Spin chains with boundary inhomogeneities

We investigate the effect of introducing a boundary inhomogeneity in the transfer matrix of an integrable open quantum spin chain. We find that it is possible to construct a local Hamiltonian, and to have quantum group symmetry. The boundary inhomogeneity has a profound effect on the Bethe ansatz solution.

It was recently noticed, in the context of the A (1) 1 R-matrix (corresponding to a spin chain of XXZ-type) and a trivial K-matrix, that Sklyanin's construction can be further generalized by introducing [12] a boundary inhomogeneity in the transfer matrix, as in (2.15) below. 1 The corresponding Hamiltonian is generally not expected to be local; however, by virtue of also having suitable staggered bulk inhomogeneities, the resulting Hamiltonian is in fact local. This model has some further remarkable features, including quantum group (QG) symmetry [13], a novel Bethe ansatz solution, and a continuum limit described by a noncompact CFT [14], see also [15][16][17][18][19][20][21] for the corresponding closed chain.
The goal of the present paper is to explore such models with boundary inhomogeneities more broadly, particularly by considering higher-rank R-matrices, as well as non-trivial Kmatrices. For concreteness, we focus here on the infinite family of A (2) 2n R-matrices; however, we expect that similar results hold for other trigonometric R-matrices [22][23][24] with crossing symmetry (including A (2) 2n−1 , B (1) n , C (1) n , D (1) n , D (2) n+1 ). By introducing suitable staggered bulk inhomogeneities, we find that the key features of locality of the Hamiltonian and QG symmetry appearing at rank one can still be maintained, and again find novel Bethe ansatz solutions.
The outline of this paper is as follows. In Sec. 2 we define the model by constructing its transfer matrix, and we see explicitly that its Hamiltonian is local. We briefly discuss the model's QG symmetry in Sec. 3, and we present its Bethe ansatz solution in Sec. 4. We conclude in Sec. 5 with a brief summary and a list of some interesting remaining questions.

The model
We begin with a brief review in Sec. 2.1 of the basic ingredients that are used in Sec. 2.2 to construct the transfer matrix. We derive the corresponding Hamiltonian in Sec. 2.3.

Basic ingredients
As already noted, the model is constructed from solutions R(u) of the bulk YBE [9] and solutions K R (u) of the corresponding boundary YBE [10,11]

2)
1 This was actually a side result of [12], which was primarily devoted to solving D 2 models.
For later reference, we note here the useful identity [32] . (2.14)

Transfer matrix
We consider the following open-chain transfer matrix for a spin chain of length N [12] whose key difference with respect to the transfer matrix in [3] is the shift by u 0 in the argument of T , which can be regarded as a boundary inhomogeneity. We shall see that this seemingly minor change in the transfer matrix in fact has a profound impact on the model. The monodromy matrices are given as usual by where {θ l } are bulk inhomogeneities. The right K-matrixK R 0 (u) in (2.15) satisfies a generalized boundary YBE which, compared with (2.2), has a shift by u 0 in the R-matrix whose argument has the sum of rapidities. The generalized boundary YBE (2.17) can be mapped to the standard one (2.2) by performing the shifts u → u−u 0 /2 and v → v−u 0 /2, and identifyingK R (u−u 0 /2) = K R (u). Hence, we setK with K R (u) given by (2.10). Setting [12] the transfer matrix (2.15) can be shown to satisfy the commutativity property which is the hallmark of quantum integrability.
In terms of theK-matrices, the identity (2.13) reads where f (u) is given by (2.14).
An important observation is that the presence of a boundary inhomogeneity affects the crossing relation of the transfer matrix. Indeed, the crossing relation now becomes i.e. there is an additional u 0 -dependent shift.
For generic values of boundary and bulk inhomogeneities, the transfer matrix (2.15) does not generate a local Hamiltonian (i.e., whose range of interactions is independent of N). Following [12], we henceforth set these inhomogeneities to Note that the boundary inhomogeneity u 0 is the half-period of the R-matrix (2.3), and the bulk inhomogeneities are staggered 2 . The same transfer matrix but with no bulk or boundary inhomogeneities (u 0 = θ l = 0), to which we refer as the "homogeneous case", was investigated in [25,26].

Hamiltonian
We define the N-site Hamiltonian by 3 Note that this is the usual recipe for a closed-chain, rather than an open-chain, Hamiltonian.
(For an open chain, usually t(0) ∝ I [3], hence the definition (2.24) reduces to t ′ (0). However, here t(0) is not proportional to I, thus these two definitions are not equivalent; and the latter definition does not yield a local Hamiltonian.) Using (2.21) and the identity we obtain, after a long calculation, the following Hamiltonian for even values of N > 2 2 Staggered models date back at least to [33]. 3 We henceforth suppress displaying the dependence of the transfer matrix on the inhomogeneities, which are given by (2.23).
where we have introduced the following short-hand notations and a prime denotes differentiation with respect to the spectral parameter u. Note that the range of interactions in this Hamiltonian does not exceed 4 sites. For the case N = 2, we obtain I .

(2.28)
A similar computation for odd values of N > 1 gives The range of interactions again does not exceed 4 sites. We conclude that the Hamiltonian is local.

Quantum group symmetry
The transfer matrix (2.15) with inhomogeneities (2.23) has the QG symmetry U q (B n−p ) ⊗ U q (C p ), corresponding to removing the p th node from the A 2n Dynkin diagram, as follows from arguments similar to those for the homogeneous case [25]. 4 The "left" algebra B n−p (with p = 0, 1, . . . , n − 1) has generators j = 1, . . . , n − p , (3.1) and the "right" algebra C p (with p = 1, 2, . . . , n) has generators where e ij are the elementary (2n + 1) × (2n + 1) matrices with elements (e ij ) αβ = δ i,α δ j,β . The coproducts for the "left" generators are given by and the coproducts for the "right" generators are given by These expressions for the coproducts are the same as in [25] (where many further details can also be found), except for the relative minus sign in ∆(E ± (r) p

) (3.4).
Due to the relative minus sign in ∆(E ± (r) p ) (3.4), this coproduct does not obey the standard co-associativity property. Indeed, which suggests that there is instead an underlying quasi-Hopf algebra structure [13,34]. We define the higher coproducts for E ± (r) p recursively by The N-fold coproducts of the "left" and "right" generators commute with the transfer matrix (2.15), (2.23) , t(u) = 0 , j = 1, . . . , p , p = 1, . . . , n , (3.7) for both values γ 0 = ±1. At least for real values of the anisotropy parameter η, the rich degeneracies in the spectrum of the transfer matrix are completely accounted for by its QG symmetry, as in the homogeneous case [25].
The energy is given, in view of (2.24) and (4.2), by where As in the homogeneous case [26], the Dynkin labels [a p ] of the representations of the "right" algebra C p (with p = 1, 2, . . . , n) are given by Given the cardinalities of the Bethe roots of each type (namely, m 1 , . . . , m n ) for an eigenvalue Λ (m 1 ,...,mn) (u), Eqs. (4.17)-(4.18) determine the Dynkin labels of the corresponding "left" and "right" representations, from which one can deduce (e.g., using LieART [36]) their dimensions, and therefore the eigenvalue's degeneracy.
We emphasize that in the homogeneous case [26], the Q-functions are given by but in the presence of the boundary inhomogeneity (2.23), the Q-functions are instead given by (4.5), with a cosh instead of sinh in the second factor. Consequently, it is not just the "left-hand-side", but also the "right-hand-side" of the Bethe equations (4.10)-(4.12) that is affected by the boundary inhomogeneity, contrary to the conventional wisdom that the boundary affects only the former. Note also that, contrary to what usually happens for open spin chains, the power appearing in the left-hand-side of the first Bethe equation (4.10) is N instead of 2N.

Conclusions
We have seen that an integrable open quantum spin chain with a boundary inhomogeneity (2.15) can have a local Hamiltonian (2.26)-(2.29), as well as QG symmetry (3.7) that accounts for rich degeneracies in the spectrum. The presence of a boundary inhomogeneity affects the crossing relation (2.22), and has a profound effect on the Bethe ansatz solution, most notably on the Q-functions (4.5).
We have focused here on a boundary inhomogeneity whose value is half the period of the R-matrix, and with corresponding staggered bulk inhomogeneities (2.23). It may be interesting to consider other choices of boundary and bulk inhomogeneities, especially if they give rise to local Hamiltonians. Although we have also focused here on the infinite family of models constructed with A (2) 2n R-matrices, we expect that similar results hold for other models with crossing symmetry, such as those considered in [25,26]. In the simpler A (1) 1 case [12], the Hamiltonian can be formulated in a beautiful compact form [14] in terms of Temperley-Lieb (TL) operators [37]. It would be interesting if the Hamiltonians obtained here (2.26)-(2.29) could be reformulated in a similar way in terms of some sort of generalized TL operators, at least for the "extremal" cases p = 0, n, where the QG symmetry is U q (B n ), U q (C n ), respectively. The continuum limit of the A (1) 1 model [12] is described [14] by a noncompact CFT; it would be very interesting if a similar phenomenon occurs for the higher-rank models introduced here.