GeneralizedT–Qrelations and the open spin-sXXZ chain with nondiagonal boundary terms

Abstract

We consider the open spin-s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we derive a generalized form of T -Q relation involving more than one independent Q(u), which we use to propose the Bethe-ansatz-type expressions for the eigenvalues of the transfer matrix. At most two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values η = iπ/2, iπ/4, . . .. We also provide numerical evidence for the completeness of the Bethe-ansatz-type solutions derived, using the s = 1 case as an example.

Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields

“…It is known that the integrable systems without U(1) symmetry have many applications in the open string theory and the stochastic process of nonequilibrium statistics. Therefore, many interesting works of high spin models with nondiagonal boundary reflections have been done [30][31][32][33][34][35].…”

Exact spectrum of the spin-s Heisenberg chain with generic non-diagonal boundaries

“…Many efforts had been made to approach this nontrivial problem. However, in a long period of time, the Bethe ansatz solutions could only be obtained for either constrained boundary parameters [29] or special crossing parameters [30][31][32][33] associated with spin- 1 2 chains or with spin-s chains [52][53][54][55].…”

“…However, in a long period of time, the Bethe ansatz solutions could only be obtained for either constrained boundary parameters [20] or special crossing parameters [21] associated with spin- 1 2 chains or with spin-s chains [33,34,35,36]. Recently, based on the fundamental properties of the R-matrix and the K-matrices for quantum integrable models, a systematic method for solving the eigenvalue problem of integrable models with generic boundary conditions, i.e., the off-diagonal Bethe ansatz (ODBA) method was proposed in [37] and several long-standing models [37,38,39] were then solved.…”

Exact solution of the alternating XXZ spin chain with generic non-diagonal boundaries

“…Many efforts had been made [26][27][28][29][30][31][32][33][34][35][36][37][38][39] to approach this nontrivial problem. However, in a long period of time, the Bethe Ansatz solutions could only be obtained for either constrained boundary parameters [26] or special crossing parameters [27] associated with spin- 1 2 chains or with spin-s chains [40][41][42][43]. Recently, a method for solving the eigenvalue problem of integrable models with generic boundary conditions, i.e., the off-diagonal Bethe Ansatz (ODBA) method was proposed in [44] ( [45] for the details) and then several long-standing models [44,46,47] were solved.…”

“…The high spin chains with periodic and diagonal boundaries have been extensively studied in the literature [13,14,57,58,59]. So far the Bethe Ansatz solutions of the models with nondiagonal boundaries were known only for some special cases such as the boundary parameters obeying some constraint [40] or the crossing parameter (or anisotropy constant) η taking some special value (e.g., roots of unity) [41,42,49]. Moreover, the XXZ chain can be generalized to integrable alternating spin chain [59,60,61,62,63], i.e., an inhomogeneous chain with spin s at odd sites and spin s ′ at even sites.…”