In analogy with a symmetry-restoring treatment of rotational invariance for deformed Hamiltonians, we discuss the case of a separable monopole pairing Hamiltonian in the quasiparticle basis. The results are compared to estimates of a conventional treatment based on the quasiparticle random phase approximation.

Physical Review CAug 1, 198820 citations

Nuclear Physics A

Collective treatment of the pairing Hamiltonian

Daniel R. Bes, R.A. Broglia, R.P.J. Perazzo, Krishna Kumar

Published Mar 1, 1970• 75 citations

Publisher

DOI: 10.1016/0375-9474(70)90677-9

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T=0

. The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final-state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS--Bose-Einstein condensate crossover: Here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum that is well established both theoretically and experimentally and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.

A modified pairing treatment is developed for handling the

N

-boson problem with repulsive particle-particle interaction. The nonpairing portion of the many-boson Hamiltonian is eliminated, to first order, by a canonical transformation. The resultant Hamiltonian contains an additional attractive particle-particle interaction, whenever there is a Bose condensation in the k=0 single-particle state. The Bose condensate plays the same role in mediating this interaction that the phonon field does in mediating the attractive interaction between electrons in a superconductor. The strength of the condensate-induced attractive interaction is approximately double that of the direct repulsive interaction. The region of

k

space over which the attractive interaction operates depends on

{N}_{0}

, the number of bosons in the condensate, this region vanishing when

{N}_{0}

does. For convenience, the problem is formulated in terms of hyperspin, in analogy with the isospin treatment of superconductivity. The ground state is characterized by two order parameters ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2}$, and an energy $\ensuremath{\Gamma}$ (proportional to

{N}_{0}

), which can be obtained by solving a set of coupled integral equations. It is shown that there is no gap, at low energies, in the single-particle excitation spectrum. Aside from these single-particle excited states, there are no other collective oscillations. The hyperspin formulation is extended to finite temperatures. There are, however, certain difficulties with this extension.