Integrability of an extended -wave pairing Hamiltonian

Abstract

We introduce an integrable Hamiltonian which is an extended d + id-wave pairing model. The integrability is deduced from a duality relation with the Richardson-Gaudin (s-wave) pairing model, and associated to this there exists an exact Bethe ansatz solution. We study this system using the continuum limit approach and solve the corresponding singular integral equation obtained from the Bethe ansatz solution. We also conduct a mean-field analysis and show that results from these two approaches coincide for the ground state in the continuum limit. We identify instances of the integrable system where the excitation spectrum is gapless, and discuss connections to non-integrable models with d + id-wave pairing interactions through the mean-field analysis.

Non-adiabatic Dynamics in d + id-Wave Fermionic Superfluids

“…2), it is clear that these two diagrams are quite similar to each other : (i) for quenches corresponding to ∆ f ∆ i we find a gapless steady state in which the pairing amplitude vanishes; (ii) for quenches when ∆ f ∼ ∆ i the pairing amplitude asymptotes to a constant and (iii) for quenches such that ∆ f ∆ i the pairing amplitude oscillated periodically and its time dependence is described by the Jacobi elliptic function. The Laxmechanism gives the asymptotic dynamical phases from the mean-field ground state which is exact in thermodynamical limit 37 and the change of coupling, confirms presence of all three dynamical phases. This implies that the mean-field dynamics of the extended d + id model obtained by computations are perfectly controlled and the fluctuations must cancel exactly at all levels.…”

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…By starting from Richardson's general solution for the Gaudin algebra [Eq. (8)] and varying over the free parameters in the wave function, it is possible to scan over all possible Gaudin models (XXX and XXZ), allowing more freedom and as such a better approximation to the energy compared to a variational approach based solely on the XXX model.…”

“…One class of these systems is the class of Richardson-Gaudin (RG) integrable systems, which can be derived from a generalized Gaudin algebra [1,2]. The pairing model in the reduced BCS approximation, used to describe superconductivity, has been shown to be RG integrable [3], as has the p x + ip y pairing Hamiltonian [4,5], the central spin model [6], factorizable pairing models in heavy nuclei [7], an extended d + id pairing Hamiltonian [8], and several atom-molecule Hamiltonians such as the inhomogeneous Dicke model [9,10]. For these models, diagonalizing the Hamiltonian in an exponentially scaling Hilbert space can be reduced to solving a set of nonlinear equations scaling linearly with system size.…”

Ground-state energies of the open and closed p + ip-pairing models from the Bethe Ansatz

“…However in the discrete case, all the Bethe roots condense at the origin [1] when the parameters reach the Moore-Read line. This discrepancy between the integral approximation and the discrete case is not present in models such as the Richardson swave pairing model and the d + id-wave pairing Hamiltonian [19,20] where a similar approach of integral approximation is adopted. In the case of the twolevel Richardson s-wave pairing model [19], the solution curve of the integral approximation evolves until it closes and forms a loop as some governing parameters approach a ground-state phase boundary line, meanwhile the discrete Bethe roots do not condense and their distribution is predicted by the solution curve within small error in the limit.…”