Solution of the classical Yang–Baxter equation with an exotic symmetry, and integrability of a multi-species boson tunnelling model

Abstract

Solutions of the classical Yang-Baxter equation provide a systematic method to construct integrable quantum systems in an algebraic manner. A Lie algebra can be associated with any solution of the classical Yang-Baxter equation, from which commuting transfer matrices may be constructed. This procedure is reviewed, specifically for solutions without skew-symmetry. A particular solution with an exotic symmetry is identified, which is not obtained as a limiting expansion of the usual Yang-Baxter equation. This solution facilitates the construction of commuting transfer matrices which will be used to establish the integrability of a multi-species boson tunneling model. The model generalises the well-known two-site Bose-Hubbard model, to which it reduces in the one-species limit. Due to the lack of an apparent reference state, application of the algebraic Bethe Ansatz to solve the model is prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of operator identities and tensor product decompositions.

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

“…This translates to the fact that (7) and (38) are equivalent. It is worth mentioning that the difference between the closed and open model is that in the closed model, it is possible for the number of Bethe roots M to be less than L, while in the open model we must have exactly L Bethe roots [8,21]. The energy is given by (6),…”

Ground-state energy of a Richardson-Gaudin integrable BCS model

“…Setting β = 0 in (5), the method in [11] will suffice to derive the BAE as the quadratic relations for the COE coincide with the open case. The challenge, in the case when β = 0, is to generalize the previous method using polynomials to accommodate the quadratic term in (5),…”

“…Integrability of the open model was established in [10] through use of the boundary Quantum Inverse Scattering Method. An alternative derivation, which is less technical, was later provided in [11].…”

Integrable spin-${\frac{1}{2}}$ Richardson–Gaudin XYZ models in an arbitrary magnetic field

“…Furthermore, while the XXX and XXZ models in arbitrary fields could be mapped to the antisymmetric models of Eqs. (3)(4) in the limit of a vanishing external field, this was done by making use of the additional U (1) symmetry lacking in the XYZ model [7]. As such, the fact that the presented XYZ model can not be mapped to an elliptic antisymmetric model reinforces how the non-skew-symmetric models differ from the symmetric ones, irrespective of the spin-1 2 restriction.…”