BEC-BCS crossover in a p+ip-wave pairing Hamiltonian coupled to bosonic molecular pairs

We analyse a p+ip-wave pairing BCS Hamiltonian, coupled to a single bosonic degree of freedom representing a molecular condensate, and investigate the nature of the BEC-BCS crossover for this system. For a suitable restriction on the coupling parameters, we show that the model is integrable and we derive the exact solution by the algebraic Bethe ansatz. In this manner we also obtain explicit formulae for correlation functions and compute these for several cases. We find that the crossover between the BEC state and the strong pairing p+ip phase is smooth for this model, with no intermediate quantum phase transition.

💡 Research Summary

The paper investigates a two‑dimensional p + ip‑wave pairing BCS Hamiltonian that is coupled to a single bosonic mode representing a molecular condensate. The authors first recall the three ground‑state phases of the uncoupled p + ip model—weak coupling, weak pairing, and strong pairing—characterized by the filling fraction (x_{C}=N_{C}/L) and the effective coupling (g=GL). When the bosonic mode is added, the total filling becomes (x=x_{b}+x_{C}) with (x_{b}=N_{b}/L). For (g<1) the ground state is a pure Bose‑Einstein condensate (BEC) with zero energy, while for (g>1) the ground state can be a mixed BEC/Cooper‑pair state, the Read‑Green state, or a strong‑pairing state depending on the value of (x).

A central result is the identification of an integrable manifold in the parameter space of the full Hamiltonian. By imposing the constraints (\delta=-F^{2}G) and (K=FG) (with (F) a free parameter), the Hamiltonian can be rewritten in terms of the generators of a (U_{q}(sl(2))) algebra and a q‑deformed boson algebra. Using the six‑vertex R‑matrix that solves the Yang‑Baxter equation, the authors construct a monodromy matrix (T(\lambda)) and the associated transfer matrix (t(\lambda)=\operatorname{tr}T(\lambda)). The commutativity (