Algebraic Bethe Ansatz

We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinorvector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RTT algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices. 1 e-mail: karakhan@yerphi.am 2 e-mail:Roland.Kirschner@itp.uni-leipzig.de


Introduction
The Inverse Scattering Method for quantum integrable systems [1,2,3,4] applies basically to symmetries of all Lie algebras types. The case of the A series has been studied in much detail, because of its applications to well known models and because of its simplicity compared to other cases. In last years the Yangian algebras of the B, C, D types and the corresponding Algebraic Bethe Ansatz (ABA) attracted increasing interest, e.g. [5,6,7,8,9,10,11].
Yangians of the orthogonal and symplectic types allow for the linear evalutation only by additional constraints on the universal enveloping of the corresponding Lie algebra. The spinor representations, where the Yangian generators are constructed from the underlying algebra C (see eq.(2.7) below), obey these constraints, i.e. the Yang Baxter RLL relation involving the fundamental R matrix with orthogonal or symplectic symmetry and the spinor L operators holds [12].
Reshetikhin [13] proposed to approach the ABA in the orthogonal case by replacing in the auxiliary space the fundamental (vector) representation by the spinor representation and to apply the fusion procedure. The ABA for the so(3) case has been shown to be related to the well known sℓ(2) ABA. Our investigation is oriented along this concept.
We consider the Yang-Baxter RLL relation in the tensor product of the fundamental and two copies of the spinor representations. It is used as the defining relation for the involved R operator intertwining the spinor representations. The spinorial R operator has been obtained in the orthogonal case in [16] and considered in [12,6]. The uniform formulation of the orthogonal and symplectic cases and of the superalgebra case have been given in [17]. The result of the spinorial R operator has been given in the expansion in invariants on the tensor product of two copies of the algebra C built as contraction of anti-symmetrized (orthogonal case) or symmetrized (symplectic case) products of its generators.
In this paper we propose an alternative approach. We use instead the invariant built from the product of one generator out of each of the spinor algebras and its powers. We obtain the spinorial R operator in terms of the Euler Beta function. We consider low rank orthogonal cases. Here the spinor representations are finite dimensional. This results in characteristic polynomials obeyed by the invariant from which the explicit form of projection operators can be read off. The general result of the spinorial R operator is reduced to a finite sum, which is derived from the spectral decomposition of the invariant.
We observe coincidences of the spinorial and known fundamental (vector) R matrices. The structure of the spinorial R in even dimensions (so(2m), D series) appears much simpler than the odd dimensional case (so(2m + 1), B series).
In the even-dimensional cases the separation of the tensor product of spinor representation spaces, where R acts, into chiral parts results in the corresponding separation of R. Both parts of R obey the Yang-Baxter relations separately. The explicit matrix forms of R have a lot of zero entries. This also appears in the corresponding monodromy matrices.
The observed coincidences of the spinorial R matrices imply coincidences of the corresponding RT T (spinorial Yangian) algebras and fundamental RT T (ordinary Yangian) algebras and as a consequence relations of the corresponding ABA.
The paper is organized as follows: In sect. 2 we recall the Yang-Baxter relations with orthogonal and symplectic symmetry, in particular the ones involving the fundamental and the spinor representations. In sect. 3 we describe our approach to the spinor-spinor R-matrix and compare with the Shankar-Witten approach. In sect. 4 we consider the relation between spinorial and vector monodromy matrices using the fusion procedure. In sect. 5 we analyze the spinorial R matrices for so(4) and so(6) and the generated RT T algebras. The spinorial R matrices for so(3) and so(5) are analyzed in sect. 6.

The spinorial R-matrix
The Yang-Baxter relation for the fundamental R matrix reads R looks similar for the cases of orthogonal and symplectic symmetry [14], [15], 3) The choices ǫ = +1 and ǫ = −1 correspond to the so(n) and sp(2m) cases respectively. The index range is a i , b i = 1, ..., n or 1, ..., 2m. ε ab denotes the metric tensor which is symmetric in the orthogonal and anti-symmetric in the symplectic case. The Yang-Baxter RLL relation with the above R is fulfilled by the linear form of the L-operator in the spinor representation case, where the matrix elements G a b are built as from the underlying algebra C generated by c a obeying the commutation relations The linear ansatz for L-operator (2.5) implies the so(n) or sp(2m) Lie algebra relations, 8) and the symmetry condition, as well as the additional constraint which specifies the Yangian linear evaluation. The Yang-Baxter relation is formulated in the tensor product of three representation spaces. We are concerned with the fundamental (vector) space V (2m-or 2m + 1-dimensional) and the spinor space S (of dimension 2 m in the orthogonal and infinite dimensional, oscillator Fock space, in the symplectic case).
In order to specify the representation involved we shall show all indices explicitly, a, b for V and α, β for S. Then (2.4) appears as Along with this Yang-Baxter relation we consider Here R(u) stands for the spinorial R-operator. We shall use also the modifications of the above RLL relations usually referred to as the check form. The R matrix is multiplied by the permutation operator interchanging the auxiliary space factors,Ř 12 = P 12 R 12 . For example, the check form corresponding to (2.12) isŘ In [17], following the approach of [16], the spinorial R-operator was obtained in the formŘ (2.14) Here c 1 symbolizes the anti-symmetrization in the orthogonal and the symmetrization in the symplectic case. The coefficients are derived from an iterative relation as A 0 (u), where A 0 and A 1 are arbitrary functions of the spectral parameter. Both the even (sum restricted to even k) and the odd parts obey (2.13). Note that in the orthogonal case c a , the generators of C, are fermionic and can be expressed in terms of the Dirac gamma matrices, c a = 1 √ 2 γ a .

The alternative approach toŘ
The spinorial R operator can be regarded as an element of C 1 ⊗ C 2 (2.7) or as an operator acting in the product of two copies of the spinor space, S 1 ⊗ S 2 . c a 1 and c a 2 (2.7) are the corresponding basic elements.
The equation (2.13) takes the form We consider it as the defining relation for the wanted spinorial R matrix and expand in powers of v. The terms proportional to v 2 are canceled, linear terms lead to the symmetry conditionŘ where Canceling the common factor d 2 4 − z 2 in both sides one obtainš from which one deduces the wanted spinorial R operatoř . (3.7)

Comparison with the Shankar-Witten form
Comparing to the previous treatments [16,6,17] we have a simpler line of arguments and a compact general form of the spinorial R operator. The direct comparison of the two expressions for spinorial R-operator seems difficult due to the complicated connection between the invariants Again the difference between orthogonal and symplectic cases consists in the presence of the sign factor ǫ. In the orthogonal case so(d) (ǫ = +1) the series in I k terminates at k = d + 1, while for sp(d) (ǫ = −1) the series does not terminate. For a proof of the recurrence relation (3.8) we refer to [17], where the generating function method is used for the calculation of (anti-) symmetrized products of c a . It follows the same line as for eq. (5.6) of that paper:

Symplectic cases of low rank
It is instructive to specify the general solution for spinorial R-matrix for symplectic algebras of low rank. For sp(2) one can realize the algebra C in terms of a pair of operators of multiplication and differentiation as Thus we haveŘ sp(2) .
The sp(4) case corresponds to two such pairs.

The orthogonal case
In the orthogonal case, due to the Clifford relation for c a (Dirac gamma matrices), the general expression for the spinor-spinor R-matrix in form of the Euler Beta-function can be transformed to a polynomial in z as well as to a finite expansion in the invariants In this case it is convenient to modify slightly the definition of the invariant z: Besides of the absence of the imaginary unit in this definition, we suppose here that the Dirac gamma matrices γ a 1 and γ b 2 related to different spaces commute, in contrast to the previous convention where we have supposed their anticommutation in order to have the unified description with the symplectic case.
Having modified the definition of z we go through the steps of the above derivation in order to check that the the result is not changed.
Consider the RLL-relatioň We multiply by γ a ± = γ a 1 ± γ 2 2 from the left and by γ b ± or γ b ∓ from the right and use Note that here an additional minus sign appears due the change in the commutativity convention. We obtaiň from which we deduce the general solution for the so(n) case:

Spinor and vector monodromy matrices
We show how the monodromy matrices with spinor auxiliary space (S 0 ) and vector auxiliary space (V 0 ) are related by fusion. The gamma matrices are known to intertwine the vector and the product of spinor representations. Formally, the following analysis can be extended to the symplectic case. However, that case is connected with the problem of defining infinite sums over the spinor indices. We restrict ourselves to the orthogonal case, which will be considered in the remaining part of the paper. The general vector-vector monodromy matrix is defined by the fundamental R matrix (2.2): We are interested in the diagonalization of the trace of this matrix, or t(u) = tr 0 T(u), because it is the generating function of the integrals of motion of a periodic quantum spin chain.
Using the L-operator (2.5) one can construct another monodromy matrix acting in the tensor product of the same quantum space V 1 ⊗ . . . ⊗ V N and the spinor auxiliary space S 0 . We consider also the related monodromy matrix Because of the inversion relation, it can be expressed by the inverse of the previous monodromy matrix It is not hard to check that the fusion of two conjugated L-operators (2.5) gives the vector R-matrix (2.2). Indeed, We have used that in the orthogonal case (2.5) takes the form Consider the product We pick the last L factor in T and the first in T and use (4.5) to continue the calculation Proceeding with the next factors in T and T we obtain finally In this way we obtain the fusion relation between spinorial and vector monodromy matrices. The fusion of two spinorial monodromy matrices by trace over the auxiliary spinor space S 0 results in the vector monodromy matrix with V 0 as auxiliary space.
In general by this relation the spinorial RT T algebra, the generators of which are contained in the matrix T , is mapped to the ordinary RT T algebra, the generators of which are matrix elements of T. Moreover, it provides a way to solve the spectral problem for the trace of the ordinary monodromy matrix by solving the spectral problem for a trace involving the spinorial monodromy matrix. The entries of the inverse-transposeT ij (u) = T −1 N +1−j,N +1−i (u) to the monodromy matrix T defined over the fundamental (vector) auxiliary space used in [10] is given by the quantum minors divided by the quantum determinant. In contrast, due to the inversion relation for L (4.4) the inverse of spinor monodromy matrix T (u) is given by the same matrix with the shifted spectral parameter.

Even-dimensional orthogonal algebras
In sect. 3 we have obtained the spinorialŘ-matrix in form of the Euler Beta-function for an arbitrary orthogonal algebra. Now we shall consider low rank examples corresponding to the D series. The universal expression (3.12) results in explicit forms using the corresponding characteristic polynomial in the invariant z and the resulting spectral decomposition. Details about the invariant z and relations following from its characteristic polynomial are considered in the Appendix.

The so(4) case
In this case we have the characteristic polynomial In other words, in the case d = 4 any function of z is represented by a polynomial of fourth degree. The roots of W 4 are the eigenvalues of z and we are lead to the spectral decompositionŘ Here the sum goes over the roots of W 4 : z k = 0, ±1, ±2 and P k are projection operators on the corresponding eigenspace, so(4) (u|z). It separates into two parts given by even and odd functions of z, respectively. Both parts satisfy the RRR Yang-Baxter relation separately.
z is acting on S 1 ⊗ S 2 and the permutation operator is given by Consider now both parts of the spinorial R matrix in more detail. The simpler parť is given by an odd function of z. The corresponding R-matrix without check, R (1) = P 12Ř (1) , is even in z, and is trivial, i.e. diagonal and independent of the spectral parameter, with the following non-vanishing entries: The RLL Yang-Baxter relation (4) reduces here to the identity We have also the Yang-Baxter RRR relation in the trivial form The part of the spinorial spinorial R matrix even in ž corresponds to the R-matrix without check also even in z, The latter is represented by the matrix with the following non-vanishing entries: It is important to notice that the two different partsŘ (1) so(4) (u) andŘ (2) so(4) (u), which are odd and even functions of z correspondingly, are distinguished by chirality. Indeed these solutions are proportional to the chiral projectors Π + and Π − , respectively, where One deduces from (3.8) the useful formulae This chiral property ensures the consistency of these solutions: both of them intertwine a pair of L-operators i.e. obey the RLL-relation linear in R, but also satisfy the trilinear RRR-relation not only separately, but also in arbitrary combination, due to their orthogonality. The chiral projectors Π ± separate the 16-dimensional representation space S 1 ⊗ S 2 into two eight-dimensional chiral subspaces. In particular, the subspace corresponding to Π + is spanned by eight eigenvectors of z corresponding to the eigenvalues ±1, while the six eigenvectors, corresponding to the zero eigenvalue of z as well as vectors corresponding to the eigenvalues ±2 span the other chiral subspace. In terms of the projectors of the eigenspaces this reads as Note that due to the definite chirality of R so(4) (u) and R (2) so(4) (u) the expressions like so(4) (v)γ b 2 does not vanish and at 2v + 1 = u results in the fusion to the L matrix (2.5),

ABA for the so(4) case
The observed simple structure of the spinorial R so(4) matrix is helpful for the diagonalization of the trace of the spinorial monodromy matrix or in components The explicit form of the matrix R Its diagonal 4 × 4 matrix components are diagonal , and have to be regarded as the elements of the Cartan subalgebra. The off-diagonal elements have to be regarded as lowering generators correspondingly. Consequently, the monodromy matrix (5.7) defined as an ordered matrix product of the factors (5.2) preserves the block form, with Cartan elements T α α (u) and rising C 1 (u) = T 4 1 (u), C 2 (u) = T 3 2 (u) and lowering B 2 (u) = T 2 3 (u), B 1 (u) = T 1 4 (u) generators. We have obtained a representation of the spinorial RT T algebra and its decomposition into a representation of two subalgebras of the sℓ(2) type Yangian. This becomes more evident by a similarity transformation with the 4 × 4 matrix V V = e 11 + e 24 + e 33 + e 42 , V −1 = V. (5.10) One calculates easily and with the 8 × 8 block-matrices Disregarding in these blocks the lines and columns with zero entries, we recognize in each of them the fundamental R matrix of sℓ (2). Consider now the general RTT relation substituting for T instead of the particular monodromy matrix (5.7) a generic algebra valued 4 × 4 matrix. Whereas in the sℓ(4) case the corresponding relation results in the first step in 256 non-trivial relations, here the relation with the so(4) spinorial R matrix (5.6) leads to 64 non-trivial relations from the matrix elements with the indices belonging to sets (1,4) and (2,3), to further relations of the form 0 = 0 and , most interesting, to such with non-trivial r.h.s. and zero l.h.s or vice versa. For example, for (α 1 , α 2 ) = (1, 1), (γ 1 , γ 2 ) = (1, 2) gives Combining all these relations one deduces, that the general so(4) spinor monodromy matrix has the form (5.9), i.e.
Thus also in the case of arbitrary representations in the quantum space the so(4) monodromy matrix generates an algebra equivalent to two independent sℓ(2) Yangian algebras and the spectral problem for its trace leads to the ordinary sℓ(2) ABA.
In particular, this applies to the spinor-vector monodromy matrix (4.3). Recall, that it produces the so(4) vector-vector monodromy matrix by fusion (4.8), We have to substitute β = 1 for so (4).
Taking into account that T andT = T −1 have the form (5.9), one can calculate explicitly the matrix elements of the vector monodromy in terms of products of the matrix elements of the spinor mondromy. In particular for the trace we obtain where we have abbreviated T α β (u + 1 2 ) = L 1 (u + 1 2 ) . . . L N (u + 1 2 ) α β andT = T −1 (u + 3 2 ). The vector-vector so(4) monodromy matrix is given by sums of products of the spinorvector transfer matrix elements which obey sℓ(2) type Yangian relations.
At the end of this subsection we consider the RT T algebra generated by R (1) so(4) (u). Due to its chirality and the diagonal character it leads to a trivial solution for T (u) of the RTT-relation. For instance, (α 1 , α 2 ) = (1, 1), (γ 1 , γ 2 ) = (1, 2) and (γ 1 , γ 2 ) = (2, 1) by (5.5) leads to . The remaining RTT-relations imply that the most general monodromy matrix T (u) intertwined by R (1) so(4) (u) is given by the diagonal matrix: which has vanishing rising and lowering generators. We summarize the above results: Proposition 1. The spinorial R matrix with so(4) symmetry R (2) (acting in the chiral subspace of the Π − projection) generates the spinorial RT T algebra decomposing into two subalgebras of the sℓ(2) Yangian type. The spinorial RT T algebra generated by R (1) (acting in the chiral subspace of the Π + projection) is a trivial commuting algebra. In this way, the ordinary sℓ 2 ABA allows to construct solutions of the ABA of for the spinorial Yangian of so(4) type.

ABA for the so(6) case
In this case we again deal with two spinor-spinor R-matrices (5.20) and (5.21) acting in the two chiral subspaces.
We summarize the results on the so(6) case: The spinorial R matrices with so(6) symmetry of both chiral sectors can be separated into blocks of the fundamental sℓ(4) R matrix form. In each case the resulting RT T algebras are equivalent to two copies of the Yangian algebra of sℓ(4) type.
The spectral problem of the trace of the so(6) monodromy matrices can be treated on the basis of the known nested ABA for the sℓ(4) Yangian.

The so(8) case
The characteristic polynomial in this case is given by (5.29) and the universal expression (3.12) for the spinorial R matrix is reduced tǒ It decomposes into two parts of opposite chirality, both obeying the Yang-Baxter relations. We observe that one of the two chiral parts is simpler: R so (8) is linear in the spectral parameter and contains two invariant tensors, while R (2) so (8) is quadratic in u and contains three tensors.

Odd-dimensional orthogonal algebras
In the odd-dimensional cases the characteristic polynomial reduces to W d = k (z − z k ) (8.9), where the product runs over m + 1 of the 2m + 1 eigenvalues of z. This leads to the decomposition of the Euler Beta-function in (3.12): where the summation goes over the m + 1 roots z k = (−1) k 2k+1 2 of the characteristic polynomial W d .

The so(3) case
The charactristic polynomial is We have the decompositioň . Unity and the permutation are given by Up to a redefinition of the spectral parameter the spinorial so(3) R matrix coincides with the fundametal R matrix with sℓ(2) symmetry. Let us compare this expression also with the fundamental R matrix (2.2) at ǫ = −1 and n = 2 corresponding to the case sp(2). This 4 × 4 matrix has the following non-vanishing entries: , and can be rewritten as R(u) = 2(u + 1)( u 2 I 12 + P 12 ), i.e. it coincides with the so(3) spinor-spinor R-matrix after rescaling 2u → u 2 . The spinorial RT T relation for the so(3) monodromy matrix (4.3) coincides with the one for the vector sℓ(2) mondromy matrix. Indeed, denoting one obtains that the commutation relations between A, B, C and D are the same as in sℓ (2) case, because the spinor-spinor R-matrix (6.3) intertwining them, coincide up to u → 2u. The coincidence of the spinorial R matrix of so(3) symmetry with the well known Yang formula for sℓ(2) was a central point in the classical paper by Reshetikhin [13]. The fusion relation (4.8) results in expressions of the 9 elements of the vector monodromy matrix T in terms of the 4 spinorial RT T generators in T (u) for all representations admitting this relation. The study of [10] resulted in relations among the matrix elements of T, in particular the three elements in the upper triagle are expressed in terms of one of them. This confirms that (4.8) establishes the equivalence of the spinorial so(3) type RT T algebra and the ordinary Yangian of sℓ(2) type.

The so(5) case
The characteristic polynomial is The spinorial R matrix acting in S 1 ⊗ S 2 can be written aš The permutation is given by with the eigenspace projectors , Their properties P 2 where the elements of the matrix K = 4P 5 2 can be written as In this matrix after rescaling 2u → u one can recognize the sp(4) vector-vector R-matrix (2.2) with the parameter β = 4 2 − (−1) = 3. We summarize the results on so(5): Proposition 3. The spinorial R matrix with so(5) symmetry coincides with the ordinary fundamental R matrix with sp(4) symmetry. The diagonalization of the traces of the corresponding spinorial so(5) monodromy and of the ordinary sp(4) monodromy is done by the same nested ABA relations. Moreover, the fusion relation (4.8) allows to treat the diagonalization of the trace of the ordinary so(5) monodromy matrix on the basis of the sp(4) nested ABA.
The nested ABA for sp (4) has been formulated in [9].

Conclusions
We have presented a new approach to the spinor R matrix of orthogonal or symplectic symmetry. Comparing with the conventional approach, the derivation is simpler and the result has the compact form of the Euler Beta function of the invariant z.
In the orthogonal case, relying on the characteristic polynomial of z, we obtain explicit expressions of the spinorial R matrices of low rank cases.
By the fusion argument we relate the spinor and vector monodromy matrices. Studying the low rank examples, we observe coincidences of spinor R matrices with some fundamental R matrices. This implies relations for the monodromy matrices, the corresponding RTT algebras and of the Algebraic Bethe Ansatz for the spectral problem of traces of spinor and vector monodromy matrices.
For the so(2m) cases (D series) the spinor R matrices have a particular simple structure. We have the decomposition into two chiral parts, where both obey the Yang-Baxter relations. Moreover, the R matrix of each chiral part is sparse with many zeros and transforms to two independent blocks.
Coincidence relations between spinor and fundamental R matrices are to be expected also at higher ranks. The simplicity of the so(2m) case compared to so(2m+1) will persist.
Like any invariant operator acting on Σ the operator P 12 for so(2m) has the spectral expansion over the projection operators on the eigenspaces of z P 12 = m k=−m (−1) k(k−1)/2 P k . (8.5) Note that P 0 and P 1 always contribute with plus sign. This formula is closely related to the well known formula In order to emphasize the pairwise appearance of the projection operators in the evendimensional case we rewrite the expression (8.5) for the permutation as P 12 = m+1 k=0 (−1) k (P 2k + P 2k+1 ). (8.8)

Odd-dimensional space
If the dimension of the space is odd d = 2m+1, the dimension of S 1 ⊗S 2 and the number of z eigenvectors remains the same 2 2m , but the operator z contains an additional term which shifts its eigenvalues by one half and breaks the symmetry between positive and negative eigenvalues. With the shifted eigenvalues the relation W d = 0 (8.4) holds. Further, ε a 1 ,...,a 2m+1 causes relations between even and odd invariants, I k (z) = const I 2m+1−k (z). As a consequence we obtain the reduction of W d to The product runs over only m + 1 of the 2m + 1 roots of z. Consider, how the eigenspaces of z change when increasing the dimension from d = 2m to d = 2m + 1. The adjacent z multiplets for d = 2m symmetric under the permutation P 12 (e.g. P 0 and P 1 ) are unified into a multiplet for d = 2m + 1 with the eigenvalue given by arithmetical mean (P 1 2 ) by addition of 1 2 or − 1 2 . Similarly, the adjacent antisymmetric multiplets (say P −2 and P −1 ) are unified (into P − 3 2 ). Then the dimension formula (8.7) holds due to the Pascal triangle recurrence relation and accordingly the characteristic polynomial is given by W d (8.9). The dimensions of the subspaces corresponding to the roots of W d are given by 2m + 1 2k , and their sum is the dimension of Σ, m k=0 2m + 1 2k = 2 2m .