On Generalized $Q$-systems

We formulate $Q$-systems for the closed XXZ, open XXX and open quantum-group-invariant XXZ quantum spin chains. Polynomial solutions of these $Q$-systems can be found efficiently, which in turn lead directly to the admissible solutions of the corresponding Bethe ansatz equations.


Introduction
An inconvenient truth about quantum integrable models -well-known to experts but seldom acknowledged -is that the corresponding Bethe ansatz (BA) equations (to which exact solutions of such models invariably reduce) are very difficult to solve. Various approaches to solving BA equations have been investigated, see e.g. [1,2] and references therein. Significant further progress on this problem was recently achieved in [3], which formulated so-called Qsystems, whose polynomial solutions can be found efficiently; the zeros of the fundamental Q-function are the sought-after Bethe roots. The SU(2)-invariant Q-system was an essential ingredient in the recent computation of torus partition functions [4,5], which exploited also techniques from algebraic geometry.
The Q-systems in [3] were restricted to rational BA equations for closed spin chains with periodic boundary conditions. The purpose of this paper is to generalize the SU(2)-invariant Q-system [3] in two different directions: from rational to trigonometric, and from closed to open. These new Q-systems will be used to compute partition functions for trigonometric vertex models and for vertex models with boundaries [6].
The outline of this paper is as follows. In Sec. 2, we review the Q-system from [3] for the closed XXX spin chain with periodic boundary conditions. However, we provide an alternative derivation based on [7], which is convenient for deriving generalizations. In Sec. 3, we formulate a Q-system for the closed XXZ spin chain with periodic boundary conditions. We then turn to open spin chains. In Sec. 4, we formulate a Q-system for the SU(2)-invariant open XXX spin chain. A Q-system for the quantum-group-invariant open XXZ spin chain [8] is formulated in Sec. 5. We conclude in Sec. 6 with a brief summary and a list of some interesting open problems.

Closed XXX Q-system
In this section we review the Q-system [3] whose polynomial solutions provide the full spectrum of the closed XXX spin chain of length N with periodic boundary conditions, whose Hamiltonian is given by First the model is introduced and its solution by the algebraic BA method is recalled, together with the physicality conditions for the Bethe roots. We then describe the relevant Q-system and construct its solution explicitly. We show that polynomial solutions are in one-to-one correspondence with the physical solutions.

Review of the algebraic BA solution
The closed spin-1/2 XXX spin chain and its solution can be succinctly formulated with the help of an SU(2)-invariant solution of the Yang-Baxter equation given by the 4 × 4 R-matrix (see e.g. [9]) R(u) = (u − i 2 )I + iP , (2.2) where P is the permutation matrix, I is the identity matrix, and u is the spectral parameter. For N sites with periodic boundary conditions, one can introduce the monodromy matrix M and the transfer matrix T as T(u) = tr 0 (M 0 (u)) , M 0 (u) = R 01 (u) R 02 (u) . . . R 0N (u) . and generates conserved charges in involution, including the Hamiltonian of the system (2.1).
We are interested in the eigenvectors and eigenvalues of the transfer matrix. The former can be generated from the all spin-up reference state satisfy the T Q-relation where Q encodes the Bethe roots {u i }: As follows from the definition of the transfer matrix and its commutativity property (2.4), T (u) is a polynomial in u, and is thus regular at u j . The T Q-relation (2.8) then leads to the BA equations for the roots: For roots with multiplicities, we have further equations [10,11]. Since repeated roots do not seem to appear in this model (see e.g. [2]), we assume that roots never coincide.
We call a solution of the BA equations physical if the corresponding Bethe vector (2.6) is an eigenvector of the transfer matrix. Unfortunately, not all solutions of the BA equations are physical. Solutions that contain the roots ± i 2 can be unphysical, i.e. they might solve the BA equations, but there is no related eigenvector of the transfer matrix. We define a solution {u 1 , . . . , u M } of the BA equations to be admissible, if all roots are finite and pairwise distinct; and, if they are of the form { i 2 , − i 2 , u 1 , . . . , u M −2 } (which we call a singular solution), then the further constraint is satisfied. It was shown in [12] that admissibility implies physicality, and the converse follows from Lemmata 2 and 4 of [7]. Hence, admissibility and physicality are equivalent. The number N (N, M) of admissible solutions of the BA equations with M ≤ N/2 has been conjectured to be given by (see e.g. [2]) Alternatively, it was observed in [3] that the polynomial solutions of a Q-system on an appropriately chosen diagram can be computed efficiently, and correctly account for the physical solutions. In the remainder of this section, we provide an alternative derivation of these results based on [7], which we will subsequently use to generalize this Q-system.

Q-system
For given values of N and M, the Q-functions Q a,s are defined on a Young tableau with the indices referring to the vertex (a, s), where the a-axis is vertical and the s-axis is horizontal, see Fig. 1. These Q-functions satisfy the QQ-equations, which are formulated around a face . The relevant diagram for the closed XXX spin chain with the boundary conditions, Q 2,s = 1, Q 1,s≥M = 1, is displayed in Fig. 1. The initial condition leads to a unique solution of the Q-system. The degree of the polynomial Q a,s (u) is given by the number of boxes in the Young tableau to the right and top of the vertex (a, s). Let us see how we can proceed column-by-column and express all Q-functions in terms of Q 0,0 and Q 1,0 . The QQ-equation for (a, s) = (1, 0) can be solved easily where we have introduced the discrete derivative defined by The function Q 1,1 is automatically a polynomial of degree M − 1. The equation for (a, s) = (0, 0) gives Making use of (2.15) and (2.17), it follows that or Recognizing the RHS of the above equation as the RHS of the T Q-relation (2.8), one obtains Polynomiality of Q 0,1 is equivalent to the polynomiality of T , which leads to the BA equations (2.10).

Q a,s in terms of Q and P
We now show that the polynomiality of the remaining Q-functions is equivalent to the admissibility of {u 1 , . . . , u M }. To this end, we define a function P (u), such that Using this parametrization for Q 0,0 , one can easily show that where prime denotes discrete derivative (2.16). Repeating the calculations starting from (a, s) = (1, 1) and (a, s) = (0, 1), we arrive at This can be iterated further where the superscript (n) denotes the n th discrete derivative. In short, all Q-functions can be expressed in terms of P and Q. Clearly, if P is a polynomial, then all Q-functions are polynomial. In the following we show that the polynomiality of P is in fact equivalent to the polynomiality of Q 0,2 . We also derive that polynomiality of P is equivalent to the admissibility of the roots {u 1 , . . . , u M }.

Construction of P
We construct P as in [7] by generalizing the approach in [13] (which implicitly assumes that all Bethe roots are regular) to the case of a singular solution. String configurations have roots that differ by i: u i 1 − u i 2 = i, and it is well known (see e.g. [7]) that the only exact string solution consists of one pair of singular roots u 1 = i 2 and u 2 = − i 2 . In the presence of such singular roots, the Q-function takes the following form (2.25) We start by dividing (2.21) by Q + Q − . We need to write in the form i.e. we need to "integrate" R in the discrete sense. To this end, we perform a partial fraction decomposition of (2.26) where π is a polynomial of order N − 2M, the polynomials q ± have degree less thanQ, while a ± are constants. Using the relation which follows from the T Q-relation (2.8), one can investigate all the singularities explicitly.
In particular, the RHS of (2.29) has no singularities at the zeros ofQ, implying for some polynomial q(u). The coefficients a ± can be determined from the residues of (2.29) at u = ∓ i 2 : . (2.31) The polynomial π can always be written as where ρ is a polynomial. Clearly, ρ is defined up to a constant. Changing this constant by a modifies P as P + aQ. This additional term, however, disappears from R and Q 0,0 , thus is irrelevant for us.
In the absence of singular roots, we haveQ = Q and a ± = 0; hence, the polynomial P = ρQ + q satisfies (2.27), which implies the required Eq. (2.21), see also [13]. In the presence of singular roots, the "integration" of R in (2.27) requires to "integrate" u −1 , appearing in (2.28). To this end, we define the function p(u) by In view of the fact we see that the function P satisfying (2.27) takes the form It is a polynomial if and only if a + = −a − , i.e. when is satisfied. Here we used the T Q-relation (2.8) to eliminate T (± i 2 ) in (2.31). Clearly this is the admissibility condition for singular solutions (2.11). Thus, we have just proven that polynomiality of P is equivalent to the admissibility of the roots.
Even if P is not a polynomial, the relation (2.27) implies (2.21), which leads to This implies that P also satisfies the T Q-relation Thus P and Q are the two independent solutions of this second order difference equation, and (2.21) is the corresponding Wronskian relation. It has been known (see e.g. [14,15]) that the two independent solutions of the T Q-relation are both polynomial iff the Bethe state (2.6) is an eigenstate of the transfer matrix.
Finally, let us investigate the polynomiality of Since Q 0,1 is a polynomial 2 , we investigate the regularity of the remaining part at u = 0.
Since Q(± i 2 ) = 0 we can see that which are not zero. We now focus on the pole contributions at u = 0. They can only come from the terms proportional to p(u), which have poles at u = −i(n + 1 2 ) for any integer n ≥ 0, with residues −1. Thus the singular parts can arise as where we have omitted regular terms in ǫ. The singular part of Q 0,2 (ǫ) is then − 1 ǫ (a + + a − )Q( 3i 2 )Q(− 3i 2 ) whose vanishing implies a + = −a − , i.e. the polynomiality of P . We can thus conclude that the following four properties are equivalent: which follows from the T Q-relation (2.8) and (2.25), and which has vanishing residues at the zeros ofQ by virtue of the BA equations.

An example
We conclude this subsection with an elementary explicit example of how to use the Q-system (2.13)-(2.14) to obtain all the Bethe roots for modest values of N ≡ L and M using a simpleminded implementation in Mathematica. (A more general and sophisticated code is provided in [3].) We begin by defining the functions Q 0,0 (u) and Q 1,0 (u) where the coefficients c k are to be determined. We also define the functions Q 1,n (u) and which follow from (2.13) with a = 1 (setting Q 2,s = 1) and a = 0, respectively. The functions Q 1,n (u) are evidently polynomials in u; the key point is that the functions Q 0,n (u) must also be polynomials in u. In order to ensure the latter requirement (the so-called zero-remainder conditions), we first use the built-in symbol PolynomialRemainder to define the polynomials y n (u) Let us consider as an example the case L = 6, M = 2. The above code generates 9 solutions, in agreement with (2.12). For each of these solutions, one can obtain the corresponding Bethe roots by solving for the zeros of Q 1,0 (u). For example, for the first solution

Closed XXZ Q-system
In this section we present a generalization of the Q-system for the closed XXZ spin chain of length N with periodic boundary conditions, whose Hamiltonian is given by We show that there is a notion of polynomial solutions of the QQ-equations, which determine the spectrum of the closed XXZ model.

Review of the algebraic BA solution
The XXZ spin chain is related to the trigonometric generalization of the rational R-matrix The monodromy and transfer matrices can be introduced by the analogous formulae to the XXX case (2.3). The off-diagonal elements of the monodromy matrix (2.6) can be used as creation and annihilation operators. The B operators, by acting on the all spin-up reference state (2.5), create eigenstates of the transfer matrix. The eigenvalue T (u) of the transfer matrix T(u) satisfies the T Q-relation where now Alternatively, we can switch to the variable t = e u . By construction, T (u) is a polynomial of t and t −1 , regular at t j = e u j , which implies the BA equations Singular BA solutions appear also for the XXZ spin chain, and the admissibility of the solution {− η 2 , η 2 , u 1 , . . . , u M −2 } with pairwise distinct and finite roots can be formulated similarly to the XXX case as [16] We conjecture that the number N (N, M) of admissible solutions of the BA equations with M ≤ N/2 is given by In the following we introduce a Q-system whose polynomial (in t and t −1 ) solutions account for the physical solutions. We work for generic η, i.e. when q = e η is not a root of unity.

Q-system and its solution
For the closed XXZ case, as in the XXX case, Q 2,s = 1 and the nontrivial Q-functions are Q 1,s and Q 0,s ; however, there is no condition on Q 1,s≥M . We now regard the Q-functions as functions of the argument t = e u . Moreover, shifts now denote f ± (t) = f (tq ± 1 2 ), and QQ-equations are formulated around each face as The initial conditions are Both of these Q-functions are polynomial in the variables t and t −1 . As in the XXX case, we introduce the analogue of P , and then proceed to express all Q-functions in terms of P and Q.
The QQ-equation for (a, s) = (1, 0) leads again to the discrete derivative of Q: However, contrary to the XXX case, the order of Q 1,1 is the same as that of Q. 6 Since the QQ-equations for the XXZ case (3.8) are the same as for the XXX case (2.13), the solutions are the same, too. In particular, formulas such as (2.18)-(2.20) are exactly the same, and polynomiality of Q 0,1 is equivalent to the polynomiality of T , which gives the BA equations (3.5). In proceeding as before, we search for a function P that satisfies With this P and Q, all Q-functions can be written as in the XXX case (2.24) except that the superscript (n) denotes the n th discrete derivative obtained from multiplicative shifts in t, with f ± (t) = f (tq ± 1 2 ) and f ′ (t) = f + (t) − f − (t). The construction of the function P , once written in terms of the shifts, literally repeats the steps in the XXX case. One first shows that the only singular solutions are t = q ± 1 2 . One then separates the singular solutions as Q(t) = (t − t −1 ) + (t − t −1 ) −Q (t), and performs a partial fraction decomposition of R(t) as where π(t) is polynomial. From the singularity structure, one can obtain r + = r + and r − = −r − for some polynomial function r(t), together with .
( 3.15) The important new step now is the discrete integration of (t − t −1 ) −1 = 1 2 ( 1 t−1 + 1 t+1 ). To this end, we define the function p q (t) by where ψ q (x) denotes the q-deformed digamma function [17,18], which satisfies (3.17) Another step of the XXX case that requires special care when generalizing to the XXZ case, which was already addressed in [13], is the deformed discrete integration of π into ρ such that ρ ′ = π . (3.18) The non-constant terms can be integrated using t n q n/2 −q −n/2 ′ = t n for n = 0. However, a constant term cannot be integrated into a polynomial, and requires instead a function c q (t) such that c ′ q (t) = 1. We have c q (t) = log t log q , (3.19) up to an additive constant. Hence, the function P finally takes the form with ρ 0 a polynomial, and α = π 0 log q a constant, where π 0 is the constant term of π. For N odd α vanishes [13], but it can be non-zero for N even. Thus we see that even in absence of strings, P is not always a polynomial, but what we will call a 'quasi-polynomial', i.e. a polynomial plus log t times a polynomial.
Quasi-polynomiality of P (in t, t −1 ) requires a + = −a − , which is equivalent to the admissibility of the Bethe roots (3.6). As for the polynomiality of the function Q 0,2 (and Q 0,n ), one sees that the log part of P (n)± is α log(t)Q (n)± so that it always cancels out in Q 0,n . Then one discovers that the polynomiality of the function Q 0,2 is also equivalent to the a + = −a − condition. Thus, (almost) similarly to the XXX case, the following statements are equivalent: (i) P is a quasi-polynomial, (ii) all Q a,s -functions are polynomial, (iii) the roots {u 1 , . . . , u M } are admissible solutions of the BA equations, (iv) the Bethe vector is an eigenvector of the transfer matrix.
Let us briefly comment on the root of unity case q = e iπ/p with integer p ≥ 2. In this case, another exact string besides ±η/2 becomes possible, namely, a complete string of length p [19,20,21,22,23], due to the periodicity of sinh in the imaginary direction. Thus the construction of the function P would involve more q-deformed digamma functions located at the center of these new exact strings, and the quasi-polynomiality of P would be equivalent to the cancellation of multiple constants. It would a priori require more work to show that their cancellation are equivalent to the QQ-relations, as we should expect.

An example
We now present an explicit example of using the Q-system (3.8)-(3.9) to compute Bethe roots. The functions Q 0,0 (t) and Moreover, we define y n (t) by where we have inserted t L factors to ensure that both the numerator and denominator are polynomials in t. We then solve for the coefficients c k that make y n (t) vanish for all values of t. From experience, it is enough to consider n = 1, 2, . . . , M. As before, let us consider the case N ≡ L = 6 and M = 2, and we now further set η = log(2). The above code generates 15 solutions, in agreement with (3.7). We can solve for the corresponding Bethe roots similarly as before, for example

Open XXX Q-system
We turn now to the open XXX spin chain of length N, with Hamiltonian which is SU(2) invariant. After reviewing its BA solution, we propose a corresponding Q-system, and argue that all the Q's are polynomial if and only if the Bethe state is an eigenstate of the transfer matrix.

Review of the algebraic BA solution
The transfer matrix T(u) is given by [24] T(u) = tr 0 U 0 (u) , where M 0 (u) is the monodromy matrix in (2.3), and M 0 (u) is given by The R-matrix is again given by (2.2). Its boundary equivalent, the K-matrix, is the identity in the case considered here. By construction, the open-chain transfer matrix (4.2) has the commutativity property and it also has the crossing symmetry T(−u) = T(u) . We denote the matrix elements of U 0 (u) (4.2) as follows .

(4.6)
The reference state (2.5) is annihilated by C(u), and is an eigenstate of A(u) and D(u), with The Bethe states are defined by whereû j is omitted. Moreover, T (u) is given by the T Q-relation (4.10) and the coefficients F j of the "unwanted" terms are given by We again write f ± (u) = f (u ± i 2 ), as in the closed XXX case. Note that both Q and T are even functions of u Q(−u) = Q(u) , T (−u) = T (u) . Substituting u = u j in the T Q-relation (4.10), we see that the LHS vanishes, and we obtain (u j −u k +i)(u j +u k +i) = 0 , j = 1, . . . , M .
(4.13) If u j = ± i 2 , then these equations are equivalent to the BA equations (4.14) The BA equations have the reflection symmetry u j → −u j , while keeping the other u's (i.e. u k with k = j) unchanged. Hence, without loss of generality, we henceforth assume that ℜe(u j ) > 0, or ℜe(u j ) = 0 and ℑm(u j ) ≥ 0.
Roughly speaking, if {u 1 , . . . , u M } satisfy the BA equations (4.14), then all F j = 0; i.e., the "unwanted" terms in the off-shell relation (4.9) vanish, hence the Bethe state (4.8) is an eigenstate of the transfer matrix, with corresponding eigenvalue T (u). However, there are some important caveats. We argue in Appendix A that certain "exceptional" solutions of the BA equations (namely 0 and ± i 2 ) do not lead to eigenstates of the transfer matrix. Moreover, we make the standard assumption (supported by numerical evidence, see e.g. [23]) that the Bethe roots are pairwise distinct, i.e. u j = u k if j = k.
We therefore define an admissible solution {u 1 , . . . , u M } of the BA equations (4.14), such that all the u j 's are finite, not equal to ± i 2 or 0, and pairwise distinct (no two are equal), and each u j satisfies either ℜe(u j ) > 0 (4. 15) or ℜe(u j ) = 0 and ℑm(u j ) > 0 .

Q-system
We propose the following Q-system where the nontrivial Q-functions for given values of N and M are again defined on the Young tableau in Fig. 1, with the boundary conditions Q 2,s = 1 , Q 1,s≥M = 1, and with the initial condition In contrast to the Q-system for periodic XXX (2.13), there is an extra factor of u on the LHS of (4.18), and the Q's are even functions of u. The degree of the polynomial Q a,s (u) is doubled with respect to the periodic XXX case (namely, twice the number of boxes in the Young tableau to the right and top of the vertex (a, s)). 8 We claim that all the Q's are polynomial if and only if the Bethe state |u 1 . . . u M (4.8) is an eigenstate of the transfer matrix T(u) (4.2).
Before entering into the proof, let us quickly check that this Q-system indeed leads to the correct BA equations for {u 1 , . . . , u M }. We write the QQ-equations for (a, s) = (0, 0): 20) and for (a, s) = (1, 0): Since Q 2,0 = Q 2,1 = 1, the latter reduces to Performing the shifts u → u ± i 2 in (4.22) and evaluating at u = u j , we obtain since Q(u j ) = 0. Moreover, evaluating (4.20) at u = u j gives Substituting (4.23) into the above relation gives which coincides with the BA equations (4.13).

Q a,s in terms of Q and P
We now solve the Q-system (4.18) in terms of Q(u) and a function P (u), such that polynomiality of P (u) implies polynomiality of all the Q's. We define P (u) by 9 where Q 0,0 is given by (4.19). It follows from (4.20) and (4.22) that where we have used the following compact notation for discrete derivatives with certain 1/u factors . .
Similarly, we obtain (4.29) and in general Since both Q(u) and P (u) are even functions of u, it follows that DQ and DP are also even functions of u. Hence, if P (u) is a polynomial function of u, then the RHS of the second equation in (4.27) is divisible by u, thus Q 0,1 is polynomial; and, from (4.30), we similarly conclude that all the Q's are polynomial.
We observe, similarly to the closed-chain case, that the T Q-equation (4.10) together with the definition of P (4.26) imply (4.31) It follows that P is also a solution of the T Q-equation Hence, (4.26) can be regarded as the Wronskian relation obeyed by the two solutions Q and P of the T Q-equation (4.10).

Construction of P
We now construct the function P (u) for a set {u 1 , . . . , u M }, and argue that P (u) is polynomial if and only if {u 1 , . . . , u M } is an admissible solution of the BA equations.
The construction of the P -function for the open chain is similar to that for the closed chain, but with some significant differences. In the presence of one singular root i 2 and one zero root 0, the Q-function takes the form 10 (4.33) We define the function R(u) which is related to P (u) defined in (4.26) by Decomposing (4.34) in partial fractions, we obtain where π is a polynomial of order 2N − 4M + 1, q ± are polynomials of degree less than that ofQ, and a ± , b ± , c ± are constants. Note that a ± arise from the presence of the singular root, while b ± , c ± are due to the presence of the zero root. From the T Q-relation (4.10), we obtain (4.37) We now evaluate the LHS of (4.37) using (4.36), and consider the values of u where singularities could arise. The RHS of (4.37) has no singularities at the zeros ofQ (recall that T (u) is regular for values of u corresponding to admissible Bethe roots), hence for some polynomial q(u). From the residues of (4.37) at u = ∓ i 2 , we obtain . (4.39) Since Q(u) is an even function of u, we conclude -in significant contrast from the closedchain case -that a + = a − ≡ a. From the residues of (4.37) at u = ∓i, we can obtain expressions for c ± and b ± , and we find that which is consistent with the constraints coming from the residues of (4.37) at u = 0 (note that the presence of a zero root implies that T (u) has a double pole at u = 0).
We write the polynomial π in (4.36) as where ρ is a polynomial. Recalling the definition of the function p(u) (2.33), we see that It immediately follows from (4.35) that the P -function is given by which is a polynomial iff a = b = 0. That is, the P -function is polynomial iff there is no singular root and no zero root, in which case {u 1 , . . . , u M } is admissible. Moreover, Q 0,2 is polynomial iff a = b = 0.
The proof of the Q-system (4.18) is now complete, since we have argued as before that the following statements are equivalent: (i) P is a polynomial, (ii) all Q a,s -functions are polynomial, (iii) the roots {u 1 , . . . , u M } are admissible solutions of the BA equations, (iv) the Bethe vector is an eigenvector of the transfer matrix.

An example
We now present an explicit example of using the Q-system (4.18)- (4.19) to compute Bethe roots. The code is similar to the one in Sec. 2.2.3. The functions Q 0,0 (u) and Q 1,0 (u) are now given by We define y n (u) and solve for the coefficients c k exactly as in Sec. 2.2.3.
For the case N ≡ L = 6, M = 2, the above code generates 9 solutions, in agreement with (4.17). For example, for the first solution, the corresponding Bethe roots are given by 0.301932, 1.26627.

Open quantum-group-invariant XXZ Q-system
We now generalize the preceding results to the open quantum-group-invariant XXZ spin chain, whose Hamiltonian is given by [8] For simplicity, we restrict to generic values of q = e η .

Review of the algebraic BA solution
The transfer matrix is now given by [24] T(u) where the R-matrix is again given by (3.2), and the left and right K-matrices (solutions of boundary Yang-Baxter equations) are given by the diagonal matrices We define the elements of U 0 (u) (5.2) as follows .

(5.4)
The reference state (2.5) is annihilated by C(u), and is an eigenstate of A(u) and D(u) The Bethe states are again defined by The off-shell equation is where T (u) is given by the T Q-relation and F j is given by Substituting u = u j in the T Q-equation (5.8), we see that the LHS vanishes, and we obtain sinh(u j − u k + η) sinh(u j + u k + η) = 0 , j = 1, . . . , M .
If u j = ± η 2 , then these equations are equivalent to the BA equations The BA equations have the reflection symmetry u j → −u j (while keeping the other u's unchanged), as well as the periodicity u j → u j + iπ.

Q-system
We propose the following Q-system 11 where the nontrivial Q-functions for given values of N and M are again defined on the Young tableau in Fig. 1, with the boundary conditions Q 2,s = 1 , Q 1,s≥M = 1, and with the initial condition (5.16) In contrast to the periodic XXZ case (3.8), there is an extra factor (t 2 − t −2 ) ∝ sinh(2u) on the LHS of (5.15). This Q-system indeed leads to the BA equations (5.11), as can be seen by following the same logic (4.20) -(4.25) of the rational case. The degree of the polynomials (highest power of t ≡ e u ) is the same as in the open XXX case.

Q a,s in terms of Q and P
We now define P by where Q 0,0 is given by (5.16). Similarly to the rational case (4.30), we find that the Q-system (5.15) implies that where D is now defined by , . .

Construction of P
The construction of P parallels the rational case. In the presence of one singular root (u = η 2 , t = q 1 2 ) and one zero root (u = 0, t = 1; the case u = iπ 2 , t = −1 is similar), the Q-function becomes We now define R(t) as 22) which is related to P (5.17) by R = ( P Q ) ′ . We decompose R as follows where π is polynomial. From the T Q-relation (5.8), we obtain From the singularity structure of this equation and the fact that Q(t) and T (t) are invariant under t → t −1 , we again obtain r + = r + , r − = −r − , (5.25) where r(t) is a polynomial in t and t −1 , and The expression (5.23) for R can therefore be rewritten in the form where p q (t) is defined in (3.16), cf. (4.42), and ρ ′ = π. 12 Since R = ( P Q ) ′ , we conclude that P is given by which is a polynomial in t and t −1 iff a = b = 0. That is, as in the rational case, P is polynomial iff there is no singular root and no zero root, in which case {u 1 , . . . , u M } is admissible. Moreover, Q 0,2 is polynomial iff a = b = 0.

Conclusions
Our main results are Q-systems for the closed XXZ (3.8)- (3.9), open XXX (4.18)- (4.19) and open quantum-group-invariant XXZ (5.15)-(5.16) quantum spin chains. Polynomial solutions of these Q-systems can be found efficiently, which in turn lead directly to the admissible solutions of the corresponding BA equations.
Numerous applications of these results are possible. In conjunction with techniques from algebraic geometry, these Q-systems allow the exact computation of partition functions for trigonometric vertex models and for vertex models with boundaries [6].
We restricted here to open spin chains with SU(2) or U q (SU(2)) symmetry. It would be both interesting and useful to formulate Q-systems for open spin chains with other integrable boundary conditions, as well as for integrable models based on R-matrices for higher-rank algebras and/or higher-dimensional representations. eigenstate of the transfer matrix. Indeed, let us define a renormalized B-operator such that lim ǫ→0B ( i 2 + ǫ) is finite and non-singular. 13 Bethe states created with this renormalized operator satisfy an off-shell relation similar to (4.9), except with F j replaced bỹ We find that lim ǫ→0F1 exists and is nonzero for u 1 = ± i 2 + ǫ. Hence, the corresponding Bethe state is not an eigenstate of the transfer matrix.
The BA equations (4.14) do have solutions with a pair of singular Bethe roots, e.g. u 1 = i 2 and u 2 = ± i 2 , which must be discarded since |u 1 | and |u 2 | are not distinct.