Separation of variables in the open XXX chain

Abstract

We apply the Sklyanin method of separation of variables to the reflection algebra underlying the open spin-1 2 XXX chain with non-diagonal boundary fields. The spectral problem can be formulated in terms of a T Q-equation which leads to the known Bethe equations for boundary parameters satisfying a constraint. For generic boundary parameters we study the asymptotic behaviour of the solutions of the T Q-equation.

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model.…”

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].…”

Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries

“…So far, there have been several well-known methods for deriving the Bethe ansatz (BA) solutions of quantum integrable models: the coordinate BA [14][15][16], the T-Q approach [17][18][19][20][21], the algebraic BA [22][23][24][25][26][27], the analytic BA [28], the functional BA [29,30] or the separation of variables method [31][32][33][34] and many others . However, there exists a quite unusual class of integrable models which do not possess the U(1) symmetry (whose transfer matrices contain not only the diagonal elements but also some off-diagonal elements of the monodromy matrix and the usual U(1) symmetry is broken, i.e., the total spin is no longer conserved).…”

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

“…Several long-standing models [29,[31][32][33][34][35][36] have since been solved. It should be noted that besides ODBA [37,38] some other methods such as the q-Onsager algebra method [39][40][41][42], the separation of variables (SoV) method [43][44][45][46][47][48][49] and the modified algebraic Bethe ansatz method [50][51][52][53] were also used to obtain the eigenstates of the XXZ spin chains with generic boundary conditions. Remarkably, ODBA allows us to obtain eigenvalues of the U(1)-broken models associated with higher-rank algebras such as the su(n) spin chain with generic integrable boundary The paper is organized as follows.…”