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Register a new userThe Goldstone bosons in the pairing Hamiltonian: the path integral approach
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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.
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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.
The clockwork mechanism has recently been proposed as a natural way to generate hierarchies among parameters in quantum field theories. The mechanism is characterized by a very specific pattern of spontaneous and explicit symmetry breaking, and the presence of new light states referred to as `gears. In this paper we begin by investigating the self-interactions of these gears in a scalar clockwork model and find a parity-like selection rule at all orders in the fields. We then proceed to investigate how the clockwork mechanism can be realized in 5D linear dilaton models from the spontaneous symmetry breaking of a complex bulk scalar field. We also discuss how the clockwork mechanism is manifest in the scalar components of 5D gauge theories in the linear dilaton model, and build their 4D deconstructed analogue. Finally we discuss attempts at building both 4D and 5D realizations of a non-abelian scalar clockwork mechanism, where in the latter we consider scenarios in which the Goldstone bosons arise from 5D scalar and 5D gauge fields.
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.
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.
Expanding upon previous work, using the path-integral formalism we derive expressions for the one-particle reduced density matrix and the two-point correlation function for a quadratic system of bosons that interact through a general class of memory kernels. The results are applied to study the density, condensate fraction and pair correlation function of trapped bosons harmonically coupled to external distinguishable masses.
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