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We address the problem of the bosonization of finite fermionic systems with two different approaches. First we work in the path integral formalism, showing how a truly bosonic effective action can be derived from a generic fermionic one with a quartic interaction. We then apply our scheme to the pairing hamiltonian in the degenerate case proving that, in this instance, several of the features characterizing the spontaneous breaking of the global gauge symmetry U(1) occurring in the infinite system persist in the finite system as well. Accordingly we interpret the excitations associated with the addition and removal of pairs of fermions as a quasi-Goldstone boson and the excitations corresponding to the breaking of a pair (seniority one states in the language of the pairing hamiltonian) as Higgs modes. Second, we face the more involved problem of a non-degenerate single particle spectrum, where one more kind of excitations arises, corresponding to the promotion of pairs to higher levels. This we do by solving directly the Richardson equations. From this analysis the existence emerges of critical values of the coupling constant, which signal the transition between two regimes, one dominated by the mean field physics, the other by the pairing interaction.
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While coupled cluster theory accurately models weakly correlated quantum systems, it often fails in the presence of strong correlations where the standard mean-field picture is qualitatively incorrect. In many cases, these failures can be largely ameliorated by permitting the mean-field reference to break physical symmetries. Symmetry-broken coupled cluster, e.g. Bogoliubov coupled cluster, theory can indeed provide reasonably accurate energetic predictions, but the broken symmetry can compromise the quality of the resulting wave function and predictions of observables other than the energy. Merging symmetry projection and coupled cluster theory is therefore an appealing way to describe strongly correlated systems. Independently, two different but related formalisms have been recently proposed to achieve this goal. The two formalisms are contrasted in this manuscript, with results tested on the Richardson pairing Hamiltonian. Both formalisms are based on the disentangled cluster representation of the symmetry-rotated coupled cluster wavefunction. However, they differ in the way that the disentangled clusters are solved. One approach sets up angle-dependent coupled cluster equations, while the other involves first-order ordinary differential equations. The latter approach yields energies and occupation probabilities significantly better than those of number-projected BCS and BCS coupled cluster and, when the disentangled clusters are truncated at low excitation levels, has a computational cost not too much larger than that of BCS coupled cluster. The high quality of results presented in this manuscript indicates that symmetry-projected coupled cluster is a promising method that can accurately describe both weakly and strongly correlated finite many-fermion systems.
We derive the exact $T=0$ seniority-zero eigenstates of the isovector pairing Hamiltonian for an even number of protons and neutrons. Nucleons are supposed to be distributed over a set of non-degenerate levels and to interact through a pairing force with constant strength. We show that these eigenstates (and among them, in particular, the ground state) are linear superpositions of products of $T=1$ collective pairs arranged into $T=0$ quartets. This grouping of protons and neutrons first into $T=1$ collective pairs and then into $T=0$ quartets represents the distinctive feature of these eigenstates. This work highlights, for the first time on the grounds of the analytic expression of its eigenstates, the key role played by the isovector pairing force in the phenomenon of nuclear quarteting.
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 Richardsons equations are classified in terms of a number $v_l$, 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 $Gtoinfty$ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values $G_{rm cr}^{+}$ and $G_{rm cr}^{-}$ of the coupling which can exist on a single particle level with no pair degeneracy. Notably $G_{rm cr}^{+}$ and $G_{rm cr}^{-}$ merge when the number of single particle levels goes to infinity, where they coincide with the $G_{rm cr}$ (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of $G_{rm cr}$ 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 $G>G_{rm cr}^{-}$, come close to the BCS predictions, whereas the latter display a divergence at $G_{rm cr}$, signaling the onset of a long range off-diagonal order in the system.
As a first step to derive the IBM from a microscopic nuclear hamiltonian, we bosonize the pairing hamiltonian in the framework of the path integral formalism respecting both the particle number conservation and the Pauli principle. Special attention is payed to the role of the Goldstone bosons. We construct the saddle point expansion which reproduces the sector of the spectrum associated to the addition or removal of nucleon pairs.
We apply the functional bosonization procedure to a massive Dirac field defined on a 2+1 dimensional spacetime which has a non-trivial boundary. We find the form of the bosonized current both for the bulk and boundary modes, showing that the gauge field in the bosonized theory contains a perfect-conductor boundary condition on the worldsheet spanned by the boundary. We find the bononized action for the corresponding boundary modes.
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