Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary

Abstract

Abstract. We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated.

Modified Algebraic Bethe Ansatz: Twisted XXX Case

“…We prove the conjecture about the multiple action of the modified creation operator on the pseudo-vacuum state given in [7]. In fact, we prove this property independently of the action on the pseudovacuum state and go beyond the proofs done for other models [1,11,12]. Moreover, we consider two different bases for solving the spectral problem of this family of models and we relate the two solutions by a modified quantum Wronskian equation.…”

Elliptic BCS-Richardson model and the modified algebraic Bethe ansatz

“…For some complicated r-matrices [34], or in the cases when the vector v is not the eigenvector of the generating function of the quantum Hamiltonians [35,36], yet another modification of ABA is needed. It concerns the modification of the Bethe equations and the formula for the spectrum of the generating function of quantum integrals.…”

Quadratic operator relations and Bethe equations for spin-1/2 Richardson–Gaudin models

“…Moreover, cases for which the U(1) symmetry is explicitly broken by an in-plane component of the magnetic field can also make such a proper pseudo-vacuum non-existant [9,16]. A variety of techniques have been developed over the years to deal with models lacking such an explicitly known reference state: Separation of Variables [21,22,23,24,25,26], Off-Diagonal Bethe Ansatz [27,28,29,30,31] or the Modified Bethe Ansatz [9,16,32,33,34,35,36,37,38]. However, by circumventing the construction of eigenstates, the approach proposed in this work makes all the possible cases equally simple, at least when exclusively looking for the eigenenergies.…”