Boundary energy of the general open XXZ chain at roots of unity

Abstract

We have recently proposed a Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with general integrable boundary terms (containing six free boundary parameters) at roots of unity. We use this solution, together with an appropriate string hypothesis, to compute the boundary energy of the chain in the thermodynamic limit.

Finite-size correction and bulk hole-excitations for special case of an open XXZ chain with nondiagonal boundary terms at roots of unity

“…This can be really challenging even for the diagonal (Dirichlet) case [34,49]. Further, solution for more general XXZ model involving multiple Q(u) functions [12,13], can also be utilized in similar capacity to explore these effects. Last but not least, excitations due to other objects that we choose to ignore here, such as special roots/holes and so forth can also be explored for these models in order to make the study more complete.…”

“…However, obtaining exact solutions for this model has been a rather challenging and elusive task for many years. Various progress have been made in obtaining solutions for this model, either using the Bethe ansatz approach for diagonal [1]- [4], constrained nondiagonal [5]- [9] and nondiagonal cases at roots of unity [10]- [13], or using the representation theory of the q-Onsager algebra for general nondiagonal cases [14]. Approaches based on boundary Temperley-Lieb algebra and its representations have also been presented recently, from which the spectral properties of the chain have been studied [15].…”

Truncation identities for the small polaron fusion hierarchy

“…To actually compute eigenvalues of the transfer matrices further steps have to be taken: for anisotropies being roots of unity the truncated fusion hierarchy can be be analyzed following the steps which have been established for the XXZ chain [33][34][35] where additional constraints on the boundary fields may arise. For generic anisotropies the situation is more complicated: in the ungraded XXZ chain a (factorized) Bethe ansatz for the Q-function given in terms of finitely many parameters such as (7.9) was possible only if the boundary parameters satisfy a constraint [18-20, 23, 28].…”

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

“…For the boundary parameters out of this region, stable boundary bound states exist in the ground state [41,42,43,44]. However, the energy is indeed a smooth function about the boundary parameters as demonstrated in the diagonal boundary field case [44,45]. (3)An interesting fact is that the contributions of a + , a − , β to the energy are completely separated and the surface energy does not depend on θ ± at all (same effect was also obtained in [46] where the surface energy and the finite size correction were derived for some constraint boundary parameters), which indicate that the two boundary fields behave independently in the thermodynamic limit.…”