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Solutions of the Gaudin Equation and Gaudin Algebras

Three well-known solutions of the Gaudin equation are obtained under a set of standard assumptions. By relaxing one of these assumptions we introduce a class of mutually commuting Hamiltonians based on a different solution of the Gaudin equation. Application of the algebraic Bethe ansatz technique to diagonalize these Hamiltonians reveals a new infinite dimensional complex Lie algebra.


Introduction
The Bardeen-Cooper-Schrieffer (BCS) pairing model Hamiltonian was diagonalized by Richardson in 1963 in an algebraic way [1]. The step operators introduced in Richardson's solution belong to an infinite dimensional complex Lie algebra which is usually referred to as the rational Gaudin algebra. It is one of the three algebras which emerged from Gaudin's work during the 1970's [2] who introduced a new class of integrable models starting with the following operators: where w α ij are complex numbers to be determined and t α i are the generators of N , mutually commuting SU (2) algebras. The latter obey the standard SU (2) commutation relations: where t ± i ≡ t 1 i ± it 2 i . Gaudin showed that the operators in (1) mutually commute if and only if the coefficients satisfy the equations: w α ij w γ jk + w β ji w γ ik − w α ik w β jk = 0 (4) for all distinct triples (i, j, k) and for all permutations of the upper indices (0, 1, 2). There are three solutions to equation (4) under the following assumptions: i) The coefficients w α ij are antisymmetric under the exchange of the indices i and j w α ij + w α ji = 0, ii) each coefficient w α ij can be expressed as a function of the difference between two real parameters u i and u j , and iii) each h i commutes with the operator T = j t 0 j so that the z component of the total SU (2) is conserved. Gaudin's solutions are given as follows: which are commonly referred to as rational, trigonometric and hyperbolic solutions, respectively. Here, p is a real parameter 1 . The operator which is obtained by substituting the rational solution into (1) is called rational Gaudin magnet Hamiltonian: Similarly, substituting the trigonometric and hyperbolic solutions into (1) we obtain which are called trigonometric and hyperbolic Gaudin magnet Hamiltonians, respectively. Application of the algebraic Bethe ansatz method to the Gaudin magnets yields the rational, trigonometric and hyperbolic Gaudin algebras [3,4]. These are infinite dimensional complex Lie algebras which are related to particular solutions of the classical Yang-Baxter equation [5]. As a result, each algebra admits a one-parameter family of mutually commuting Hamiltonians H(λ). Here λ is a complex parameter which is usually referred to as the spectral parameter. These operators may be identified as the integrals of motion of a quantum system as well as the traces of transfer matrices of a 1 These solutions are not completely unrelated. Gaudin equation (4) is satisfied for all complex values of p. But only for the real and pure imaginary values of p Hamiltonians in (1) are Hermitian. On the other hand, substituting ip in place of p, one can convert trigonometric and hyperbolic solutions into each other. For this reason, we restrict ourselves with the real values of p. Also note that in the limit p → 0 trigonometric and hyperbolic solutions go to the rational solution.
vertex model. Main objective in both cases is to diagonalize them simultaneously and this is achieved by either functional or algebraic Bethe ansatz techniques [6]. Richardson-Gaudin methods have recently found many applications in quantum many-body physics [7].

Further Solutions of the Gaudin Equation
In this paper we investigate a different solution of the Gaudin equation (4). Under the three assumptions listed in the introduction, all possible solutions are enumerated above. To find new solutions we need to relax one or more of these assumptions. We keep the constraints (ii) and (iii) but generalize the constraint (i) to where q is a real parameter 2 . This solution is given by The operators which are obtained by substituting this new solution in (1) will be denoted by h Since (13) is a solution of Gaudin equation (4), these operators mutually commute: In the limit q → 0, the new solution given by (13) approaches Gaudin's rational solution (6). As a result, The solution presented here is a different solution of Gaudin equation and the Hamiltonians given by (14) for q = 0 cannot be obtained from the rational, trigonometric or hyperbolic Gaudin magnet Hamiltonians. To see that, let us assume for a moment that there exists an operator S such that where h ( * ) i represents any one of the Hamiltonians given in equations (9)- (11). Suppose we take the sum of both sides of (17): Antisymmetry of (5) implies for rational, trigonometric and hyperbolic magnet Hamiltonians. Hence the sum on the left hand side is equal to zero. On the other hand, the sum on the right hand side of (18) is not equal to zero. Instead the new condition given in (12) implies i.e. an operator S satisfying (17) does not exist for q = 0.

Gaudin Algebras
If one applies the algebraic Bethe ansatz technique to diagonalize the Gaudin magnet Hamiltonians which correspond to the solutions (6), (7), or (8), one uses the step operators belonging to the rational, trigonometric or hyperbolic Gaudin algebras, respectively. Below we will show that if the algebraic Bethe ansatz method is employed to diagonalize the Hamiltonian of (14), then a different algebra emerges in a completely analogous way. Before introducing this algebra, however, we would like to outline the basic features of rational Gaudin algebra and its relation to the rational Gaudin magnet Hamiltonian, as an illustration of the technique. Trigonometric and hyperbolic Gaudin algebras and their relation to the trigonometric and hyperbolic Gaudin magnet Hamiltonians follow very similar lines. Rational Gaudin algebra is generated by three families of operators J + (λ), J − (λ) and J 0 (λ) parameterized by a complex number λ. Commutators are given by Commutators of J + (λ), J − (λ) and J 0 (λ) at the same value of the complex parameter are given by taking the limit µ → λ. From these commutation relations it is easy to show that form a one-parameter family of mutually commuting operators: Starting from a lowest weight vector and using J + (λ) as step operators, one can diagonalize these operators simultaneously. Lowest weight vector |0 > by definition satisfies for every λ. Here W (λ) is a complex-valued function. The state |0 > itself is an eigenvector of H(λ): where the prime denotes derivative with respect to λ. In addition is an eigenvector of H(λ) if the quantities ξ 1 , ξ 2 , . . . , ξ n satisfy the set of Bethe ansatz equations (For an analytic solution of these equations see e.g. References [8] and [9]). The corresponding eigenvalues will be There exists a realization of the rational Gaudin algebra in terms of the SU(2) generators of (2), given by Here u 1 , u 2 , . . . , u N are arbitrary real numbers which are all different from each other and N is a nonnegative integer 3 . Assuming that the eigenvalues of the Casimir operator of the j th SU (2) are s j (s j + 1), the corresponding W (λ) is given by In this realization H(λ) is given by We see that H(λ) has simple poles on the real axis. Residues of H(λ) at the points λ = u i are proportional to the rational Gaudin magnet Hamiltonians given in (9): Equation (23) where E (r) and h (r) Here |0 > is the lowest weight state and ξ α are the solutions of the Bethe ansatz equation (28). We next search for an operator H (q) (λ), the residue of which gives us the operators h (q) i (cf. Equation (33)). We wish to write such an operator using an algebra similar to that in (21). Below we show that such an algebra exists and its generators J 0 q (λ), J + q (λ) and J − q (λ) satisfy the commutation relations given as [J 0 q (λ), J 0 q (µ)] = [J ± q (λ), J ± q (µ)] = 0. We observe that, in the limit q → 0, commutators in (39) approach the commutators of Gaudin algebra given by (21). It should nevertheless be emphasized that the algebra of (39) is not a q-deformed algebra, but an ordinary Lie algebra. We can show that form a one-parameter commutative family: We can simultaneously diagonalize them starting from a lowest weight vector satisfying where W q (λ) is a complex valued function. It can be shown that |0 > is an eigenvector of H (q) (λ) with the eigenvalue In general, one can write a Bethe ansatz form for the eigenvectors of H (q) (λ) as |ξ 1 , ξ 2 , . . . , ξ n >≡ J + q (ξ 1 )J + q (ξ 2 ) . . . J + q (ξ n )|0 > .
For this to be an eigenvector of H (q) (λ), the complex numbers ξ 1 , ξ 2 , . . . , ξ n must satisfy the following system of equations: In this case, eigenvalue associated with the state |ξ 1 , ξ 2 , . . . , ξ n > is given by This algebra admits two realizations in terms of the SU (2) generators of (2). These are given by and In the limit q → 0, the first realization goes to the realization of the rational Gaudin algebra given by (30) whereas the second realization vanishes. In the first realization, Equation (47), the operator H (q) (λ) becomes We see that H (q) (λ) has simple poles on real axis at the points λ = u i . It is easy to show that −1/2 times the residue of H (q) (λ) at the point λ = u i is Since − → t i · − → t i commutes with every h As a result, the operators h (q) i are automatically diagonalized together with H (q) (λ). As in the case of the rational Gaudin algebra, one can read off the eigenvalues of h (q) i from the eigenvalues of H (q) (λ): where and h (q) Details of the derivation of these eigenvalues are given in the Appendix.

Linear r-Matrix Structure
The rational, trigonometric and hyperbolic Gaudin algebras have a linear r-matrix structure. This means that one can write down the commutators of these algebras in the following matrix form: where I is the 2 × 2 identity matrix, r(λ − µ) is the r-matrix described below and L(λ) is given by J 0 (λ), J + (λ) and J − (λ) are generators of the rational, trigonometric or hyperbolic Gaudin algebras. In (56), the term [L(λ) ⊗ I, I ⊗ L(µ)] is equal to a 4×4 matrix whose elements are various commutators between the generators J 0 (λ), J + (λ) and J − (λ). On the other hand r(λ−µ) is a M at 2 (C)⊗M at 2 (C) matrix valued function of λ−µ which carries the information about the structure constants of the algebra. For instance, for the rational Gaudin algebra, the r-matrix is given by where P is the permutation matrix on C 2 ⊗ C 2 , which is equal to It is customary to introduce the following three mappings where a, b are 2 × 2 matrices and I is the 2 × 2 identity matrix. Then r ij are defined as follows: Since r-matrix carries information about the structure constants of the algebra, Jacobi identity leads to the equality [r 13 , r 23 ] + [r 12 , r 13 ] + [r 12 , r 23 ] = 0 (64) which is the well known classical Yang-Baxter equation (for solutions of this equation see [10], for a review see [11], [12]). Equation (64) guarantees the mutual commutativity of the operators H(λ) which can now be written as Trigonometric and hyperbolic Gaudin algebras also have the r-matrix structure described above and their r-matrices are also solutions of the classical Yang-Baxter equation.
In order to see if one can repeat the same procedure for the algebra we introduced in (39), we study a matrix of the form We then substitute it into the equation assuming the existence of such an r-matrix. One can compute the left-hand side of this equation and then set all the components equal to zero to find the components r ij of this r-matrix. For instance, the (21) and (13) elements are equal to respectively. Setting these elements equal to zero requires setting the coefficients of all the algebra elements equal to zero. The coefficient of J − q (λ) in the first equation and the coefficient of J + q (µ) in the second equation gives respectively. But these equations are incompatible unless q = 0. Repeating the same calculations for the other components, one can conclude that there is no r-matrix which satisfies (67). Instead of the L q (λ) matrix in (66), one may try to find a more general form of L q (λ) matrix for which an r-matrix can be found to satisfy (67). But we were unable to find such an L q (λ) matrix 4 . On the other hand we were able to show that one can write the commutators of the algebra introduced in (39) in the following form: Here L q (λ) is given by (66) and the r q -matrix is given by P is the permutation matrix introduced in (59). As in the case of the r-matrices of rational, trigonometric and hyperbolic Gaudin algebras, the r q -matrix carries information about the structure constants of the algebra 4 For instance, for a matrix of the form However, one can similarly show that there is no solution to (67) for this Lq(λ) either. of (39). r ij q matrices, defined by similar maps as in Equations (60)-(63), satisfy the following equation: We see that the r-matrix structure of the algebra introduced in (39) is not the same as the r-matrix structures of the rational, trigonometric and hyperbolic Gaudin algebras. Nevertheless, it shares with the other three algebras, the crucial property of admitting a one-parameter family of mutually commuting operators H (q) (λ). Similar to (65), one can write this operator as the trace of the square of L q (λ): Note that in the limit q → 0, the algebra of (39) approaches the rational Gaudin algebra. In this limit both r q and r −q go to the r-matrix of the rational Gaudin algebra given in (58) and from (71) we recover the classical Yang-Baxter equation (64).

Conclusion
In this article, by relaxing one of the conditions imposed by Gaudin, we presented a different solution to the Gaudin equation and wrote down the corresponding set of mutually commuting Hamiltonians. We also identified the related infinite dimensional Lie algebra. This algebra allows a one parameter family of mutually commuting Hamiltonians H (q) (λ) parameterized by a complex spectral parameter. We diagonalized these Hamiltonians using the algebraic Bethe ansatz method and constructed the eigenvectors and eigenvalues. The procedure is parallel to the other known Gaudin algebras. A different generalization of the Gaudin algebras was recently introduced in Ref. [13] that is based on the rational, trigonometric, and hyperbolic solutions of the Gaudin equations. Our generalization is based on a different solution of the Gaudin equation obtained under different assumptions. funds granted by the Wisconsin Alumni Research Foundation. We also acknowledge support through NSF-TUBITAK Joint Research Project TBAG-U/84(103T113).