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Models solvable by Bethe Ansatz 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Models solvable by Bethe Ansatz 1

Petr P. KULISH*

St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia

*Corresponding Author:
Petr P. KULISH
St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27
191023 St. Petersburg, Russia
E-mail: [email protected]

Received date: December 20, 2007; Revised date: March 10, 2008

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

Diagonalization of integrable spin chain Hamiltonians by the quantum inverse scattering method gives rise to the connection with representation theory of different (quantum) algebras. Extending the Schur-Weyl duality between sl2 and the symmetric group SN from the case of the isotropic spin 1/2 chain (XXX-model) to a general spin chains related to the Temperley-Lieb algebra TLN(q) one finds a new quantum algebra Uq(n) with the representation ring equivalent to the sl2 one.

Introduction

The development of the quantum inverse scattering method (QISM) [1, 2, 3, 4] as an approach to construction and solution of quantum integrable systems has lead to the foundations of the theory of quantum groups [5,6, 7, 8, 9].

The theory of representations of quantum groups is naturally connected to the spectral theory of the integrals of motion of quantum systems. In particular, this connection appeared in the combinatorial approach to the question of completeness of the eigenvectors of the XXX Heisenberg spin chain [10] with the Hamiltonian

image (1.1)

where image are the Pauli matrices.

Three algebras are connected to this system: the Lie algebra sl2 of rotations, the group algebra imageof the symmetric group SN and the infinite dimensional algebra image the Yangian [7], with the corresponding R-matrix image whereimage is the 4 × 4 permutation matrix flipping the two factors of image

The Yangian is the dynamical symmetry algebra which contains all the dynamical observables of the system. It is important to note that the algebras image and image are related by the Schur- Weyl duality in the representation space image This follows from the fact that image andimage are each other’s centralizers in this representation space. As a consequence, since the Hamiltonian commutes with the global generators of image : image is an element of image This can also be seen from the expression of HXXX in terms of the permutation operators, which are the generators of the symmetric group SN

image

An analogous situation arises in the anisotropic XXZ chain

image(1.2)

which commutes [11] with the global generators of the quantum algebra Uq(sl(2)) [5]. Here the role of the second algebra is played by the Temperley-Lieb algebra TLN(q), whose generators

image in the spaceimage coincide with the constant R-matriximage

image (1.3)

As in the case of the XXX spin chain, the Hamiltonian (1.2) can be expressed in terms of the generator (1.3) of the algebra image

image

The dynamical symmetry algebra of the XXZ chain is the quantum affine algebra image

The eigenvectors for both models can be constructed by the coordinate Bethe Ansatz (see [24]) or by an algebraic Bethe Ansatz [1,2,3]. The latter one follows from the main relation of the QISM for the auxiliary L-operator (see Sec. 2)

image (1.4)

where the indices a and j refer to the corresponding auxiliary and quantum spaces image

The Temperley-Lieb algebra is a quotient of the Hecke algebra (see section 3) and allows for an R-matrix representation in the space image for any n = 2, 3, . . .. There is corresponding spectral parameter depending R-matrix obtained by the Yang - Baxterization process. Consequently, it is possible to construct an integrable spin chain [12]. The open spin chain Hamiltonian is the sum of the image generators image

image(1.5)

where Xj act nontrivially on image and as the identity matrix on the other factors of H. The aim of this work is to describe the quantum algebra image which is the symmetry algebra of such spin system and to show that the structures (categories) of finite dimensional representations of these algebras image and image coincide. In this case image and image are each other’s centralizers in the space image We consider the general case when the complex parameterimage is not a root of unity.

Let us note that the relation between TLN(q) and integrable spin chains was actively used in many works and monographs (see for example [13,14,15,16,17,18,19] and the references within). However, the authors used particular realizations of the generators Xj , related to some Lie algebras (or quantum algerbas). Characteristic property of the latter ones was the existence of one-dimensional representation in the decomposition of the tensor product of two fundamental representations imageThen Xj was proportional to the rank one projector on this subspace, and the symmetry algerba was identified with the choosen algebra. We point out that the symmetry algerba Uq(n) is bigger and its Clebsch - Gordan decomposition of image has only two summands similar to the sl(2) caseimage

The dual Hopf algebra image was introduced as the quantum group of nondegenerate bilinear form in [20,21]. The categories of co-modules of image and their generalisations were studied in [22,23] where it was shown that the categories of co-modules of imageare equivalent to the category of co-modules of the quantum group SLq(2).

Bethe Ansatze

Using the L-operator (1.4) a new set of variables (operators in the space image depending on the parameter ¸) is introduced by an ordered product of image as 2 × 2 matrices on the auxiliary space image according to the QISM [1]-[4]

image(2.1)

The entries of the monodromy matrix T(¸) are new variables. The commutation relations of the new operators (A(¸), . . . ,D(¸)) can be obtained from the local relaltion for the L-operator at one site:

image (2.2)

where R-matrix is image and it acts on the tensor product of two auxiliary spaces image while the index j refers to the space of spin quantum statesimage The relation for T(λ) is of the same form [1]-[4]

image

where image

HXXX:

image

where image Multiplying the RTT-relation byimage and taking the trace over two auxiliary spaces one gets commutativity property of transfer matrix t(λ):

image

The operator B(μ) is a creation operator of the eigenvectors we are looking for. These operators act on a vacuum state (a highest weight vector) Ω. The Hamiltonian is extracted from the transfer matrix t(λ) which is a generating function of mutually commuting integrals of motion. The vector Ω is the tensor product of states corresponding to spin up at each site of the chain:

image

Using the explicit form of the L-operator and the definition of the monodromy matrix T(λ) it is easy to get the relations

image

where image It follows also from quadratic relation ofimage that

image

and a similar relation for D(λ) and the product of B(μj ). Sum of these relations acting on the vacuum ­ gives the eigenvector of the transfer matrix t(λ)

image

under the condition that the parameters μk satisfy the Bethe equations (k = 1, 2, . . . ,M)

image

The eigenvalue is

image

This construction of the eigenvectors of quantum integrable models was coined as algebraic Bethe Ansatz (ABA) [1]-[3].

Originally these eigenvectors of the XXX spin chain were found by H. Bethe at 1931 as a linear combination of one magnon eigenstates using the local operators image

image

It is easy to see that image is an eigenvector of HXXX with the eigenvalueimage However, the condition of periodicity i.e. the requirement that image is also an eigenvector of the shift operator: image results in the quantization of z

image

These yields N − 1 states, m = 1, 2, . . . ,N − 1 (because the state with z = 1 belongs to the vacuum multiplet: image Multimagnon states are given by a Bethe sum or (coordinate Bethe Ansatz)

image

where the coefficients (amplitudes) A(P) are defined by the elements P of permutation group SM, quasimomenta zPj and the two magnon S-matrix [24]

image

Constructed eigenstates image are highest weight vectors for the global symmetry algebra sl2 with generators

image

The proof is purely algebraic and it follows from the RTT-relation and the asymptotic of the monodromy matrix [10]

image

Hecke and Temperley-Lieb Algebras

Both algebras image andimageare quotients of the group algebra of the braid group BN generated by(N − 1) generatorsimage their inversesimage and subject to the relations (see [25])

image(3.1)

The Hecke algebra image is obtained by adding to these relations the following characteristic equations obeyed by generators

image(3.2)

It is known that image is isomorphic to the group algebraimage Consequently, irreducible representations of the Hecke algebra, as that of SN, are parametrized by Young diagrams. By virtue of (3.2) we can write ˇR using the idempotents image

image(3.3)

Substituting the expression (3.3) for ˇR in terms of X, into the braid group relations (3.1) one gets relations forimage

image(3.4)

Requiring that each side of (3.4) is zero we obtain the quotient algebra of the Hecke algebra, the Temperley-Lieb algebra image It is defined by the generatorsimageand the relations image

image(3.5)

The dimension of the Hecke algebra is N!, the same as the dimension of the symmetric group SN, the dimension of imageis equal to the Catalan number imageIn connection with integrable spin systems we will be interested in representations of image on the tensor product space image.One representation is defined by an invertibleimage matriximage which can also be seen as an n2 dimensional vectorimage [17]. We use the notationimageand view this matrix also as an n2 dimensional vector image The generators Xj can be expressed as

image(3.6)

where we explicitely write the indices corresponding to the factors in the tensor product space H. It is easy to see, that the second relation (3.5) is automatically satisfied and the first one determines the parameter image

image (3.7)

An obvious invariance of the braid group relations (the Yang - Baxter equation) (3.1) in this representation with respect to the transformation of the R-matrix

image.

results in the following transformation of the matrix imageIf one uses an R-matrix depending on a spectral parameter (Yang - Baxterization of image

image(3.8)

where imagethen relation (3.5) can be written as

image

In terms of constant R-matrices (generators of image this relation has the formimage

image(3.10)

Replacing in (3.9) the expression imageor in (3.5) substituting X =image yields the vanishing of the q-antisymmetriser

image(3.11)

Thus the irreducible representations of image are parametrized by Young diagrams containing only two rows with N boxes.

The constructed representation (3.3), (3.6) is reducible. The decomposition of this representation into the irreducible ones will be discussed in the next section.

Quantum Algebra image

According to the R-matrix approach to the theory of quantum groups [26], the R-matrix defines relations between the generators of the quantum algebra imageand its dual Hopf algebra, the quantum group A(R). In this paper the emphasis will be on the quantum algebra image and its finite dimensional representations Vk, k = 0, 1, 2, . . . . The generators of image can be identified with the L-operator (L-matrix) entries and their exchange relations (commutation relations) follow from the analogue of the Yang-Baxter relations (2.2) (withouht spectral parameter)

image(4.1)

where the indices a1 and a2 refer to the representation spaces Va1 and Va2 , respectively, and index q refers to the algebra image. Hence the equation (4.1) is given in End image

In general the L-operator is defined through the universal R-matrix, where a finite dimensional representation is applied to one of the factors of the universal R-matrix

image(4.2)

image(4.3)

image

Furthermore, the universal R-matrix satisfies Drinfeld’s axioms of the quasi-triangular Hopf algebras [7,25]. In particular,

image(4.4)

Thus, choosing the appropriate representation space as the first space, one obtains the co-product of the generators of image from the following matrix equation

image(4.5)

The case when imageis of particular interest and it will be presented below in detail. To this end the generators of image are denoted byimageand the L-matrix is given by

image

Multiplying two L-matrices with entries in the corresponding factors image we obtain

image (4.7)

or explicitly for the generators

image

etc. The central element in image is obtained from the defining relation (4.1)

image(4.11)

image

However this central element is group-like: imageIt is proportional to the identity in the tensor product of representations. The analogue of the imageCasimir operator can be obtained according to [26] using imageas image

In the case when Va, Vq are the three dimensional space imageand the b matrix is taken from the references [15,16]

image(4.12)

c2 is written as

image

Parameters p and q are related: imageFor the explicit L-operator and its 3 × 3 blocks we get c2 = qIq where Iq is the identity operator on imageThe form of the generators imagewhich corresponds to the choice of the b matrix (4.12), follows from the expression for the ˇR and L-matrices imagewhere P is the permutation matrix.

For example, we have

image(4.13)

If we choose b as in equation (4.12) the R-matrix commutes with imagewhere h = diag(1, 0,−1). As a highest spin vector (pseudovacuum of the corresponding integrable spin chain [3]) we choose image[12, 16]. If we act on the tensor product of these vectors imagewith the coproduct of the lowering operators imagewe obtain new vectors. By looking at the explicit forms of the operators imagein the space imagewe can convince ourselves that the vectorsimage are linearly independent. Together with the vectors image and

image

they span an 8 dimensional subspace. The vector imageis a linear combination of the vectors imageThe vector imagespans a one dimensional invariant subspace. Thus we have the following decomposition

image

This decomposition can also be obtained using the projectorsimage expressed in terms of b matrix (vector) (4.12). Due to the commutativity of the R-matrix imagewith the co-product (4.5), (4.7) the corresponding subspaces imageare invariant. Similarly, using imageone can get the decomposition of

image(4.15)

This type of decomposition is valid for any n.

The result of applying the co-product Δ, given by (4.5), on the generators of the quantum algebra image several times can also be presented in the matrix form

image(4.16)

where imageetc. In general case of the tensor representation of TLN(q), with the spaceimageat each site, the generators of the algebraimagecommute with the generators (4.16) of the global (diagonal) action of the quantum algebra Uq(n) in the space imageThis follows from the relation

image

and the possibility due to the co-associativity of the coproduct to write the product of of Laqj as

image

Hence,

image

Thus, the algebras Uq(n) and TLN(q) are each other’s centralizers in the space H. The tensor representation of TLN(q) in H decomposes into irreducible factors whose multiplicities are given by the dimensions of the irreducible representations of the algebra Uq(n), corresponding to the same Young diagrams

image(4.17)

In this decomposition the index k parametrizes the Young diagrams with two rows and N boxes and multiplicities are given by the dimensions of the corresponding irreducible representations

image(4.18)

of the algebras Uq(n) and TLN(q), respectively. As for the finite dimensional irreducible representations of the Lie algebra sl2, V0(n) = C is the one-dimensional (scalar) representation and the fundamental representation of the algebra Uq(n) is n dimensional, image. The dimensions of other representations follow from the trivial multiplicities of the factors in the decomposition of the tensor product of the Vk(n) and the fundamental representation V1(n) into two irreducible factors, as for the image,

image(4.19)

Thus, for the dimensions pk(n) = dim Vk(n) the following recurrence relation is valid

image(4.20)

with the initial conditions p−1(n) = 0, p0(n) = 1, whose solutions are Chebyshev polynomials of the second kind

image(4.21)

The multiplicity image or the dimensions of the subspacesimagein (4.17) is the number of paths that go from the top of the Bratteli diagram to the Young diagram corresponding to the representation image Ifimageis the partition of N, image then image

image (4.22)

The subspaces invariant under the diagonal action of the quantum algebra Uq(n) on the space H, can be obtained using the projectors (idempotents), which can be expressed in terms of the elements of the Temperley-Lieb algebra TLN(q). Using the R-matrix depending on a spectral parameter, the projector P(+) N on the symmetric subspace can be written in the following way [4,27]

image(4.23)

Similar construction can be done with the underlying Lie algebra sl(3). Then the corresponding q-antisymmetrizer P(4) − which defines a quotient of the Hecke algebra is [4,27]

image(4.24)

This form follows from the intertwiner of four monodromy matricesimage

image(4.25)

where imageMultiplying by appropriate product of the permutation operators Pk k+1 one can get the expression in terms of the baxterized Hecke generators

image

The q-antisymmetrizer (4.24) is obtained by fixing shifts of the spectral parameters image [4,27].

Theorem 4.1. Consider the quotient of the Hecke algebra imageby the ideal I generated by the q-antisymmetrizers P(4) − ,

image

The tensor product representation of image(q) in the spaceimage3.with the qantisymmetrizersimage of rank 1 define the quantum algebraimageas the centralizer algebra of image

Let us mention that although the spectrum of the spin chains related to the general Temperley- Lieb R-matrix was found by the fusion procedure and a functional Bethe Ansatz [12], it would be nice to get the corresponding eigenvectors. Also the subject of reconstructing algebras from their representation ring structure is actively discussed in the literature (see e.g [28]).

Acknowledgement

It is a pleasure to thank the organizers for having arranged this nice Baltic-Nordic Workshop. The useful discussions with P. Etingof, A. Mudrov and A. Stolin are highly appreciated. This research was partially supported by RFBR grant 06-01-00451.

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