Thermodynamic limit and surface energy of the XXZ spin chain with arbitrary boundary fields

Abstract

In two previous papers [26,27], the exact solutions of the spin-1 2 chains with arbitrary boundary fields were constructed via the off-diagonal Bethe ansatz (ODBA). Here we introduce a method to approach the thermodynamic limit of those models. The key point is that at a sequence of degenerate points of the crossing parameter η = η m , the off-diagonal Bethe ansatz equations (BAEs) can be reduced to the conventional ones. This allows us to extrapolate the formulae derived from the reduced BAEs to arbitrary η case with O(N −2 ) corrections in the thermodynamic limit N → ∞. As an example, the surface energy of the XXZ spin chain model with arbitrary boundary magnetic fields is derived exactly. This approach can be generalized to all the ODBA solvable models.

Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms

“…These relations, together with other properties, allow us to construct an off-diagonal (or inhomogeneous) T − Q relation (4.9) of the eigenvalue of the transfer matrix and the associated BAEs (4.18). When the boundary parameters satisfy one constraint (4.23), the resulting T − Q relation is reduced to the conventional one (4.20), which might allow one to use the method developed in [88] to study the thermodynamic properties (up to the order of O(N −2 )) of the model for generic values of η via the conventional thermodynamic Bethe ansatz methods [86,87]. Taking the limit ǫ, ǫ ′ → +∞, the corresponding K-matrices become diagonal ones and the resulting T − Q relation is then reduced to that in [66].…”

“…This method has been proven to be very successful in the derivation of the surface energy of the XXZ spin chain with arbitrary boundary fields [88].…”

Surface energy of the one-dimensional supersymmetric t − J model with unparallel boundary fields

“…Due to the U(1)-symmetry-broken for the case with unparallel boundary fields, the conventional methods have failed to solve the model for a long time [26]. The exact solution of the one-dimensional supersymmetric t − J model with unparallel boundary fields [27,28] was obtained recently, which will enable one further to study the thermodynamic limit and surface energy [29][30][31] of the model.…”

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

“…Subsequently, the nested-version of ODBA for the models associated with su(n) algebra [62], the application to the integrable models beyond A-type [63] and the thermodynamic analysis based on the ODBA solutions [64] were developed. We remark that two other promising methods, namely, the q-Onsager algebra method [65] and the separation of variables (SoV) method [47][48][49][50]66] were also used to approach the spin-1 2 chains with generic integrable boundaries.…”