hamiltonian in a space of two degenerate time-reversed fermions. We perform a two-fold analysis by working both in the fermionic space of these collective pairs and in a space of corresponding elementary bosons. On the fermionic side, we monitor the variations which occur, with increasing the pairing strength, in the structure of both these collective pairs and the lowest eigenstates. On the bosonic side, after reviewing a fermion-boson mapping procedure, we construct exact images of the fermion eigenstates and study their wave function. The analysis allows a close examination of the phenomenon of pair condensation in a finite Fermi system and gives new insights into the evolution of the lowest (seniority-zero) excited states of a pairing Hamiltonian from the unperturbed regime up to a strongly interacting one.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
The problem of one pair of identical nucleons sitting in
N single particle levels of a potential well and interacting through the pairing force is treated introducing, in the Hamiltonian formalism, even Grassmann variables. The eigenvectors are analytically expressed solely in terms of these with coefficients fixed by the eigenvalues and the single particle energies. For these a specific model is needed: in the case of the harmonic oscillator well, for any strength of the pairing interaction, an accurate expression is derived for both the collective eigenvalue and for those trapped in between the single particle levels. Notably the latter are labelled through an index upon which they depend parabolically.
Background: The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system.Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number
A it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated.Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles
A on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory.Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number
A of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit.Conclusions: While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. In the present paper, the zero-pairing limit of a HFB state carrying an arbitrary number of particles has been worked out and shown to lead to drastic differences and consequences depending on the closed-shell or open-shell nature of the system in that limit. From a general perspective, the present analysis demonstrates that HFB theory does not reduce to HF theory when the pairing field is driven to zero in the HFB Hamiltonian matrix.