Universal Bethe ansatz solution for the Temperley-Lieb spin chain

We consider the Temperley-Lieb (TL) open quantum spin chain with"free"boundary conditions associated with the spin-$s$ representation of quantum-deformed $sl(2)$. We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.


Introduction
The generators {X (1) , . . . , X (N −1) } of the unital Temperley-Lieb (TL) algebra T L N [1], can be used to define the Hamiltonian of an open quantum spin chain of length N with "free" boundary conditions This type of model has been the subject of many investigations. For simplicity, we focus here on the models associated with U Q (A 1 ) = U Q sl (2). The generators X (i) have been constructed for any value of spin s [2,3]. The s = 1/2 case is the well-known quantumgroup-invariant spin-1/2 XXZ chain [4]. The s = 1 case is the quantum deformation [2] of the pure biquadratic spin-1 chain [5,6,7]. These models are integrable; and closedchain versions with periodic boundary conditions have been investigated for s > 1/2 using inversion relations [8,9], numerically [10], by coordinate Bethe ansatz [5,6,11], and by analytical Bethe ansatz [12]. Additional results can be found in [13,14,15,16,17] for the open chain, and in [18,19,20] for the closed chain. TL models associated with higher-rank algebras have also been investigated [21,22,23].
Despite these and further efforts, a number of fundamental problems related to these models, such as the formulation of an algebraic Bethe ansatz solution, have remained unsolved. Moreover, the analytical Bethe ansatz solution proposed in [12] does not give the complete spectrum.
The goal of this paper is to address some of these problems. We construct the transfer matrix corresponding to the Hamiltonian (1.2), and we determine its eigenvalues using analytical Bethe ansatz. We prove that the transfer matrix has quantum group symmetry, which accounts for the degeneracies of the spectrum. We propose an algebraic Bethe ansatz construction of the Bethe states, which (when on-shell) we conjecture are highest-weight states of the quantum group. The scalar product between an off-shell Bethe state and an on-shell Bethe state is also considered, and we conjecture that it can be given in terms of a determinant formula; the square of the norm, i.e., the scalar product between on-shell Bethe states, follows as a limit 1 . We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin.
Although most of this paper concerns the open TL chain, we briefly consider the closed TL chain with periodic boundary conditions in an appendix. There we revisit the analytical Bethe ansatz computation in [12], and show how the proposed solution can be improved so as to obtain the complete spectrum. In contrast with the case of the open chain, the solutions of the closed-chain Bethe equations are not universal, as the Bethe roots depend on the value of the spin.
The outline of this paper is as follows. In section 2 we describe the construction of the Hamiltonian (1.2) and the corresponding transfer matrix. In section 3 we use analytical Bethe ansatz to determine the eigenvalues of the transfer matrix and the corresponding Bethe equations. In section 4 we show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. In section 5 we present our proposals for the algebraic Bethe ansatz solution and scalar products. We briefly discuss these results and remaining problems in section 6. We treat the closed TL chain in Appendix A.

Transfer matrix
We begin this section by describing in more detail the construction of the Hamiltonian (1.2). We then construct the corresponding transfer matrix, which is the generating function of the Hamiltonian and the higher local conserved commuting quantities, and we review some of its important properties.
We consider the TL open quantum spin chain corresponding to the spin-s representation of U Q sl (2). The X (i) appearing in the Hamiltonian (1.2) are operators on (C 2s+1 ) ⊗N defined by where X is a (2s + 1) 2 by (2s + 1) 2 matrix (an endomorphism of C 2s+1 ⊗ C 2s+1 ) with the following matrix elements [2] where m 1 , m 2 , m ′ 1 , m ′ 2 = −s, −s + 1, . . . , s, and s = 1 2 , 1, 3 2 , . . .. In other words, X (i) is an operator on N copies of C 2s+1 , which acts as X on copies i and i + 1, and otherwise as the identity operator, where I is the identity operator on C 2s+1 . These operators satisfy the TL algebra (1.1), where c is given by We assume throughout this paper that Q has a generic value.
The Hamiltonian (1.2) is integrable for any value of spin s. In the notation of [12], the corresponding R-matrix is given by [28] where P is the permutation matrix on C 2s+1 ⊗ C 2s+1 . Indeed, the Yang-Baxter equation is satisfied. This R-matrix has the unitarity property where R 21 = P 12 R 12 P 12 = R t 1 t 2 12 , and ω(u) is defined as This R-matrix also has crossing symmetry where V is an anti-diagonal matrix with elements The model (1.2) is an open spin chain.
For an integrable open spin chain, the transfer matrix is given by [29] where T 0 (u) andT 0 (u) are the monodromy matrices T 0 (u) = R 0N (u) · · · R 01 (u) ,T 0 (u) = R 10 (u) · · · R N 0 (u) . (2.12) Moreover, the K-matrices (endomorphisms of C 2s+1 ) satisfy the boundary Yang-Baxter equations and [30] where M is the diagonal matrix given by The Hamiltonian (1.2) corresponds to the special case with quantum-group invariance [30] K − = I , and therefore the transfer matrix (2.11) takes the simpler form Indeed, the Hamiltonian is related to the transfer matrix as follows The higher conserved quantities can be obtained by taking higher derivatives of the transfer matrix. These quantities commute with each other by virtue of the commutativity property [29] [t(u) , t(v)] = 0 . (2.20) The transfer matrix also has crossing symmetry [31] t(u) = t(−u −1 q −1 ) . (2.21)

Analytical Bethe ansatz
We now proceed to determine the eigenvalues of the transfer matrix (2.17) by analytical Bethe ansatz [31,32,33,34]. To this end, it is convenient to introduce inhomogeneities {θ j }, i.e. to consider instead the inhomogeneous transfer matrix where As noted in [12], the R-matrix (2.5) degenerates into a one-dimensional projector at u = q −1 , Hence, we can use the fusion procedure [35,36], as generalized to the case of boundaries in [37], to obtain the fusion formula wheret(u; {θ j }) is a fused transfer matrix, and the scalar function f (u) is given by a product of quantum determinants where g(u) is given by Using the fact that the fused transfer matrix vanishes when evaluated at q −1 θ i , i.e.
it follows from (3.4) that the fundamental transfer matrix (3.1) satisfies a set of exact functional relations Let us denote the eigenvalues of t(u; {θ j }) by Λ(u; {θ j }). From (2.21) it follows that the eigenvalues have crossing symmetry (3.10) and from (3.8) it follows that the eigenvalues obey the functional relations To solve these equations, we introduce the functions which have the properties It is easy to see that equations (3.10) and (3.11) are satisfied by where Q(u) is any crossing-invariant function From the form (2.5) of the R-matrix and the commutativity property (2.20), it follows that Λ(u; {θ j }) must be a Laurent polynomial in u (with a finite number of terms). We assume that Q(u) is also a Laurent polynomial, and is given by where the so-called Bethe roots {u 1 , . . . , u M } are still to be determined, which is consistent with (3.15). Obviously Q(u k ) = 0, which means that both terms in the expression (3.14) for Λ(u; {θ j }) have a simple pole at u = u k . (We assume that the Bethe roots are distinct, and are not equal to 0 or ∞.) The corresponding residues must cancel (since Λ(u; {θ j }) must be finite for u not equal to 0 or ∞), which implies the so-called Bethe equations Since we no longer need the inhomogeneities, we now set them to unity θ j = 1.
To summarize, we have argued that the eigenvalues Λ(u) of the transfer matrix t(u) (2.17) are given by 18) and the Bethe roots are given by the Bethe equations .

(3.19)
These equations take a more symmetric form in terms of the rescaled Bethe rootsũ k ≡ u k q 1/2 : . (3.20) Note that the results (3.18) -(3.20) do not depend on the value of s; in particular, they coincide with the well-known results for the case s = 1/2 [4,31]. This is consistent with the TL equivalence (see e.g. [1,7,38,39,40]), which suggests that the spectrum (but not the degeneracies) of the TL Hamiltonian (1.2) is independent of the representation.
We have verified numerically for small values of N and s that every distinct eigenvalue of the transfer matrix can be expressed in the form (3.18). See Tables 1 -3, and note that all (2s + 1) N eigenvalues are accounted for.    In closing this section, we note that the eigenvalues of the Hamiltonian (1.2) are given by

Quantum group symmetry
In this section we demonstrate the quantum-group invariance of the transfer matrix, and we discuss the implications of this symmetry for the Bethe ansatz solution.

Symmetry of the transfer matrix
Let us denote by R ± the asymptotic limits of the R-matrix R(u) (2.5) and let us similarly denote by T ± 0 the asymptotic limits of the monodromy matrix T 0 (u) (2.12) Regarding T ± 0 as a (2s + 1) × (2s + 1) matrix in the auxiliary space, its matrix elements T ± ij (which are operators on the quantum space (C 2s+1 ) ⊗N ) define a quantum group, which has been identified in [41] as U q (2s + 1). 2 We shall demonstrate that each of these matrix elements commutes with the transfer matrix The proof, which is similar to the one in [42] (see also [43]), requires two lemmas: Proof. We recall the fundamental relation Taking asymptotic limits of u 1 yields which further implies Therefore, where the first equality follows from (4.6), and the second equality follows from (4.7). We have therefore shown the commutativity property Furthermore, where the second line follows from unitarity (2.7). Substituting into (4.9) we obtain the desired result (4.4).
Proof. We write the unitarity condition (2.7) as and then use crossing symmetry (2.9) to obtain By taking asymptotic limits and noting that V 2 = (−1) 2s I, we obtain (4.14) Moreover, from (4.12) we obtain R ± 12 R ∓ t 1 t 2

12
= I, which implies that Substituting into (4.14), we obtain which can be rearranged to give the desired result (4.11).
We are now ready to prove the main result (4.3), which is equivalent to the following Proposition.
Proof. Recalling the expression (2.17) for the transfer matrix, we obtain In passing to the third line, we have used the fact M 1 M 2 , R ± 12 = 0 as well as the first lemma (4.4). Then In passing to the second line we have made the identifications A 12 = M −1 1 (R ± 12 ) −1 M 1 and Z 2 = M 2 T 2 (u)T 2 (u). Finally, we obtain In passing to the second line we have used the second lemma (4.11).

Degeneracies and multiplicities
The U q (2s + 1) symmetry of the transfer matrix implies that its eigenstates form representations of this algebra. The space of states has the decomposition (see e.g. [40,12,44,41,15] and references therein) where the summation is over even (odd) integers for even (odd) N, respectively; V k are representations of U q (2s + 1); and ν k are the multiplicities. The dimensions of the representations are given by where p k (x) are Chebyshev polynomials of the second kind, which are defined by the recurrence relations The multiplicities are given by 3

Algebraic Bethe ansatz
We present here several conjectures related to the algebraic Bethe ansatz solution of the TL chain. The conjecture for the off-shell equation has been proved for s = 1 [45], while the other conjectures have been checked numerically (up to M = 3, N = 6 and s = 3 2 ).

Off-shell equation
In order to implement the algebraic Bethe ansatz, we need to choose a convenient representation in the auxiliary space for the double-row monodromy matrix T 0 (u)T 0 (u). We choose where each entry acts on the quantum space (C 2s+1 ) ⊗N . Let us also introduce the (2s + 1)−dimensional reference state where  .7) are known [29]; for the s = 1 case a proof will be reported in a separate paper [45]. where T + ij are quantum group generators defined in section 4.1. That is, on-shell Bethe states are highest-weight states of the quantum group, in the sense that they are eigenstates of the diagonal elements of T + , and are annihilated by the lower triangular elements of T + . This would help account for the observations in section 4.2 that the degeneracies and multiplicities are given by group theory.

Scalar products
Let us suppose that {u 1 , . . . , u M } are Bethe roots. We propose that the scalar product between the on-shell state u 1 , . . . , u M | and an arbitrary off-shell state |v 1 , . . . , v M is given by .(5.11) The formula (5.11) was proved in [46] for the s = 1 2 case with (diagonal) boundary fields (see also [47] for the XXX chain). Performing the limit v k → u k , we obtain the square of the norm, namely, where G is a M × M matrix with elements . (5.13)

Discussion
We have considered the TL open quantum spin chain associated with the spin-s representation of quantum-deformed sl (2). We have constructed the transfer matrix (2.17), and we have seen that its eigenvalues (3.18)  We have proposed an algebraic Bethe ansatz construction of the Bethe vectors (5.4) and (5.5), and the corresponding off-shell equations (5.6) and (5.7), respectively. Remarkably, despite the fact that the auxiliary space has dimension greater than 2 for s > 1/2, a single creation operator suffices to construct all the Bethe states -no nesting is needed. We have also proposed a determinant formula for the scalar products between off-shell and on-shell Bethe states (5.11). Remarkably, these results are also universal in the sense that they depend on the value of the spin only through a constant factor. It is important to find proofs for these conjectures. So far, we have been able to prove only the off-shell equations for s = 1 [45].
We have seen that appropriate boundary conditions are necessary for the TL model to have a universal solution. Indeed, for periodic boundary conditions, the solution (A.11) is no longer universal. This solution has the unusual feature that it has a twist that is "dynamically" generated (i.e., the twist is not a fixed parameter of the model, as is typically the case). An algebraic Bethe ansatz solution for this model remains to be found. It may be interesting to consider generalizations of the open TL chain (1.2) which are still integrable but have boundary terms that break the quantum group symmetry [48,49].

Acknowledgments
The work of RN was supported in part by the National Science Foundation under Grant PHY-1212337, and by a Cooper fellowship. RP thanks the São Paulo Research Foundation (FAPESP), grants # 2014/00453-8 and # 2014/20364-0, for financial support. We also acknowledge the support by FAPESP and the University of Miami under the SPRINT grant #2016/50023-5.

A Closed TL chain
Let t(u) now denote the transfer matrix for the closed TL chain with periodic boundary conditions where the monodromy matrix T 0 (u) is given by (2.12). To determine the eigenvalues of t(u), we follow the same approach used in Section 3 to analyze the open chain. Hence, we consider the inhomogeneous transfer matrix where T 0 (u; {θ j }) is given by (3.2). Using the fusion procedure, we arrive at the functional relations where F (u) is now given by (cf. (3.9)) The corresponding eigenvalues Λ(u; {θ j }) therefore obey the same functional relations To solve these equations, we introduce the functions where the twist parameter κ is still to be determined. We observe that Hence, the functional relations (A.5) are satisfied by where Q(u) is now given by Setting the inhomogeneities to unity θ j = 1, we conclude that the eigenvalues Λ(u) of the closed transfer matrix (A.1) are given by and the Bethe roots are given by the Bethe equations .

(A.11)
A similar solution was proposed in [12], except with a trivial twist (i.e., with κ = 1). Such a twist is not expected, since the transfer matrix (A.1) corresponds to periodic boundary conditions. Nevertheless, from numerical studies (see below), we find that a nontrivial twist (κ = 1) is necessary in order to obtain the complete set of eigenvalues from the Bethe ansatz solution. The presence of an effective twist was already noted in earlier work, see e.g. [10,15,50,51].
We remark that the twist is characterized by an integer in Z N . Indeed, we observe from (2.5) that R(1) = ω(q) P. Hence, from (A.1) we obtain In particular, the twist is characterized by an integer l ∈ Z N We have verified numerically for small values of N and s that every distinct eigenvalue of the transfer matrix (A.1) can be expressed in the form (A.10). See Tables 7 and 8, and note that all (2s + 1) N eigenvalues are accounted for. Note also that, in contrast with the case of the open chain, the solutions of the closed-chain Bethe equations (A.11) are not universal: the Bethe roots depend on the value of the spin s (cf . Tables 1-3).