Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.


Introduction
In the pantheon of anisotropic integrable quantum spin chains, one model stands out for its high degree of symmetry: the U q sl(2)-invariant open spin-1/2 XXZ quantum spin chain, whose Hamiltonian is given by [1] 1) where N is the length of the chain, σ are the usual Pauli spin matrices, and q = e η is an arbitrary complex parameter. As is true for generic quantum integrable models, the Hamiltonian is a member of a family of commuting operators that can be obtained from a transfer matrix [2]; and the eigenvalues of the transfer matrix can be obtained in terms of admissible solutions {λ k } of the corresponding set of Bethe equations [3, 2, 1] 1 sh(λ k − λ j + η) sh(λ k + λ j + η) , k = 1 , 2 , . . . , M , M = 0 , 1 , . . . , N 2 , where k denotes the integer not greater than k.
When the anisotropy parameter η takes the values η = iπ/p with integer p ≥ 2, and therefore q = e η is a root of unity, several interesting new features appear. In particular, the symmetry of the model is enhanced (for example, an sl(2) symmetry arises from the so-called divided powers of the quantum group generators); the Hamiltonian has Jordan cells [4,5,6]; and the Bethe equations (1.2) admit continuous solutions [7], in addition to the usual discrete solutions (the latter phenomenon also occurs for the closed XXZ chain [8,9,10,11,12]).
We have recently found [7] significant numerical evidence that the Bethe equations have precisely the right number of admissible solutions to describe all the distinct (generalized) eigenvalues of the model's transfer matrix, even at roots of unity.
We focus here on the related problem of constructing, via the algebraic Bethe ansatz, all 2 N (generalized) eigenvectors of the transfer matrix. For generic q, the construction of these eigenvectors is similar to the one for the simpler spin-1/2 XXX chain: to each admissible solution of the Bethe equations, there corresponds a Bethe vector, which is a highest-weight state of U q sl(2) [1,13,14]; and lower-weight states can be obtained by acting on the Bethe vector with the quantum-group lowering operator F . However, at roots of unity q = e iπ/p with integer p ≥ 2, we find that there are two additional features: i. Certain eigenvectors must be constructed using the continuous solutions noted above.
These solutions contain p equally-spaced roots (so-called exact complete p-strings), whose centers are arbitrary, see Prop. 3.1 for more details. This construction is a generalization of the one proposed by Tarasov for the closed chain at roots of unity [12].
ii. We propose that the generalized eigenvectors can be constructed using similar string configurations of length up to p − 1, except the centers tend to infinity. We refer to Prop. 4.3 for more details.
We demonstrate explicitly for several values of p and N that the complete set of (generalized) eigenvectors can indeed be obtained in this way.
The outline of this paper is as follows. In section 2 we briefly review results and notations (specifically, the construction of the transfer matrix, the algebraic Bethe ansatz, and U q sl(2) symmetry) that are used later in the paper. In section 3 we work out in detail the construction noted in item i above with the result formulated in Prop. 3.1, see in particular Eqs. (3.7) and (3.26). In section 4 we describe the construction noted in item ii above with the final result in Prop. 4.3, see in particular Eq. (4.44). These two constructions are then used in section 5 to construct all the (generalized) eigenvectors for the p = 2 root of unity case with N = 4, 5, 6, as well as selected eigenvectors with N = 7, 9. We present all the (generalized) eigenvectors for various values of p > 2 and N in section 6. We conclude with a brief discussion in section 7. Some ancillary results are collected in four appendices. In Appendix A, we explicitly describe the action of U q sl (2) in tilting modules at roots of unity. In Appendix B, we present numerical evidence for the string solutions used in section 4 for constructing generalized eigenvectors. In Appendix C, we derive a special off-shell relation (similar to the one found by Izergin and Korepin [15] for repeated Bethe roots), which we use in Appendix D to derive an off-shell relation for generalized eigenvectors.

Preliminaries
The transfer matrix and algebraic Bethe ansatz for the model (1.1) follow from the work of Sklyanin [2], which was already reviewed in [7]. However, we repeat here the main results, both for the convenience of the reader and also to explain a useful change in notation (see (2.8) and subsequent formulas).

Transfer matrix
The basic ingredients of the transfer matrix are the R-matrix (solution of the Yang- respectively. The R-matrix is used to construct the monodromy matrices T a (u) = R a1 (u) · · · R aN (u) ,T a (u) = R aN (u) · · · R a1 (u) .
Finally, the transfer matrix t(u) is given by [2] t(u) = tr a K + a (u) U a (u) , (2.4) where U a (u) = T a (u) K − a (u)T a (u) .

U q sl(2) symmetry
For generic q, the quantum group U q sl(2) has generators E , F , K that satisfy the relations These generators are represented on the spin chain by (see e.g. [16]) The transfer matrix has U q sl(2) symmetry [17] [ 22) and the Bethe states satisfy As reviewed in [7], the Bethe states are U q sl(2) highest-weight states of spin-j representations and dimension For the root of unity case q = e iπ/p , the generators satisfy the additional relations The Lusztig's "divided powers" [18] are defined by (see e.g. [19]) The generators e, f, h obey the usual sl(2) relations The transfer matrix also has this sl(2) symmetry at roots of unity.
The space of states of the spin chain is given by the N -fold tensor product of spin-1/2 representations V 1/2 . As already reviewed in [7], for q = e iπ/p , this vector space decomposes into a direct sum of tilting U q sl(2)-modules T j characterized by spin j, where the sum starts from j = 0 for even N and j = 1/2 for odd N . The multiplicities d 0 j of these T j modules are given by [20] where d j is given by and t(j) = 1 for (j mod p) > p−1 2 , 0 for (j mod p) < p−1 2 . (2.33) The dimensions of the tilting modules are given by [7] dim T j = 2j + 1,

General structure of the tilting modules
For our analysis, we need an explicit structure and the U q sl(2) action on the tilting modules T j that appear in the decomposition (2.30). The structure of the tilting U q sl(2)-modules was studied in many works [1,21,18,22,20]. The tilting U q sl(2)-modules T j in (2.30) for 2j + 1 less than p or divisible by p are irreducible and isomorphic to the spin-j modules (or V j in our notations). 3 Otherwise, each T j is indecomposable but reducible and contains V j as a submodule while the quotient T j /V j is isomorphic to V j−s(j) , where s(j) is defined in (2.35). Both the components V j and V j−s(j) are further reducible but indecomposable: V j has the unique submodule isomorphic to the head (or irreducible quotient) of the V j−s(j) module, and V j−s(j) has the unique submodule isomorphic to the head of the V j−p module. We denote the head of V j by j . Then, the sub-quotient structure of T j in terms of the irreducible modules j can be depicted as where arrows correspond to irreversible action of U q sl(2) generators and we set j = 0 for j < 0.
To compute dimensions dim j of the irreducible subquotients in (2.36), we note the relation dim j = 2j + 1 − dim j − s(j) that follows from the discussion above (2.36). It is then easy to check the following formula for dimensions 4 by induction in r ≥ 0: Note that the highest-weight vector in the irreducible module j has S z = j.
We shall refer to the four irreducible subquotients in (2.36), starting from the top j−s(j) and going around clockwise, as the "top" T j , "right" R j , "bottom" B j , and "left" L j nodes, respectively. We refer the interested reader to Appendix A for the description of the basis and U q sl(2)-action in T j .

Bethe states for exact complete p-strings
For η = iπ/p with integer p ≥ 2 (so that q = e η is a root of unity), the Bethe equations (1.2) admit exact solutions consisting of p λ's differing by η, e.g.
where v is arbitrary. Such solutions have been noticed in the context of (quasi) periodic chains [8,9,10,11,12], and were called in [9] "exact complete p-strings." Such solutions do not lead to new eigenvalues of the transfer matrix, and therefore, we do not regard such solutions as admissible. Nevertheless, Bethe states corresponding to such solutions are necessary in order to construct the complete set of states when one or more tilting modules are spectrum degenerate [7].
The Bethe states (2.9) corresponding to such solutions are naively null, since as already noticed by Tarasov for the (quasi) periodic chain in [12,23,24]. 5 We proceed, following [12] (see also [10]), by regularizing the solution and taking a suitable limit. Therefore, we now define and we consider the limit µ → 0. Given a usual Bethe state |λ 1 . . . λ M (2.9), we define the operators 6

5)
5 For the closed chain, the corresponding product of B operators is a component (top-right corner) of a fused [25,26] monodromy matrix; and, for η = iπ/p, this fused monodromy matrix becomes block diagonal, and therefore the top-right corner becomes zero. (See Proposition 5 parts (i) and (ii) in [23], and Lemmas 1.4 and 1.5 in [24].) The same logic applies to the open chain, in view of the open-chain generalization [27] of the fusion procedure. 6 For simplicity, we assume here that the λ i 's are fixed and do not depend on µ. In principle, the analysis presented here could be generalized by not making any assumptions about the λ i 's at the outset, which in fact is the approach taken in [12] for the closed chain. However, the result of such an analysis is that, in order to obtain an eigenvector of the transfer matrix, the λ i 's must indeed be solutions of the Bethe equations with µ → 0. 7 The operator (3.5) is well defined, since as well as the corresponding new states and where the transfer matrix t(u) and the B operators (including those used in the construction of the Bethe state |λ 1 . . . λ M of course) should be understood to be constructed with generic anisotropy η instead of η 0 , and x 1 , . . . , x p are still to be determined. To this end, we obtain the off-shell relation for this state (c.f. (2.11)) 8) and the limit µ → 0 remains to be performed. Evidently, there are now two kinds of "unwanted" terms.
It is easy to see from (2.12) that X(u), which appears in the first line of (3.8), is given by where Q(u) is given by (2.13), and the E ± (u) are defined by where In the second line of (3.11), we keep explicitly only the first term in the expansion around µ = 0 and neglect contributions that vanish when µ vanishes. We see that E ± (u) → 1 in the limit µ → 0, and therefore X(u) → Λ(u).
Similarly, from (2.14) we find that Y m , which appears in the second line of (3.8), is given by and therefore Y m → Λ λm as µ → 0. Hence, the "unwanted" terms of the first kind in (3.8) vanish provided that λ 1 , . . . , λ M satisfy the usual Bethe equations (1.2) at η = η 0 . (The factor 1/µ in the second line of (3.8) is canceled by the contribution from p−1 r=0 B(v +rη + µx r+1 ) which vanishes as fast as O(µ) for µ → 0, as we noticed above.) Finally, again from (2.14) we find that Z r , which appears in the third line of (3.8), is given by and we have again neglected contributions that vanish when µ vanishes. We find while for r = 0 the above result continues to hold except with while for r = p−1 the above result continues to hold except with x p+1 = x 1 −p. We conclude that the "unwanted" terms of the second kind in (3.8) vanish provided that x 1 , . . . , x p satisfy for r = 0, 1, . . . , p − 1, where (3.18) and Q(u) in (2.13) is to be evaluated with η = η 0 .
In order to solve (3.17) for x 1 , . . . , x p , we now make (along the lines of [12]) the following ansatz where F (u) and G(u) are functions with periodicities η 0 and iπ, respectively, Then the boundary conditions (3.18) are satisfied, and where , (3.24) which is satisfied by 8 We have therefore proved the following proposition.
Proposition 3.1. If |λ 1 . . . λ M is an eigenstate of the transfer matrix t(u) with eigenvalue Λ(u), then for any v ∈ C the corresponding state v; λ 1 . . . λ M constructed in (3.7) using an exact complete p-string, where x r are given by (3.19), (3.23) and (3.25) using (2.13), is also an eigenstate of the transfer matrix with the same eigenvalue Λ(u).
By this proposition we see that the operator B(v) in (3.5) maps the specific eigenstate |λ 1 . . . λ M defined in (2.9) to another eigenstate of t(u). But acting with B(v) on other Bethe states does not give in general eigenstates, or saying differently the operator B(v) does not in general commute with t(u), as its definition involves Bethe roots λ i via the function Q(u).
Remark 3.2. For the particular case p = 2, the Q(u) function obeys Q(u + 2η 0 ) = Q(u), and therefore the ratio of Q(u) functions in (3.24) equals 1, which implies that H(u) can be chosen independently of {λ i }, e.g. H(u) = sh 2N (u − η 0 2 ) sh(2u − η 0 ); and therefore {x r } and thus B(v) are independent of {λ i }. This suggests that B(v) might be a symmetry of t(u) as it maps any Bethe state to another eigenstate of the same eigenvalue. We have verified numerically for p = 2 and up to N = 6 that B(v) indeed commutes with t(u) for any complex numbers u and v.
Several examples of the construction in Proposition 3.1 with p = 2 can be found in Sec. 5, see e.g. Secs. 5.2, 5.3, and 5.4. For p > 2, the first appearance of an exact complete p-string is for the case p = 3, N = 8, M = 0, see Section D.6 in [7]. We have constructed the vector || v; − (3.7) numerically for this case, with a generic value for v, and we have verified that it is an eigenvector of the Hamiltonian with the same eigenvalue as the reference state (namely, E = 3.5), yet it is linearly independent from the reference state. Moreover, it is a highestweight vector with spin j = 1, exactly as required for the right node of the tilting module T 1 (recall the structure in (2.36) and its description above), which is spectrum-degenerate with the tilting module T 4 containing the reference state.
Remark 3.3. The generalization to the case of more than one exact complete p-string is straightforward: a vector with m such p-strings is given by where B(v i ) is constructed as in (3.5) and with {x i,r } given by with the same boundary conditions on x i,r . We note that the S z -eigenvalue of (3.26) is N 2 − M + mp and thus the operators m i=1 B(v i ) describe t(u) degeneracies between S zeigenspaces that differ by a multiple of p. We stress that these degeneracies are extra to the degeneracies corresponding to the action by the divided powers of U q sl(2) that also change S z by ±p. We discuss below this new type of degeneracies. An example with two exact complete p-strings (i.e., m = 2, with p = 2) is given in Sec. 5.5.

Generalized Bethe states
The usual Bethe states (2.9) are, by construction, ordinary eigenvectors of the transfer matrix t(u). In order to construct generalized eigenvectors (which, as noted in the Introduction, appear at roots of unity), something different must be done. We recall that generalized eigenvectors |v are defined as 9 We note that a generalized eigenvector, as |v in (4.2), is defined only up to the transformation Generalized eigenvectors appear only in (direct sums of) the tilting U q sl(2)-modules T j with s(j) non-zero, i.e. in the cases where T j are indecomposable but reducible, and thus are described by the diagram in (2.36). This fact is borne out by the explicit examples in our previous paper [7], see also [4,5] and the proof for p = 2 in [16]. As we will see further from an explicit construction in this section, it is only the states in the head of T j -the top sub-quotient j − s(j) in (2.36) -on which the Hamiltonian (1.1) is non-diagonalizable. For the case p = 2, it was already shown in [16] using certain free fermion operators.

Introduction and overview
An important clue to a Bethe ansatz construction of the generalized eigenvectors can already be learned by considering the simplest case, namely a chain with two sites (N = 2). Indeed, for this case and for generic values of q, the eigenvectors of the Hamiltonian (1.1) are given by The first three vectors, which form a spin-1 representation of U q sl(2), have the same energy eigenvalue E 1 = 1 2 [2] q , while the fourth vector (a spin-0 representation) has the energy eigenvalue E 0 = − 3 2 [2] q . For p = 2 (i.e., q = e iπ/2 = i), the vectors |v 2 and |v 4 evidently coincide (and E 1 = E 0 = 0), signaling that the Hamiltonian is no longer diagonalizable. A generalized eigenvector of the Hamiltonian with generalized eigenvalue 0 can be constructed from the q → i limit of an appropriate linear combination of these two vectors, e.g., Let us now consider the corresponding Bethe ansatz description. For generic q, the vector |v 4 is given by where a depends on η such that a(iπ/2) = 0. As q approaches i (i.e., η approaches iπ 2 ), the Bethe root ν in (4.6) goes to infinity. Indeed, setting η = iπ 2 − iω 2 , we find that for ω near 0. Expanding the Bethe vector in a series about ω = 0, we observe that which is a generalized eigenvector of the Hamiltonian. Note the similarity of the constructions in (4.5) and (4.9): both involve subtracting from a (generically) highest-weight state a contribution proportional to |v 2 = F |Ω and taking the q → i limit. The generalized eigenvector |v is evidently a linear combination of the generalized eigenvector |w in (4.5) and the eigenvector |v 2 in (4.4), recall that the generalized eigenvector is defined up to the transformation (4.3).
A construction of generalized Bethe states similar to (4.9) is possible for general values of N and p. We observe from numerical studies given in App. B that, as the anisotropy parameter η approaches η 0 = iπ/p with integer p ≥ 2, the Bethe roots corresponding to a generalized eigenvalue contain a string of length p ∈ {1, 2, . . . , p − 1}, whose center (real part) approaches infinity. In more detail, such a string is a set of p roots differing by iπ/p , e.g.
with ν 0 → ∞. As we shall see below, the value of p is related to the spin j of the tilting module T j (the one containing the corresponding generalized eigenvector) by the simple formula where s(j) ∈ {1, 2, . . . , p − 1} is defined in (2.35). For p = 2, the only possibility is p = 1, i.e. an infinite real root, as already discussed. For p = 3, the only possibilities are p = 1 and p = 2, where the latter consists of the pair of roots ν 0 ± iπ/4 with ν 0 → ∞. For p = 4, we can have p = 1, 2, 3; the p = 3 case consists of a triplet of roots ν 0 , ν 0 ± iπ/3 with ν 0 → ∞, etc. The corresponding Bethe state has Bethe roots {ν ∞ k } tending to infinity in the limit, and requires a certain subtraction to get a finite vector. In a nutshell, our construction of generalized eigenvectors in a tilting module T j starts with the spin-j highestweight state that lives in the right node denoted by j in the diagram (2.36). This state can be constructed using the ordinary ABA approach as in (2.9). Then, a generalized eigenstate living in the top node j − s(j) is constructed by applying a certain p -string of B(ν k ) operators (with ν k as in (4.10) but finite ν 0 ) on the usual Bethe state in j at generic value of η, subtracting the image of F p on the spin-j highest-weight state and taking the limit η → η 0 . We give below details of the construction with our final claim in Prop. 4.3, while our representation-theoretic interpretation is given in Sec. 4.5.

General ABA construction of generalized eigenstates
With these observations in mind, let denote an on-shell Bethe vector, i.e., an ordinary eigenvector of the transfer matrix where the eigenvalue Λ(u) is given by (2.12). This state is an U q sl(2) highest-weight state with spin j = N/2 − M , see (2.24). Under the already-mentioned assumption that the top node j − s(j) of T j contains generalized eigenstates, let us construct a generalized eigenvector ||| λ (p ) ≡ |||λ 1 . . . λ M (p ) whose generalized eigenvalue is also Λ(u), where p = s(j). To this end, we now set and look for a generalized eigenvector as the limit Note that the subscripts α and β on λ α,k and λ β,k are simply labels (i.e., not indices) that serve to distinguish λ α,k from λ β,k and from λ k . Note that λ k is the Bethe solution precisely at the root of unity, when ω = 0, while λ α,k and λ β,k are a priori different functions of ω.
And we assume that, as ω → 0+, where ν ∞ j is given in (4.10) with ν 0 diverging as ν 0 = − log ω. However, the {ν j }, {λ α,k }, {λ β,k } as well as the coefficients α and β (actually certain powers of ω) are still to be determined. The B operators and the transfer matrix t(u) should again (as in Section 3) be understood to be constructed with anisotropy η instead of η 0 . Moreover, F is the U q sl(2) generator (see section 2.3) and as an operator it also depends on q = e η .
We shall see that the state ||| λ (p ) or the limit (4.15) is well defined and has the same transfer-matrix (generalized) eigenvalue as | λ in (4.12), and both states belong to the same tilting module T j , see Rem. 4.4 below. As in the usual ABA construction, the state ||| λ (p ) in our construction also has the maximum value of S z in the irreducible subquotient to which it belongs, namely, the top node j − s(j) . We know from (2.23) and (4.17) that this state On the other hand, we know from the general structure of tilting modules (2.36) that ||| λ (p ) has S z = j − s(j). It follows that p = s(j), as already noted in (4.11).
Next, we observe that for ω → 0, the vector | ν, λ α has the power series expansion: where c is some numerical factor. For p = 1, this follows from the fact that for ω → 0. We therefore set According to the off-shell relation (2.11), the transfer matrix t(u) has the following action on the off-shell Bethe vector | ν, λ α : where a hat over a symbol means that it should be omitted, i.e. and and Q ν (u) and Q α (u) are defined as Moreover, according to (2.14), we have and Similarly, the action of the transfer matrix on the off-shell Bethe vector | λ β is given by and We argue in Appendix D that, in order for ||| λ (p ) (4.15) to be a generalized eigenvector of the transfer matrix, i.e., it obeys (4.1), it suffices to satisfy the following conditions: where α and β are given by (4.21).
Recalling the expressions (4.26) and (4.27) for Λ ν i (u) and Λ λ α,i (u), we see that the conditions (4.34)-(4.35) require that { ν, λ α } be approximate solutions (as ω → 0) of the Bethe equations 10 By 'approximate solutions' we mean that the equations are satisfied up to a certain order in ω, not necessarily in all orders, i.e., we solve equations (4.37) and (4.38) in the sense of perturbation theory in the small parameter ω, until (4.34)-(4.35) are satisfied. Similarly for the condition (4.36), it requires that λ β be an approximate solution of the Bethe equations corresponding to Λ λ β,i (u) in (4.31), Let us therefore look for a solution { ν, λ α } of the Bethe equations (4.37)-(4.38) with M +p Bethe roots that approaches { ν ∞ , λ} as ω → 0, recall our assumption on the limit (4.18). We assume that for small ω this solution is given by where the coefficients {a jk , b jk } are independent of ω. To determine these coefficients, we rewrite the Bethe equations (4.37)-(4.38) in the form where BAE k is defined as the difference of the left-hand and right-hand sides. We insert (4.14) and (4.40) into (4.41), perform series expansions about ω = 0, and solve the resulting equations for {a jk , b jk }, starting from the most singular terms in the series expansions (the most singular term has obviously a finite order in ω). In practice, the conditions (4.34)-(4.35) are satisfied by keeping sufficiently many terms in the expansion (4.40).
Similarly, we can find a solution λ β of the Bethe equations (4.39) with M Bethe roots that approaches λ as ω → 0. We assume that for small ω this solution is given by and we solve for the coefficients {c jk } in a similar way. We find in practice that, by keeping sufficiently many terms in the expansion (4.42), the condition (4.36) is also satisfied. In general, λ β = λ α .
We then find by doing explicit expansion using (4.40) and (4.42), with the same number of terms in the sums as in the previous step, that Λ β (u) − Λ α (u) (recall the definitions (4.24), (4.29)) is of order ω 2p (4.43) For the choice of β in (4.21), it follows that both conditions (4.32) and (4.33) are also satisfied.
has. Note further that |v is in the same tilting module T j=N/2−M as the initial Bethe state | λ because the two states have the same eigenvalue Λ(u) of the transfer matrix t(u), and the ordinary Bethe states of the same M value are non-degenerate (with respect to t(u)) at roots of unity [7]. Indeed, if the generalized eigenstate |v would belong to another copy of T j=N/2−M not containing | λ , we could obtain by acting on |v with (p power of) the raising U q sl(2) generator E a highest-weight state, see the action in App. A, which is another Bethe state 11 , say | λ , with the same M and by construction the same eigenvalue Λ(u) as |v , which contradicts the non-degeneracy result in [7], and thus | λ ∼ | λ . Further, the weight S z = N/2 − M − p is only doubly degenerate in T j=N/2−M : |v = ||| λ (p ) and the vector F p | λ in the bottom of T j have this weight. We thus have (4.2) with |v ∼ F p | λ .

Representation-theoretic description
We give here a representation-theoretic interpretation of our construction in Prop. 4.3 by analyzing the contribution of V j 's to different tilting modules in the root-of-unity limit. Then, we also discuss the problem of counting the (generalized) eigenvectors using this analysis.
We begin with the decomposition of the spin chain at generic q where the multiplicity d j of the spin-j representation V j is defined in (2.32). It is instructive to compare this decomposition with the one (2.30) at roots of unity in terms of tilting modules T j with multiplicities d 0 j ≤ d j , see the expression in (2.31). We will consider further only those values of j for which 2j + 1 modulo p is nonzero (that is, s(j) defined in (2.35) is nonzero), i.e., when T j are indecomposable but reducible and thus contain generalized eigenvectors, recall the discussion after (4.3). The multiplicity d 0 j is then strictly less than d j . Each such T j contains V j as a proper submodule. The corresponding spin-j highest-weight state lives in the node denoted by • j in the left half of Fig. 1. This state can be constructed using the ordinary ABA approach as in (2.9). The rest d j − d 0 j = d 0 j+p−s(j) of the initial number of V j 's are not submodules but sub-quotients in another tilting module -in T j+p−s(j) (recall the discussion in Sec. 2.4.) Being 'sub-quotient' here means that the spin-j states lose the property "highest-weight" in the root-of-unity limit. These states are generalized eigenstates of t(u). They live in the node • j in the right half of Fig. 1.
We therefore expect that d 0 j of the spin-j Bethe states have a well-defined limit as q approaches a root of unity and give ordinary t(u)-eigenstates living in • j ; and on the other side, we expect irregular behavior of the d j − d 0 j Bethe states -the corresponding Bethe roots ν k go to infinity as (4.10) -such that an appropriate limit gives the generalized eigenstates living in  .20)) on the spin-(j + p − s) highest-weight state that guarantees absence of diverging terms in the limit. We sketched this in the right half of Fig. 1 where the subtraction is schematically denoted by [. . .] 2 . Note that the difference in the highest S z -eigenvalues in We give finally a comment about counting the (generalized) eigenstates. The limit of ordinary Bethe states gives as many linearly independent states as the number of admissible solutions of the Bethe equations at the root of unity, and we know [7] that there can be deviations of this number from d 0 j (it is less than d 0 j in general). Taking into account the deviations n j studied in [7] we should thus have d 0 j − n j linearly independent eigenstates and the number d 0 j+p−s − n j+p−s of linearly independent generalized eigenstates of spin-j. To construct the missing eigenstates of spin-j or highest-weight states in • j , we should use the exact complete p-strings from Sec. 3. We believe that the same complete p-strings construction can be applied to generalized eigenvectors and it recovers the total number d 0 j+p−s of generalized eigenvectors of spin-j.

Examples
We now illustrate the general construction (4.44) with several explicit examples.

p = 2
As already noted, for p = 2, the only possibility is p = 1, i.e. an infinite real root. For even N and irrespectively of the value of M , the small-ω behavior of this root is given by as in (4.7) and (4.40). We find that the construction (4.44) produces a generalized eigenvector irrespectively of the values of the O(ω 0 ) and higher-order terms. Hence, for p = 2 and even N , the generalized eigenvector ||| λ (1) corresponding to the on-shell Bethe vector | λ with any value of M is given by for some "non-universal" constant c and ν is given by (4.47). We denote by |||− (1) the result for the reference state (no Bethe roots) | λ = |Ω . For odd N , there is no solution of the form (4.47), which is in correspondence with the fact that the Hamiltonian is diagonalizable at odd N .
For example, we have explicitly computed (4.48) with | λ = |Ω for N = 4, 6, 8 using Mathematica, and we have verified that the result |||− (1) is a generalized eigenvector of the Hamiltonian (1.1), with generalized eigenvalue 0: where we use ∼ to denote equality up to some nonzero numerical factor.

p = 1
Let us first consider the case p = 1, N = 6 and M = 0. Following the procedure explained in (4.40) and immediately below, we find that the corresponding ν is given by Let us now consider p = 2. An example is the case N = 4 and M = 0, for which ν (4.40) is given by We have explicitly verified that |||− (2) is a generalized eigenvector of the Hamiltonian, with generalized eigenvalue 3/2: Another example is the case N = 6 and M = 1. This is our first example with M > 0 (and p > 2), which makes this case particularly interesting. There are 4 solutions of the Bethe equations (1.2) with p = 3, N = 6, M = 1, and let us focus here on the simplest λ = 1 2 log 2 ≈ 0.346574. By following the procedure described around (4.40)-(4.42), we obtain Note that λ α = λ β . We have explicitly verified that the corresponding vector |||λ (2) (4.44) is a generalized eigenvector of the Hamiltonian with generalized eigenvalue −3/2,

p = 4
For p = 4, we can have p = 1, 2, 3, but we illustrate here only two of these three possibilities.

p = 3
Let us now consider p = 3. An example is the case p = 4, N = 6, M = 0, p = 3, for which ν (4.40) is given by We have explicitly verified that the corresponding vector |||− (3) (4.44) is a generalized eigenvector of the Hamiltonian with generalized eigenvalue 5 √ 2/2, Even N For p = 2 and even N , the decomposition (2.30) consists of tilting modules T j of dimension 4j, where j is an integer. Recall the diagram in (2.36): each such module has a right node (or simple subquotient) R j of dimension j + 1, a bottom node B j of dimension j, a top node T j of dimension j, and a left node L j of dimension j − 1 (provided that j > 1). We use the basis and U q sl(2)-action in T j in App. A to make the following statements. The right node consists of the vectors 12 R j : |v , f |v , f 2 |v , . . . , f j |v , where |v can be either a usual Bethe state or a state constructed from an exact complete 2-string; and f is the s (2) lowering generator from U q sl(2). The bottom node consists of the vectors obtained by acting on the right node with the U q sl(2) lowering generator F The top node consists of the generalized eigenvectors where |||v (1) is given by (4.48) with | λ = |v . Finally, the left node L j consists of (ordinary) eigenvectors. We first introduce states obtained by acting on the top node with F L j : F |||v (1) , F f |||v (1) , F f 2 |||v (1) , . . . , F f j−2 |||v (1) .

(5.4)
Together with (5.1), they form a basis in the direct sum L j ⊕ R j , the states in L j are linear combinations of those inL j and R j . For later convenience, we will refer toL j instead of L j , see more details in Sec. 6 for the general case.
We note that the generalized eigenvectors appear only in the top node.
Odd N For p = 2 and odd N , the decomposition (2.30) consists of irreducible tilting modules T j = V j of dimension 2j + 1, where j is half-odd integer -indeed, the number s(j) is zero for all these j, and all T j are then irreducible following the discussion in Sec. 2.4. Starting from a highest-weight vector |v , the remaining vectors of the multiplet are obtained by applying F and powers of f . For odd N there are only ordinary eigenvectors (i.e., no generalized eigenvectors), which is in agreement with [16].

Examples
We now illustrate the above general framework by exhibiting ABA constructions of complete sets of 2 N (generalized) eigenvectors for the cases N = 4, 5, 6. For each of these cases, we have explicitly verified that the vectors are indeed (generalized) eigenvectors of the Hamiltonian (1.1) and are linearly independent. The needed admissible solutions of the Bethe equations for p = 2 are given in Appendix C of [7].
We also consider selected eigenvectors for the cases N = 7, 9 in order to further illustrate the construction in Sec. 3. We emphasize that when one or more modules in the decomposition (2.30) are spectrum-degenerate (which can occur for either odd or even N ), it is necessary to use this construction (3.7), (3.26) based on exact complete 2-strings.

N = 5
For p = 2, N = 5, the space of states decomposes into a direct sum of irreducible representations The V5 2 , with dimension 6, has the reference state |Ω as its highest weight state. As noted in Appendix D of [7], this module is spectrum-degenerate with one copy of V

N = 6
For p = 2, N = 6, the decomposition (2.30) is given by The T 3 consists of the following 12 vectors: |v , f |v , f 2 |v , f 3 |v , where |v = |Ω is the reference state. As noted in Appendix D of [7], this module is spectrum-degenerate with one copy of T 1 ; the latter has the basis (5.6) where |v = ||v 1 ; − is a generalized eigenvector constructed from an exact perfect 2-string and no other Bethe roots, and v 1 is an arbitrary number. The remaining four copies of T 1 also have the basis (

N = 7
For p = 2, N = 7, the decomposition (2.30) is given by For this case we do not enumerate all the eigenvectors, focusing instead on those constructed with exact complete 2-strings.
As noted in Appendix D of [7], V 7 2 is spectrum-degenerate with two copies of V Each of the corresponding V 1 2 , with dimension 2, has the highest-weight vector ||v 1 ; λ i.e. an eigenvector constructed from an exact perfect 2-string (v 1 is arbitrary) and the Bethe root λ. These are the first examples of the construction (3.7) that we meet involving a Bethe state other than the reference state. However, since here p = 2, then (as noted in Rem. 3.2) the {x r } used in this construction do not depend on λ.

N = 9
For p = 2, N = 9, the decomposition (2.30) is given by Again for this case we do not enumerate all the eigenvectors, focusing instead on those constructed with exact complete 2-strings.
As noted in Appendix D of [7], V 9 2 is spectrum-degenerate with three copies of V 5 2 as well as with two copies of V 1 2 . The module V 9 2 , with dimension 10, has the reference state |Ω as its highest weight state. The

Complete sets of eigenstates for p > 2
We now exhibit ABA constructions of complete sets of 2 N (generalized) eigenvectors for various values of p > 2 and N . The decomposition (2.30) consists of tilting modules T j of dimension 2j + 1 if s(j) = 0, see (2.35), and of dimension 4j + 2 − 2s(j) = 2pr, where j is an integer or half-odd integer, and we set 2j + 1 ≡ rp + s and s ≡ s(j) (6.1) for brevity. Recall the diagram in (2.36): each T j with non-zero s(j) has a right node (or simple subquotient) R j of dimension s(r + 1), a bottom node B j of dimension (p − s)r, a top node T j of dimension (p − s)r, and a left node L j of dimension s(r − 1) (provided that r > 1). We use the basis (A.4) and U q sl(2)-action in T j in App. A to make the following statements. The right node consists of the vectors where |v can be either a usual Bethe state or a state constructed from an exact complete p-string -it is a highest-weight vector; and f is the s (2) lowering "divided power" generator from U q sl (2). The bottom node consists of the vectors obtained by acting on the right node with the U q sl(2) lowering generator F The top node consists of the generalized eigenvectors where |||v (s) is given by (4.44). Finally, the left node L j consists of the (ordinary) eigenvectors l n,m . To construct the basis {l n,m } in the left node L j , we first introduce states obtained by acting on the top node with F p−s : Together with (6.2), they form a basis in the direct sum L j ⊕ R j . The vectorsl n,m do not belong to L j , they are a linear combination of l n,m and r n,m+1 :l n,m = 1 r (r n,m+1 − l n,m ), compare with the F action in App. A. We will use below the basis elementsl n,m instead l n,m .
In all the examples below, we have explicitly checked that the vectors in (6.2)-(6.5) are indeed (generalized) eigenvectors of the Hamiltonian (1.1) and are linearly independent, and thus give a basis in T j as they should. We have also verified by the explicit construction of the states that the dimensions of the nodes in T j coincide with the values given by (2.36) and (2.37) and reviewed just above. We remind the reader that all the needed admissible solutions of the Bethe equations (1.2) are given in Appendix E in [7].
Each of the three T 1 are irreducible representations of dimension 3 consisting of a highestweight vector B(λ)|Ω plus two more states obtained by lowering with F . The three admissible solutions of (1.  Each of the four T3 2 (dimension 6) has the following basis, see (6.2)-(6.5) for r = 1 (i.e. L3 2 is absent) and s = 1: • right node R3 2 consisting of the two ordinary eigenvectors |v and f |v , where |v = B(λ) |Ω ; • bottom node B3 2 consisting of the two ordinary vectors F |v and F 2 |v ; and • top node T3 2 consisting of the two generalized eigenvectors |||λ (1) and F |||λ (1) .
The corresponding five admissible solutions of (1. All together we thus find 2 6 = 64 vectors.

Discussion
We have seen that, when q is a root of unity (q = e iπ/p with integer p ≥ 2), the U q sl (2)invariant open spin-1/2 XXZ chain has two new types of eigenvectors: eigenvectors corresponding to continuous solutions of the Bethe equations (exact complete p-strings), and generalized eigenvectors. We have proposed here general ABA constructions for these two new types of eigenvectors. The construction for exact complete p-strings (3.7), (3.26) is a generalization of the one proposed by Tarasov [12] for the closed chain, while the construction of generalized eigenvectors (4.44) is new. We have demonstrated in examples with various values of p and N that these constructions are indeed sufficient for obtaining the complete set of (generalized) eigenvectors of the model.
The model (1.1) at primitive roots of unity is related to the unitary (p − 1, p) conformal Minimal Models, by restricting to the first p − 1 irreducible tilting modules (see e.g. [1]), as well as to logarithmic conformal field theories if one keeps all the tilting modules [22,20]. We expect that our results can be easily generalized to the case of rational (non-integer) values of p, which is related to non-unitary Minimal Models. Indeed, for rational p = a/b, with a, b coprime and a > b, there are two different cases q a = ±1, i.e., b even or odd. For odd b (or q a = −1 and a can be odd or even), we have obviously the same structure of the tilting U q sl(2) modules, as the structure depends only on the conditions on q and it is the same as for b = 1. The repeated tensor products of the fundamental U q sl(2) representations (or the spin-chains) are decomposed in the same way as well (replacing p by a, of course) and thus with the same multiplicities d 0 j , and therefore our construction of the generalized eigenstates should be the same but using a instead of p, i.e., the p in the p -string takes values from 1 to a − 1, etc. For even b (or q a = 1 and odd a), a more careful analysis is required. According to [18], for the case of q a = 1, the tilting modules have the same structure as in Sec. 2.4, where one should again replace p by a, and the multiplicities in the tensor products are also identical to what we had here. The only real difference will be in the values of the Bethe roots, as the spectrum of the Hamiltonian is different for different choices of a and b, and thus the continuum limit too. We also expect that similar constructions can be used for quantum-group invariant spin chains at roots of unity with higher spin and/or rank of the quantum-group symmetry. It would be interesting to consider similar constructions for supersymmetric (Z 2 -graded) spin chains, such as the U q sl(2|1)-invariant chain [28]. Of course, the algebraic Bethe ansatz would require nesting for rank greater than one, which would render the corresponding constructions more complicated.
We are currently investigating the symmetry operators -generators of a non-abelian symmetry of the transfer-matrix t(u) -responsible for the higher degeneracies of the model, which are signaled by the appearance of continuous solutions of the Bethe equations, whose corresponding eigenvectors are obtained by the construction of section 3. It would also be interesting to find a group-theoretic understanding of the construction in section 4 of generalized eigenvectors, e.g. within the context of the quantum affine algebra U q sl(2) or rather its coideal q-Onsager subalgebra at roots of unity [29].
Acknowledgments AMG thanks Hubert Saleur for valuable discussions, and the IPhT in Saclay for its hospitality. The work of AMG was supported by DESY and Humboldt fellowships, C.N.R.S. and RFBR-grant 13-01-00386. RN thanks Rodrigo Pimenta and Vitaly Tarasov for valuable discussions, and the DESY theory group for its hospitality. The work of RN was supported in part by the National Science Foundation under Grant PHY-1212337, and by a Cooper fellowship.
A Tilting U q sl(2)-modules at roots of unity We explicitly describe here the U q sl(2) action in the tilting modules T j for q = e iπ/p and integer p ≥ 2. For 2j + 1 ≤ p, these modules are irreducible of dimension 2j + 1 = s(j) ≡ s, recall our convention (2.35), and have the basis {a n , 0 ≤ n ≤ s−1} where a 0 is the highestweight vector and the action is Ka n = q s−1−2n a n , h a n = 0, (A.1) Ea n = [n] q [s − n] q a n−1 , e a n = 0, (A.2) F a n = a n+1 , f a n = 0, where we set a −1 = a s = 0.
For 2j +1 > p, the T j 's are identified 13 with projective U q sl(2)-modules from [19] denoted there by P α p−s(j),r with s ≡ s(j) and r defined from the equation 2j + 1 = rp + s(j), i.e., s an integer 1 ≤ s ≤ p − 1 and r ≥ 1, and α ≡ α(r) = (−1) r−1 . Using the identification and the known basis and action [19] in P α p−s(j),r , we give below the U q sl(2)-action in T j 's. For r > 1 and 2j + 1 is not zero modulo p, T j has the basis where we set t n,−1 = t n,r = 0, and identically in B j , while for R j the action is h r k,l = 1 2 (r − 2l)r k,l , e r k,l = l(r + 1 − l)r k,l−1 , f r k,l = r k,l+1 (A.6) where we set r n,−1 = r n,r+1 = 0, and identically in L j but with the replacement of r by r − 2 in (A.6). The U q sl(2)-action of the three other generators E, F , and K in the basis (A.4) is given by 13 The identification is easy to see using the diagram (2.36) with the formula for dimensions (2.37) and the general decomposition of the spin-chain over U q sl(2) in terms of projective covers in [20,Sec. 3.2]. terms and we imply l k,l ≡ 0 in the action. Then, the formulas for the action are the same as above.

B Large p -strings
We provide here some numerical evidence that the Bethe equations (1.2) have solutions of the form (4.10) i.e.
with ν 0 → ∞ as η → η 0 = iπ/p with integer p ≥ 2; and that the corresponding transfermatrix eigenvalues become degenerate in this limit. Such "large p -string" solutions play a key role in the construction described in section 4 of generalized eigenvectors. For convenience, in this section we set η = iπ/p with p real, and we study the limit that p approaches an integer.
B.1 p = 3 , p = 1 Let us consider the case N = 6. For p = 3, we know [7,Table 4(b)] that the transfer-matrix eigenvalue corresponding to the reference state (M = 0, j = 3 ) has degeneracy 12; while away from p = 3, we find that this degeneracy splits into 7 + 5. In view of (2.25) describing the transfer-matrix degeneracy, the corresponding two solutions of the Bethe equations must have M = 0 and M = 1, respectively. The latter solution is our p -string with p = s(3) = 1.  Table 4(b)] that the transfermatrix eigenvalue corresponding to the reference state (M = 0, j = 2) has degeneracy 6; while away from p = 3, we find that this degeneracy splits into 5 + 1. In view of (2.25), the corresponding solutions of the Bethe equations must have M = 0 and M = 2, respectively. The latter solution is our p -string with p = s(2) = 2. Figure 2(a) shows a plot in the complex plane of the latter solution for values of p near 3. We observe that, as p approaches 3, the real part increases, and the imaginary parts approach ±π/4. This solution corresponds to the generalized eigenvector |||− (2) in the tilting module T 2 discussed in sections 4.7.2 and 6.1.
Let us next consider the case N = 6. For p = 3, we know [7, Table 3(b)] that there are 4 transfer-matrix eigenvalues corresponding to solutions of the Bethe equations with M = 1, j = 2, each of which has degeneracy 6. Away from p = 3, we find that this degeneracy splits into 5 + 1. In view of (2.25), the corresponding solutions of the Bethe equations must have M = 1 and M = 3, respectively. We indeed find 4 solutions with M = 3 that consist of a real root and a 2-string, such that, as p approaches 3, the real root remains small, the center of the 2-string becomes large, and the imaginary parts of the 2-string approach ±π/4. These solutions correspond to the generalized eigenvector |||λ (2) with p = s(2) = 2 in the tilting module T 2 discussed in sections 4.7.2 and 6.3.

B.3 p = 4 , p = 1
Let us consider the case N = 4. For p = 4, we know [7,Table 4(c)] that the transfer-matrix eigenvalue corresponding to the reference state (M = 0, j = 2) has degeneracy 8; while away from p = 4, we find that this degeneracy splits into 5 + 3. In view of (2.25), the corresponding solutions of the Bethe equations must have M = 0 and M = 1, respectively. The latter solution is our p -string with p = s(2) = 1. As p approaches 4, this real Bethe root becomes large. This solution corresponds to the generalized eigenvector |||− (1) in the tilting module T 2 discussed in sections 4.8.1 and 6.4.

B.4 p = 4 , p = 3
Let us consider the case N = 6. For p = 4, we know [7,Table 4(c)] that the transfer-matrix eigenvalue corresponding to the reference state (M = 0, j = 3) has degeneracy 8; while away from p = 4, we find that this degeneracy splits into 7+1. In view of (2.25), the corresponding solutions of the Bethe equations must have M = 0 and M = 3, respectively. The latter solution is our p -string with p = s(3) = 3. Figure 2(b) shows a plot in the complex plane of the latter solution for values of p near 4. We observe that, as p approaches 4, the real part becomes large, and the nonzero imaginary parts approach ±π/3. This solution corresponds to the generalized eigenvector |||− (3) in the tilting module T 3 discussed in sections 4.8.2 and 6.5.

C Special off-shell relation
We derive here an off-shell relation for Bethe vectors of the special form B(u) j B(v j )|Ω (i.e., with an "extra" factor B(u), whose argument is the same as that of the transfer matrix t(u)), which we need in Appendix D to derive an off-shell relation for generalized eigenvectors. The proof is a generalization of the one developed by Izergin and Korepin [15] for repeated Bethe roots.

D Off-shell relation for a generalized eigenvector
We derive here an off-shell relation for the vector ||| λ (p ) (4.15), which leads to the set (4.32)-(4.36) of sufficient conditions for this vector to be a generalized eigenvector of the transfer matrix.
In order to satisfy both conditions (D.8) and (D.9), we conjecture that it suffices to have: where the limit in the first line (D.10) is supposed to be finite. Indeed, the conditions (D.10), (D.11) and (D.14) are fairly obvious. The condition (D.13) is less evident, since it is instead αΛ λ α,i (u) that appears in (D.8) and (D.9). However, some of the terms with this factor also contain the vector B(u)| ν,λ α,i which is of order ω −2p N according to (4.19). Hence, we need ω −2p N αΛ λ α,i (u) to vanish as ω → 0, which is equivalent to (D.13), since α and β are given by (4.21). 14 The condition (D.12) has a similar explanation: although αΛ ν i (u) appears in (D.8) and (D.9), some of the terms with this factor also contain the vector |ν i , . . . , which is missing the factor B(ν i ), and therefore is of order ω −2(p −1)N . Hence, we require ω −2(p −1)N αΛ ν i (u) to vanish in the limit.
Corollary D.1. As a corollary of the expression in (D.9) and if the sufficient conditions above are satisfied, the limit of (t(u) − Λ α (u)) ||| λ (p ) ω equal (t(u) − Λ(u)) ||| λ (p ) is non-zero and proportional to F p | λ . Indeed, the proportionality coefficient is (D.10) and finite nonzero by the assumption, while the limit of F p | λ β is F p | λ and it is non-zero due to our special choice of p = s(j) -it is a state in the bottom node of the tilting module T j , recall the discussion just above (4.19) and Sec. 4.5.