Particle–hole duality, integrability, and Russian doll BCS model

Abstract

We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable.We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations. PACS numbers: 02.30Ik, 74.20.Fg, 03.65.Fd I. INTRODUCTIONBasic properties of conventional superconductors can be described by the microscopic Bardeen-Cooper-Schrieffer (BCS) theory, which assumes that pairing between electrons is due to their interaction through phonons. The simplest possible Hamiltonian, known as a reduced BCS Hamiltonian, accounts only couplings between the spin up and spin down electrons having opposite momenta; moreover, these couplings are supposed to be constant. Within the BCS theory, this Hamiltonian is solved approximately by using a mean-field treatment 1,2 .It was shown by Richardson long time ago that the same Hamiltonian can be solved exactly 3 . The approach to the problem, developed by Richardson, resembles a coordinate Bethe ansatz method. The Hamiltonian eigenstates and eigenvalues are expressed through the set of energy-like quantities (rapidities). They satisfy the set of nonlinear algebraic equations, which are now widely referred to as Richardson equations. The system described by the reduced BCS Hamiltonian is closely related to the so-called Gaudin ferromagnet 4 , for which the exact solution is also known.Unfortunately, the resolution of Richardson equations is a formidable task, so that very few explicit results have been obtained so far. However, in the case of a system, which contains a limited number of pairs, the equations can be solved numerically. Nowadays, this approach is used to investigate pairing correlations in ultrasmall metallic grains at low temperatures 5 .In addition, the reduced BCS Hamiltonian is integrable and exactly solvable through the algebraic Bethe anzats (ABA) method 6-10 , so that Richardson equations can be treated as Bethe equations (BE). In Ref. 11 , the confromal

The Dicke model as the contraction limit of a pseudo-deformed Richardson-Gaudin model

“…Lastly, no physical observables can be associated with the rapidities, whereas the eigenvalue-based equations can be connected to the conserved charges and to the occupations of specific spins S z i (as will be shown in Section 3.2.3). As such, they also exhibit the symmetries associated with physical operators, which remain hidden in the rapidities [141,142].…”

Exact solution for the inhomogeneous Dicke model in the canonical ensemble: Thermodynamical limit and finite-size corrections

“…The above result for the excitation energy is similar to the well known result of Bardeen-Cooper-Schrieffer theory of superconductivity. Note that in this theory chemical potential resides exactly in the middle of the interaction band, so that the equation for the chemical potential is fulfilled automatically and therefore is dropped; however, it must be kept in the situation of a crossover from local Bose-condensed pairs to the dense condensate [29,[31][32][33][34].…”

“…However, in contrast to the Richardson model, Dicke model supports arbitrarily large number of pseudo-particles through this degree of freedom. This fact sometimes makes it not so straightforward to apply ideas relevant for Richardson-Gaudin models to the Dicke model, see, e.g., recent developments on the particlehole duality [29][30][31]. The phase diagram of the inhomogeneous Dicke model in the thermodynamical limit is much richer, since it contains larger number of controlling parameters which include pseudo-particle density, mean detuning between the spin and boson energies, as well as spin-boson coupling energy [15].…”

Eigenvalue-based determinants for scalar products and form factors in Richardson–Gaudin integrable models coupled to a bosonic mode

“…The derivation of form factors and overlaps in the eigenvalue-based formalism [16][17][18] depends heavily on the existence of a dual state and therefore a (dual) highest weight state in conjunction with a lowest weight. This dual state can be seen as a consequence of the particlehole symmetry, which was recently investigated for several RG models [28][29][30]. However, this dependency makes the generalization of results for the su(2)-models towards models containing a bosonic degree of freedom far from straightforward, because the hw(1) algebra of a bosonic mode is non compact and therefore lacks a highest weight.…”