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Bethe Ansatz for the open XXZ chain from functional relations at roots of unity

We briefly review Bethe Ansatz solutions of the integrable open spin-1/2 XXZ quantum spin chain derived from functional relations obeyed by the transfer matrix at roots of unity.


Introduction
A long standing problem has been to solve the open spin- 1 2 XXZ quantum spin chain with general integrable boundary terms, defined by the Hamiltonian 1,2 H = 1 2 N −1 n=1 σ x n σ x n+1 + σ y n σ y n+1 + ch η σ z n σ z n+1 (1) where σ x , σ y , σ z are the standard Pauli matrices, η is the bulk anisotropy parameter, α ± , β ± , θ ± are arbitrary boundary parameters, and N is the number of spins. Determining the energy eigenvalues in terms of solutions of a system of Bethe Ansatz equations is a fundamental problem, which has important applications in integrable quantum field theory as well as condensed matter physics and statistical mechanics 3 , and perhaps also string theory. (For an introduction to Bethe Ansatz, see e.g. Refs. 4,5,6.) The basic difficulty in solving (1) is that, in contrast to the special case of diagonal boundary terms (i.e., α ± or β ± → ±∞, in which case H has a U (1) symmetry) which was solved long ago 7,8,9 , a simple pseudovacuum state does not exist. For instance, the state with all spins up is not an eigenstate of the Hamiltonian. Hence, many of the techniques which have been developed to solve integrable models cannot be applied.
We observed some time ago 10,11 that, for bulk anisotropy parameter values (hence q ≡ e η is a root of unity, satisfying q p+1 = −1) and arbitrary values of the boundary parameters, the model's transfer matrix t(u) (see Sec. 2) obeys a functional relation of order p + 1. For example, for the case p = 2, the functional relation is where δ(u) and f (u) are known scalar functions which depend on the boundary parameters. (Expressions for these functions in terms of the boundary parameters in (1) are given in Ref. 18.) Similar results had been known earlier for closed spin chains. 12,13,14 By exploiting these functional relations, we have obtained Bethe Ansatz solutions of the model for various special cases of the bulk and boundary parameters: 15,16,17] The bulk anisotropy parameter has values (2); and the boundary parameters satisfy the constraint where k ∈ [−(N + 1) , N + 1] is even (odd) if N is odd (even), respectively. All of these cases have the property that the quantity ∆(u), defined by is a perfect square. Solutions for generic values of the bulk anisotropy parameter and for boundary parameters obeying a constraint similar to (4) have been discussed in Refs. 20, 21, 22.
Here we briefly review our results for the cases (i) -(iii).

Transfer matrix
The transfer matrix t(u) of the open XXZ chain with general integrable boundary terms, which satisfies the fundamental commutativity property where T 0 (u) andT 0 (u) are the monodromy matrices and tr 0 denotes trace over the "auxiliary space" 0. The R matrix is given by where η is the bulk anisotropy parameter; and K ∓ (u) are 2 × 2 matrices whose components are given by 1,2 and where α ∓ , β ∓ , θ ∓ are the boundary parameters. The Hamiltonian (1) is proportional to t ′ (0) plus a constant.
The transfer matrix also has iπ periodicity crossing symmetry and the asymptotic behavior

Case (i)
Our main objective is to determine the eigenvalues Λ(u) of the transfer matrix t(u) (6), from which the energy eigenvalues can readily be computed. The functional relations for the transfer matrix (e.g., (3)) evidently imply corresponding relations for Λ(u). Following Ref. 23, we observe that the latter relations can be written as where M(u) is the (p + 1) × (p + 1) matrix if there exists an iπ-periodic function h(u) such that To solve for h(u), we set z(u) ≡ p j=0 h(u + jη), and observe that (16) implies that z(u) satisfies a quadratic equation whose solution is evidently given by where ∆(u) is defined in (5). If the boundary parameters satisfy the constraint (4), then ∆(u) is a perfect square, and two solutions of (16) are Let us now label the corresponding matrices (15) by M (±) (u). The condition (14) implies that M (±) (u) has a null eigenvector v (±) (u), Note that the matrix M (±) (u) has the symmetry It follows that the null eigenvector v (±) (u) satisfies Sv (±) (u) = v (±) (u + η). Thus, its components can be expressed in terms of a function Q (±) (u) , with Q (±) (u + iπ) = Q (±) (u). We make the Ansatz which has the crossing symmetry Q (±) (u) = Q (±) (−u − η). Substituting the expressions for M (±) (u) (15) and v (±) (u) (23) into the null eigenvector equation (20) yields the result for the transfer matrix eigenvalues The asymptotic behavior (13) implies that M (±) = 1 2 (N − 1 ± k), where k is the integer appearing in the constraint (4). Analyticity of the eigenvalues (25) implies the Bethe Ansatz equations In short, for case (i), the eigenvalues of the transfer matrix (6) are given by (25), where h (±) (u) and Q (±) (u) are given by (19), (24) and (26). In Ref. 17, we have verified numerically that this solution holds also for generic values of η, which is consistent with Refs. 20, 21, 22; and that this solution gives the complete set of 2 N eigenvalues. To illustrate how completeness is achieved, let us consider the case N = 4. The integer k in the constraint (4) must therefore be odd, with −5 ≤ k ≤ 5. The six possibilities are summarized in Table 1.

Case(ii)
A key feature of case (i) is that the quantity ∆(u) (5) is a perfect square. We therefore look for additional such cases. For p even, we find that ∆(u) is also a perfect square if either (a) at most one of the boundary parameters is nonzero; or (b) any two of the boundary parameters {α − , α + , β − , β + } are arbitrary, the remaining boundary parameters from this set are either η or iπ/2, and θ − = θ + . For definiteness, we focus here on the subcase (b) with α ± arbitrary, β ± = η and N even. Unfortunately, the resulting z(u) (18) is not consistent. To surmount this difficulty, we use a matrix M(u) which is different from (15), namely 18 where h(u) is 2iπ-periodic. This matrix has the symmetry where S is given by (22). By arguments similar to those used in Sec. 3, we find that the transfer matrix eigenvalues are given by where h(u) is given by , and Q(u) is given by with M = N + 2p + 1; and the Bethe Ansatz equations are We have verified numerically the completeness of this solution. The other subcases (a) and (b) are mostly similar. a

Case(iii)
For p odd, we find that the quantity ∆(u) (5) is also a perfect square if at most two of the boundary parameters {α − , α + , β − , β + } are nonzero, and θ − = θ + . For definiteness, we focus here on the case with α ± arbitrary, β ± = 0 and N even. As in case (ii), the resulting z(u) (18) is not consistent.
To surmount this difficulty, we again use a matrix M(u) which is different from (15), namely 19 The exception is the subcase (a) with θ ± nonzero, for which Q(u) = 2M j=1 sh(u − u j ), which is not crossing symmetric. See Sec. 3.3 in Ref. 18. where h (1) (u) and h (2) (u) are iπ-periodic. It has the reduced symmetry where T = S 2 , and S is given by (22). (While (21) implies (34), the converse is not true.) The condition det M(u) = 0 implies that M(u) has a null eigenvector v(u), where v(u) satisfies T v(u) = v(u + 2η). Thus, its components are expressed in terms of two independent functions Q 1 (u), We make the Ansätze Substituting the expressions for M(u) (33) and v(u) (36) into the null eigenvector equation (35) yields two expressions for the transfer matrix eigenvalues, Analyticity of these expressions leads to the Bethe Ansatz equations We expect that there are sufficiently many equations to determine all the zeros {u and We have verified numerically the completeness of this solution. Similar results hold for the case with α − , β − arbitrary and α + = β + = 0, etc.
We observe that this solution represents a generalization of the famous Baxter T − Q relation 4 , which schematically has the form t(u) Q(u) = Q(u ′ ) + Q(u ′′ ) .
Such generalized T −Q relations, involving two or more independent Q(u)'s, may also appear in other integrable models.

Conclusions
We have seen that Bethe Ansatz solutions of the open spin-1 2 XXZ quantum spin chain are available for the cases (i)-(iii), for which the quantity ∆(u) (5) is a perfect square. There may be further special cases for which ∆(u) is a perfect square, in which case it should not be difficult to find the corresponding Bethe Ansatz solution. Our solution for case (iii) involves more than one Q(u). This is a novel structure, which should be further understood. The general case that ∆(u) is not a perfect square and/or that η = iπ/(p + 1) also remains to be understood.