Coordinate Bethe Ansatz for Spin s XXX Model

We compute the eigenfunctions and eigenvalues of the periodic integrable spin s XXX model using the coordinate Bethe ansatz. To do so, we compute explicitly the Hamiltonian of the model. These results generalize what has been obtained for spin 1/2 and spin 1 chains.


Introduction
The resolution of Heisenberg spin chain [1] was initiated in H. Bethe's seminal paper [2] where he used a method called now coordinate Bethe ansatz. Since this work, several new methods appeared: algebraic Bethe ansatz [3,4], functional Bethe ansatz (or separation of variables) [5] or analytical Bethe ansatz [6]. These more elaborated techniques allowed one to go further: new solvable models have been discovered and new results have been computed such as correlation functions. As a consequence, the coordinate Bethe ansatz was neglected. However, this method is the simplest one and gives a very efficient way to construct explicitly eigenfunctions, but it is believed that it works only for simple models. In this note, we show that actually it can be applied also to more complicated models as the spin s XXX model. This paper is organized as follows. In Section 2, we compute the Hamiltonian of the spin s chain we want to solve. To our knowledge, the explicit form of the entries of the Hamiltonian are written for the first time. We also compte the su(2) symmetry algebra and the pseudo-vacuum, a particular (reference) eigenstate. In Section 3, we present the coordinate Bethe ansatz and get the Bethe equations obtained previously by the algebraic or analytical Bethe ansatz. We conclude, in Section 4, on the advantages of this method and on open problems.
2 Integrable periodic spin s chain 2

.1 Hamiltonian of the spin s chain
The Hamiltonian of the periodic integrable spin s chain has been computed in [7] thanks to a fusion procedure. This Hamiltonian has been expressed as a polynomial of the invariant of su(2) (see for example also [8] for a review). For our purpose, we need to give an explicit expression of the Hamiltonian entries. Namely, the Hamiltonian is the following matrix acting on (C 2s+1 ) ⊗L H = L j=1 h j,j+1 (2.1) with the periodic condition L + 1 = 1 and the subscript (j, j + 1) stands for the two spaces where the (2s + 1) 2 × (2s + 1) 2 -matrix h acts non-trivially. We choose to enumerate the basis of C 2s+1 as follows: |s , |s − 1 , . . . , | − s , where |m ≡ |s, m denotes 1 the spin s state with s z -component equals to m. The non-vanishing entries of the matrix h may be parameterized by three integer parameters

su(2) symmetry
At each site, we have a spin s representation, and the expression of the su(2) generators in this representation reads: They obey We will note s α j , α = z, ± and j = 1, . . . , L, the generators acting on site j. Let us stress that these local operators do not commute with the Hamiltonian H given in (2.1). However, there is a global su(2) symmetry. The generators of this su(2) symmetry take the form They obey the su(2) commutation relations [S z , S ± ] = ±S ± , [S + , S − ] = 2S z . Remark that the Casimir operator C 2 = (S z ) 2 + 1 2 (S + S − + S − S + ), although central, is not proportional to the identity, since we are considering the tensor product of L spin s representations, a reducible representation.
It is a simple calculation to show that [s α j + s α j+1 , h j,j+1 ] = 0, α = z, ±. It amounts to check the following recursion relations on the coefficients β n m 1 ,m 2 : Hence, S α commutes with H. Due to this su(2) symmetry, the wave functions can be characterized by their energy, their spin and their S z component. In other words, one can diagonalize the Hamiltonian in a sector where the operators S z has a fixed value S z = Ls − m. This is done in the next section.

Pseudo-vacuum and pseudo-excitations
We wish to present the construction of the Hamiltonian eigenfunction in the framework of coordinate Bethe ansatz for spin s. The spin s = 1 2 case is the Heisenberg chain, solved in [2]. It gave the name to the method. For the case s = 1, the method has been generalized in [9].
The first step of the coordinate Bethe ansatz [2] consists in finding a particular eigenvector, called the pseudo-vacuum, for the Hamiltonian. It is usually chosen as the vector with the highest spin. In the present case, it is the unique vector in the sector S z = Ls: |∅ = |s ⊗ |s · · · ⊗ |s .
Using the explicit forms for the β's, we get h 12 |s ⊗ |s = 0. Thus, |∅ is a H-eigenvector with vanishing eigenvalue.
The second step consists in adding pseudo-excitations. These pseudo-excitations are not physical excitations (hence the name pseudo-excitation). They are obtained by acting with a creation operator e − j , conjugated to s − j , on the pseudo-vacuum |∅ . Let us remark that this operator in this finite representation does not satisfy (e − ) 2 = 0 but rather (e − ) 2s+1 = 0. This explains the supplementary difficulties to deal with s > 1 2 . Indeed, in the case s = 1 2 , no more than one pseudo-excitation can be at the same site: we have strict exclusion. In the general case of spin s, we have a weaker exclusion. More precisely, we can have up to 2s pseudo-excitations at the same site. This behavior appears already for s = 1.

Coordinate Bethe ansatz for general spin s
We define a state in the sector S z = Ls − m, where e − is conjugated to s − : This choice for e − is for later convenience. The set of such non-vanishing vectors (i.e. x j+2s > x j for 1 ≤ j ≤ L − 2s) provides a basis for this sector. As already noticed, contrarily to the usual case (s = 1 2 ), several (up to 2s) pseudo-excitations at the same site are allowed, that is to say, some x j 's can be equal. The restriction that no more than 2s particles are on the same site is implemented by the fact that (e − ) 2s = 0. Using the explicit form of e − , we can rewrite the excited states as follows: where m j is the number of times x j appears and is the binomial coefficient. Let us remark that, if m j > 2s the vector |s − m j has no meaning but the normalization α m j vanishes. Any eigenvector in the sector S z = Ls − m is a linear combination of the vectors (3.1). Then, let us introduce the vector where a(x 1 , x 2 , . . . , x m ) are complex-valued functions to be determined. As in the case of s = 1 2 , we assume a plane wave decomposition for these functions (Bethe ansatz) where S m is the permutation group of m elements and A P (k) are functions on the symmetric group algebra depending on some parameters k which will be specified below 2 . Using the fact that the states |x 1 , x 2 , . . . , x m form a basis, we can project the eigenvalue equation on these different basis vectors to determine the A P (k) parameters.
Since H is a sum of operators acting on two neighbouring sites only, one has to single out the cases where the x's obey the following constraints: • all the x j 's are far away one from each other and are not on the boundary sites 1 and L (this case will be called generic), • x j = x j+1 = · · · = x j+m 1 and x j + 1 = x j+m 1 +1 = · · · = x j+m 1 +1+m 2 for some positive integers m 1 and m 2 , As the eigenvalue problem is a linear problem, it is enough to treat the cases where at most one of the particular cases appears: more complicated cases just appear as superposition of these 'simple' cases.
Projection on |x 1 , x 2 , . . . , x m with x j + 1 < x j+1 , ∀ j, x 1 > 1 and x m < L. As usual, we start by projecting (3.2) on a generic vector |x 1 , x 2 , . . . , x m . This leads to which must be true for any choice of generic x's. Therefore, we get for the energy (using the explicit forms of the β's given in Section 2) After the change of variable the energy becomes This form for the energy is the one obtained by algebraic Bethe ansatz [10].
Projection on |x 1 , x 2 , . . . , x m with x j + 1 = x j+1 for some j. Let us consider now the projection of (3.2) when two pseudo-excitations are nearest neighbours. Using the form of the energy previously found, we get P ∈Sm This equation is trivially satisfied for s > 1 2 since using explicit values we find β 0 For the case s = 1 2 , we find a constraint between A P and A P T j (where T j is the transposition of j and j + 1). Explicitly, it is given by (3.5) with s = 1 2 . Projection on |x 1 , x 2 , . . . , x m with x j = x j+1 for some j. For s > 1 2 , we must also consider the case when several particles are on the same site. Defining S j , the shift operator adding 1 to the j th variable, we get the following relation, when two particles are on the same Using the plane waves decomposition, we get the following constraint where T j is the transposition of j and j + 1, and we have introduced the scattering matrix As in the case s = 1 2 , relation (3.5) allows us to express all the A P 's in terms of only one, for instance A Id (where Id is the identity of S m ). More precisely, one expresses P ∈ S m as a product of T i , and then uses (3.5) recursively to express A P in terms of A Id . At this point, one must take into account that the expression of P in terms of T i is not unique, because of the relations and Therefore, for the construction to be consistent, the function σ has to satisfy the relations By direct computation, we can show that (3.6) indeed satisfies these relations. We can solve the recursive defining relations for A P and we find, with a particular choice of normalisation for A Id , the following explicit form, for any P ∈ S m This scattering matrix becomes after the change of variables u = λ+is λ−is and v = µ+is Let us remark that after this change of variables the scattering matrix σ(λ, µ) does not depend on the value of spin and is similar to the one obtained for s = 1 2 .
where P m 1 ,m 2 (y; z) = m 2 n=1 α n α m 2 −n α m 1 β n 0,m 2 z m 2 −n+1 · · · z m 2 and we have used the notation (2.3) and y = (y 1 , y 2 , . . . , y m 1 ) with Relation (3.7) is implied by (3.4). The sketch of the proof goes as follows. First, one can check directly that 3 P 0,2 (S j , S j+1 ) = 0, ∀ j, since P 0,2 (S j , S j+1 ) corresponds to relation (3.4). The same is true for P 2,0 (S −1 j+1 , S −1 j ) (after multiplication by S −1 j+1 S −1 j ). Next, we rewrite (3.7) as P m 1 ,m 2 (y; z) + P m 2 ,m 1 (z; y) = 0, (3.8) where we have defined These variables are such that if P 0,2 (z j , z j+1 ) = 0 ∀ j, then we have also P 0,2 (z j , z j+1 ) = 0. Hence, a property valid for P m 1 ,m 2 (y; z) will be also valid for P m 2 ,m 1 (z; y). We first focus on If we suppose that we have variables z j such that P 0,2 (z j , z j+1 ) = 0, ∀ j, then from expression (2.2), and after some calculation, one can show that Thus, the polynomial P 0,m (z) becomes a linear function of the z j 's. Looking at the coefficient of z j and at the constant term, one checks that they identically vanish, so that P 0,m (z) = 0. Looking at the general polynomial P m 1 ,m 2 (y, z) and using the relation Thus, to prove relation (3.8), it is enough to show that R m 1 ,m 2 (y) + R m 2 ,m 1 (z) = 0. (3.9) Using the expression of M 1 and M 2 , see (2.3), it is easy to see that Two cases have to be distinguished: m 12 = 0 or m 12 > 0. In the first case, relation (3.9) is trivially satisfied. In the second case, equation (3.9) can be rewritten as which is obeyed if To prove this last relation, we use recursively (3.4) to show Taking k = i + j − 1, ℓ = 2s and using m 1 + m 2 = 2s + m 12 gives the result. Hence, relation (3.8) is satisfied if the variables are such that P 0,2 (z j , z j+1 ) = P 0,2 (y j , y j+1 ) = 0, ∀ j. This ends the proof.
This step concludes the bulk part of the problem, the other possible equations being fulfilled by linearity. It remains to take into account the periodic boundary condition. It is done through the following projection.
Projection on |1, x 2 , . . . , x m . As usual, this leads to a constraint on the parameters k j . It is not surprising since these parameters can be interpreted as momenta: we are quantizing them since we are on a line with (periodic) boundary conditions. Namely, this leads to P ∈Sm A P exp(i(k P 2 x 2 + · · · + k P m x m )) − exp(i(k P 1 x 2 + k P 2 x 3 + · · · + k P (m−1) x m + k P m L)) = 0. Now, we first perform the change of variable in the summation P → P T 1 · · · T m−1 in the second term of the previous relation. Then, using recursively relation (3.5) and projecting on independent exponential functions, we get the quantization of the momenta via the so-called Bethe equations Since these equations express the periodicity of the chain, they are equivalent to the ones obtained through projection on |x 1 , . . . , x m−1 , L (as it can be checked explicitly). Thus, we do not have any new independent equations through projections, and the eigenvalue problem has been solved (up to the resolution of the Bethe equations). Note that using the change of variables (3.3) and the expression (3.6) for the scattering matrix, equations (3.10) can be rewritten as One recognizes the usual Bethe equations of the spin s chain [10,11].

Action of su(2) generators
Since the su(2) generators commute with the Hamiltonian, from any eigenfunction Ψ m , one can construct (possibly) new eigenfunctions by application of S α , α = z, ± on Ψ m . As already mentioned, it is a straightforward calculation to check that Moreover, it is part of the ansatz to suppose that the eigenvector Ψ m is a highest weight vector of the su(2) symmetry algebra, Let us stress that for Ψ m to be an eigenvector, one has to assume that the rapidities λ j have to obey the Bethe equations. In the same way, Ψ m is a highest weight vector only when the Bethe equations are fulfilled. In the context of coordinate Bethe ansatz, there exists no general proof (for generic spin s) of it (at least to our knowledge). Note however that for spin 1 2 , the proof was given in [12]. Nevertheless, one can check the highest weight property on different cases, and we illustrate it below by the calculation of S + Ψ 1 , S + Ψ 2 and S + Ψ 3 . We also show on the last example where the proof used by Gaudin does not work anymore for spin s > 1 2 . The Ψ m vectors should be also related to the ones obtained through algebraic Bethe ansatz (ABA). Such a correspondence, for the case of spin 1 2 , has been done in [13] using an iteration trick based on the comultiplication [14,15]. Let us note that in [13] they used the relation (T 12 ) 2 = 0 which is not true anymore for s > 1 2 . Their proof must be generalized to apply in our case. Let us also notice the other method using the Drinfel'd twist [16]. Moreover, since it is known that the ABA construction leads to su(2) highest weight vectors, and assuming the same property for the coordinate Bethe approach, it is clear that the two methods should lead to the same vectors, up to a normalisation.
For instance, considering Ψ 1 , its ABA "counterpart" takes the form where T (j) (λ) is the representation of the monodromy matrix at site j: and λ 1 is the Bethe parameter. This leads to that has to be compared with Using the change of variable (3.3), it is clear that, apart from a normalisation factor, the two vectors are equal. Calculation of S + Ψ 1 and S + Ψ 2 . A direct calculation leads to which is identically zero using the Bethe equation y L = 1. Hence, Ψ 1 is indeed a highest weight vector for the su(2) symmetry.
In the same way, one can compute Using the relation and the normalisation A Id (k 1 , k 2 ) = 1, one gets where y j = e ik j , j = 1, 2. Now, from the Bethe equations y L 1 = σ(y 1 , y 2 ) and y L 2 = σ(y 2 , y 1 ), one simplifies it as Finally, the form of the scattering matrix σ ensures that the quantity within brackets {· · · } vanishes. Calculation of S + Ψ 3 . Performing the same kind of calculation on Ψ 3 , we get where 4 y j = e ik P j , j = 1, 2, 3 and σ jℓ = σ(y j , y ℓ ), 1 ≤ j = ℓ ≤ 3.
After use of the Bethe equation, y L 3 = σ 23 σ 13 , it can be recasted as Using the sum on P to relabel the variables y j , one can rewrite this equality as × (2s − 2) 1 + σ 12 + σ 23 + σ 12 σ 13 + σ 23 σ 13 + σ 12 σ 13 σ 23 (1 + σ 12 )(2s − 1) + 2s The term inside the square bracket [· · · ] in factor of (y 1 y 2 ) x 1 y x 2 3 on the one hand, and in factor of y x 1 1 (y 2 y 3 ) x 2 on the other hand, identically vanishes. It is in fact the same identity as the one used to show that S + Ψ 2 = 0. It is also the identity used by Gaudin [12] to prove, for spin 1 2 , that Ψ m , ∀ m, is a highest weight vector.
When the spin is higher than 1 2 , it remains the term in factor of |x, x , which is a state that does not exist when s = 1 2 . The square bracket in front of |x, x also identically vanishes, another identity due to the form of the scattering matrix σ, and we get S + Ψ 3 = 0.
When s = 1 this new identity is sufficient (together with the one used for S + Ψ 2 ) to prove that Ψ m , ∀ m, is a highest weight vector. However, for s > 1, to prove that S + Ψ 4 = 0, one needs to consider the state |x, x, x , that we will lead to another identity of the scattering matrix, and so on: for spin s, one needs 2s identities to prove that Ψ m , ∀ m, is a highest weight vector. Hence the difficulty to get a generic proof of it.

Conclusions
In previous studies, the eigenfunctions of the spin s chain studied in this paper were known thanks to the algebraic Bethe ansatz. This later construction allows one to compute the correlation functions [17,18,19]. Prior to that computation, the coordinate Bethe ansatz allowed Gaudin [20] to show, for spin 1 2 chains, orthogonality relations for the Bethe eigenfunctions, and prove a closure property for these functions. The explicit form of the eigenfunctions computed in this note is a first step toward a generalisation to spin s chains. The same method can also be applied to spin chains associated to higher rank algebras, for which less is known.
In the same way, the spin 1 XXX chain with open (diagonal) boundaries has been studied in [21]: there is no doubt that their results can be generalized to spin s, using the present approach. The advantage of this method lies in the fact that we do not need to solve the reflection equation before computing the spectrum. We may start with general boundary conditions and find the ones for which the method is still consistent. In this way, the boundaries which keep the model solvable are classified.
We also believe that the method presented here can be applied to solve the XXZ model with higher spin in the case of periodic boundary conditions. These cases have been treated through algebraic Bethe ansatz, see e.g. [22,23]. More interestingly, general XXZ models with open boundary conditions can also be treated in this way, see e.g. [24] where a first account has been given.
To conclude, we hope that this paper convinced the reader that the coordinate Bethe ansatz is a very powerful method and can be applied to solve a rather large class of integrable models.