Journal of Generalized Lie Theory and Applications Vol. 2 2008), No. 3, 90 200 Models solvable by Bethe Ansatz Petr P. KULISH St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 9023 St. Petersburg, Russia E-mail: kulish@pdmi.ras.ru 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 sl 2 and the symmetric group S N from the case of the isotropic spin /2 chain XXX-model) to a general spin chains related to the Temperley-Lieb algebra T L N q) one finds a new quantum algebra U q n) with the representation ring equivalent to the sl 2 one. 2000 MSC: 7B37, 6D90 Introduction The development of the quantum inverse scattering method QISM) [, 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 [0] with the Hamiltonian H XXX = N n= σ x n σ x n+ + σ y nσ y n+ + σz nσ z n+).) where σ α n α = x, y, z) are the Pauli matrices. Three algebras are connected to this system: the Lie algebra sl 2 of rotations, the group algebra C[S N ] of the symmetric group S N and the infinite dimensional algebra Ysl 2 ) the Yangian [7], with the corresponding R-matrix Rλ) = λi + ηp, where P is the 4 4 permutation matrix flipping the two factors of C 2 C 2. The Yangian is the dynamical symmetry algebra which contains all the dynamical observables of the system. It is important to note that the algebras sl 2 and C[S N ] are related by the Schur- Weyl duality in the representation space H = N C 2. This follows from the fact that sl 2 and C[S N ] are each other s centralizers in this representation space. As a consequence, since the Hamiltonian commutes with the global generators of sl 2 : S α = /2 N n= σα n, α = x, y, z, it is an element of C[S N ]. This can also be seen from the expression of H XXX in terms of the permutation operators, which are the generators of the symmetric group S N σnσ α n+ α = 2P nn+ I nn+ α Presented at the 3 rd Baltic-Nordic Workshop Algebra, Geometry, and Mathematical Physics, Göteborg, Sweden, October 3, 2007.
Models solvable by Bethe Ansatz 9 An analogous situation arises in the anisotropic XXZ chain H XXZ = N n= σ x n σn+ x + σnσ y y n+ + σz nσn+ z ) q q ) + σ z σ z 2 N).2) which commutes [] with the global generators of the quantum algebra U q sl2)) [5]. Here the role of the second algebra is played by the Temperley-Lieb algebra T L N q), whose generators Ř n := Řn,n+q) in the space H = N C 2 coincide with the constant R-matrix ωq) = q /q) Ř XXZ q) = q 0 0 0 0 ωq) 0 0 0 0 0 0 0 q.3) As in the case of the XXX spin chain, the Hamiltonian.2) can be expressed in terms of the generator.3) of the algebra T L N q) = q + q )/2) σnσ x n+ x + σnσ y y n+ + σz nσn+ z + ωq) ) σ z 2 n σn+ z ) ωq) = Ř XXZ q) + q I nn+ 2 The dynamical symmetry algebra of the XXZ chain is the quantum affine algebra U q ŝl 2). The eigenvectors for both models can be constructed by the coordinate Bethe Ansatz see [24]) or by an algebraic Bethe Ansatz [, 2, 3]. The latter one follows from the main relation of the QISM for the auxiliary L-operator see Sec. 2) L aj λ) = λi + η σa α σj α.4) 2 α where the indices a and j refer to the corresponding auxiliary and quantum spaces C 2 a, C 2 j. The Temperley-Lieb algebra is a quotient of the Hecke algebra see section 3) and allows for an R-matrix representation in the space H = N C n 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 [2]. The open spin chain Hamiltonian is the sum of the T L N q) generators X j = Ř qi H T L = N j= X j.5) where X j act nontrivially on C n j Cn j+ and as the identity matrix on the other factors of H. The aim of this work is to describe the quantum algebra U q n) which is the symmetry algebra of such spin system and to show that the structures categories) of finite dimensional representations of these algebras U q n) and sl 2 coincide. In this case U q n) and T L N q) are each other s centralizers in the space H = N C n. We consider the general case when the complex parameter q C is not a root of unity. Let us note that the relation between T L N q) and integrable spin chains was actively used in many works and monographs see for example [3, 4, 5, 6, 7, 8, 9] and the references within). However, the authors used particular realizations of the generators X j, 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
92 Petr P. Kulish fundamental representations V j V j+. Then X j 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 U q n) is bigger and its Clebsch - Gordan decomposition of V j V j+ has only two summands similar to the sl2) case C 2 C 2 = C 3 C. The dual Hopf algebra U q n) was introduced as the quantum group of nondegenerate bilinear form in [20, 2]. The categories of co-modules of U q n) and their generalisations were studied in [22, 23] where it was shown that the categories of co-modules of U q n) are equivalent to the category of co-modules of the quantum group SL q 2). 2 Bethe Ansatze Using the L-operator.4) a new set of variables operators in the space H depending on the parameter λ) is introduced by an ordered product of L aj λ) as 2 2 matrices on the auxiliary space C 2 a according to the QISM []-[4] T λ) := L an λ)l an λ)... L a λ) ) Aλ) Bλ) T λ) = Cλ) Dλ) 2.) The entries of the monodromy matrix T λ) are new variables. The commutation relations of the new operators Aλ),..., Dλ)) can be obtained from the local relaltion for the L-operator at one site: R 2 λ µ)l j λ)l 2j µ) = L 2j µ)l j λ)r 2 λ µ) 2.2) where R-matrix is R 2 λ) = λi + ηp 2 EndC 2 C2 2 ) and it acts on the tensor product of two auxiliary spaces C 2 C2 2, while the index j refers to the space of spin quantum states C2 j. The relation for T λ) is of the same form []-[4] R 2 λ µ)t λ)t 2 µ) = T 2 µ)t λ)r 2 λ µ) where T λ) := T λ) I, T 2 µ) = I T µ). One can extract 6 relations for the entries of T λ). We will use only few of them to construct algebraically eigenvectors of the Hamiltonian H XXX : Aλ)Bµ) = fλ µ)bµ)aλ) + gλ µ)bλ)aµ) Dλ)Bµ) = fµ λ)bµ)dλ) + gµ λ)bλ)dµ) Bλ)Bµ) = Bµ)Bλ) where fλ µ) = λ µ η)/λ µ), gλ µ) = η/λ µ). Multiplying the RTT-relation by R2 λ µ) and taking the trace over two auxiliary spaces one gets commutativity property of transfer matrix tλ): tλ)tµ) = tµ)tλ), tλ) := tr T λ) = Aλ) + Dλ) 2.3) The operator Bµ) 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 tλ) 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: Ω = N e +) m, σme z ±) m = ±e ±) m, σ me + +) m = 0, σme +) m = e ) m
Models solvable by Bethe Ansatz 93 Using the explicit form of the L-operator and the definition of the monodromy matrix T λ) it is easy to get the relations Cλ)Ω = 0, Aλ)Ω = aλ)ω, Dλ)Ω = dλ)ω where aλ) = λ + η 2 )N, dλ) = λ η 2 )N. It follows also from quadratic relation of Aλ), Bµ) that M M M Aλ) Bµ j ) = fλ µ j )Bµ j )Aλ) + j= j= k= gλ µ k )Bλ) j k fµ k µ j )Bµ j )Aµ k ) and a similar relation for Dλ) and the product of Bµ j ). Sum of these relations acting on the vacuum Ω gives the eigenvector of the transfer matrix tλ) M tλ) Bµ j )Ω = Λ λ {µ k } M j= ) M Bm j )Ω j= under the condition that the parameters µ k satisfy the Bethe equations k =, 2,..., M) aµ k ) M dµ k ) = fµ j µ k ) fµ k µ j ) j k The eigenvalue is M M Λλ {µ k } M ) = aλ) fλ µ j ) + dλ) fµ j λ) j= j= This construction of the eigenvectors of quantum integrable models was coined as algebraic Bethe Ansatz ABA) []-[3]. Originally these eigenvectors of the XXX spin chain were found by H. Bethe at 93 as a linear combination of one magnon eigenstates using the local operators σ α j Ψz) = N z k σ k Ω k= It is easy to see that Ψz) is an eigenvector of H XXX with the eigenvalue 2z+z 2). However, the condition of periodicity i.e. the requirement that Ψz) is also an eigenvector of the shift operator: Uσj α = σα j U, results in the quantization of z UΨz) = zψz), z N =, or z = exp2πim/n) These yields N states, m =, 2,..., N because the state with z = belongs to the vacuum multiplet: Ψ) = S Ω). Multimagnon states are given by a Bethe sum or coordinate Bethe Ansatz) Ψ{z j } M ) = M AP ) z n k P j σ k Ω {n k } P S m j= where the coefficients amplitudes) AP ) are defined by the elements P of permutation group S M, quasimomenta z Pj and the two magnon S-matrix [24] + z z 2 2z 2 )/2z z z 2 ), z j = µ j + η ) / µ j η ) 2 2
94 Petr P. Kulish Constructed eigenstates Ψ = M Bµ j )Ω, M [N/2] are highest weight vectors for the global j= symmetry algebra sl 2 with generators S α = 2 N σn, α S + Ψµ,..., µ M ) = 0 n The proof is purely algebraic and it follows from the RTT-relation and the asymptotic of the monodromy matrix [0] T λ) = λ N I + 2ηλ N α σ α a S α + O λ N 2) 3 Hecke and Temperley-Lieb Algebras Both algebras H N q) and T L N q) are quotients of the group algebra of the braid group B N generated by N ) generators Řj, j =, 2,..., N, their inverses Ř j and subject to the relations see [25]) Ř j Ř k = ŘkŘj, j k > Ř j Ř k Ř j = ŘkŘjŘk, j k = 3.) The Hecke algebra H N q) is obtained by adding to these relations the following characteristic equations obeyed by generators Řj q ) Ř j + /q ) = 0. 3.2) It is known that H N q) is isomorphic to the group algebra C[S N ]. Consequently, irreducible representations of the Hecke algebra, as that of S N, are parametrized by Young diagrams. By virtue of 3.2) we can write Ř using the idempotents P + and P P + + P = ): Ř = qp + q P = qi q + ) P := qi + X 3.3) q Substituting the expression 3.3) for Ř in terms of X, into the braid group relations 3.) one gets relations for X j, X k, j k = X j X k X j X j = X k X j X k X k 3.4) Requiring that each side of 3.4) is zero we obtain the quotient algebra of the Hecke algebra, the Temperley-Lieb algebra T L N q). It is defined by the generators X j, j =, 2,..., N and the relations νq) = q + /q): Xj 2 = q + ) X j = νq)x j q X j X k X j = X j, j k = 3.5) The dimension of the Hecke algebra is N!, the same as the dimension of the symmetric group S N, the dimension of T L N q) is equal to the Catalan number C N = 2N)!/N!N + )!. In connection with integrable spin systems we will be interested in representations of T L N q) on the tensor product space H = N C n. One representation is defined by an invertible n n matrix b GLn, C) which can also be seen as an n 2 dimensional vector {b cd } C n C n [7]. We use
Models solvable by Bethe Ansatz 95 the notation b := b and view this matrix also as an n 2 dimensional vector { b cd } C n C n. The generators X j can be expressed as X j ) cd,xy = b cd bxy Mat C n j C n ) j+ 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 q νq) = q + /q): Xj 2 = X j tr b t b, tr b t b = q + ) = νq) 3.7) q An obvious invariance of the braid group relations the Yang - Baxter equation) 3.) in this representation with respect to the transformation of the R-matrix Ř AdM AdM Ř ), M GLn, C) results in the following transformation of the matrix b MbM t. If one uses an R-matrix depending on a spectral parameter Yang - Baxterization of Řq)) Řu; q) = uřq) Řq) ) = ωuq)i + ωu)x 3.8) u where Řq) = /q)i + X, then relation 3.5) can be written as Řq ; q)řq 2 ; q)řq ; q) = 0 3.9) In terms of constant R-matrices generators of T L N q)) this relation has the form i k =, Ři = Ři,i+) Ři qi ) νq)řk q 2 I ) Ř i qi ) = 0 3.0) Replacing in 3.9) the expression Řu; q) = ωu)řq) + u ωq)i, or in 3.5) substituting X = Ř qi yields the vanishing of the q-antisymmetriser I q Ř 2 + Ř23) + q 2 Ř 2 Ř 23 + Ř23Ř2) q Ř 2 Ř 23 Ř 2 = 0 3.) Thus the irreducible representations of T L N q) 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. 4 Quantum Algebra U q n) 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 U q and its dual Hopf algebra, the quantum group AR). In this paper the emphasis will be on the quantum algebra U q and its finite dimensional representations V k, k = 0,, 2,.... The generators of U q 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) Ř 2 L a ql a2 q = L a ql a2 qř2 4.) where the indices a and a 2 refer to the representation spaces V a and V a2, respectively, and index q refers to the algebra U q. Hence the equation 4.) is given in End V a V a2 ) U q.
96 Petr P. Kulish 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 R univ = j R j) R j) 2 := R R 2 U q U q 4.2) L aq = ρ id) R univ = ρr ) R 2 4.3) where ρ : U q EndV a ). Furthermore, the universal R-matrix satisfies Drinfeld s axioms of the quasi-triangular Hopf algebras [7, 25]. In particular, id ) R = R 3 R 2 4.4) Thus, choosing the appropriate representation space as the first space, one obtains the co-product of the generators of U q from the following matrix equation id ) L aq = L aq2 L aq EndV a ) U q U q 4.5) The case when V a = C 3 is of particular interest and it will be presented below in detail. To this end the generators of U q are denoted by {A i, B i, C i, i =, 2, 3} and the L-matrix is given by A B B 3 L aq = C A 2 B 2 4.6) C 3 C 2 A 3 Multiplying two L-matrices with entries in the corresponding factors U q n) U q n) we obtain L ab ) = 3 L kb L ak = k= or explicitly for the generators 3 I L ak ) L kb I) 4.7) k= B ) = B A + A 2 B + C 2 B 3, 4.8) B 2 ) = B 3 C + B 2 A 2 + A 3 B 2 4.9) B 3 ) = B 3 A + B 2 B + A 3 B 3 4.0) etc. The central element in U q is obtained from the defining relation 4.) b L aq bl t aq = c 2 I 4.) c 2 = jkl b ) j L jkb kl L l However this central element is group-like: c 2 = c 2 c 2. It is proportional to the identity in the tensor product of representations. The analogue of the U q 2) Casimir operator can be obtained according to [26] using L + := L and L := ρ id) R 2 ) as tr q L + L = tr b b t L + L. In the case when V a, V q are the three dimensional space V a V q C 3 and the b matrix is taken from the references [5, 6] b ij = p 2 i δ i4 j = b ) ij 4.2) c 2 is written as ) c 2 = p p A 3A + C 2 B + pc 3 B 3
Models solvable by Bethe Ansatz 97 Parameters p and q are related: p 2 + + p 2 = q + q ). For the explicit L-operator and its 3 3 blocks we get c 2 = qi q where I q is the identity operator on V q = C 3. The form of the generators {A i, B i, C i, i =, 2, 3} which corresponds to the choice of the b matrix 4.2), follows from the expression for the Ř and L-matrices L = PŘ = P qi + X), where P is the permutation matrix. For example, we have B = 0 0 0 q 0 0 0 p 0, B 2 = 0 0 0 p 0 0 0 q 0 4.3) If we choose b as in equation 4.2) the R-matrix commutes with h + h where h = diag, 0, ). As a highest spin vector pseudovacuum of the corresponding integrable spin chain [3]) we choose θ =, 0, 0) t C 3 [2, 6]. If we act on the tensor product of these vectors θ θ C 3 C 3 with the coproduct of the lowering operators B j ), j =, 2, 3 we obtain new vectors. By looking at the explicit forms of the operators A j,b j,c j in the space C 3 we can convince ourselves that the vectors B i )) k θ θ, i =, 2; k =, 2, 3 are linearly independent. Together with the vectors θ θ and B i )) 4 θ θ 0 0 0 0 they span an 8 dimensional subspace. The vector B 3 )θ θ is a linear combination of the vectors B i )) 2 θ θ, i =, 2. The vector b = 00p 00 p 00) t spans a one dimensional invariant subspace. Thus we have the following decomposition C 3 C 3 = C 8 C 4.4) This decomposition can also be obtained using the projectors P +, P 3.3), 3.6) expressed in terms of b matrix vector) 4.2). Due to the commutativity of the R-matrix Řq q 2 with the co-product 4.5), 4.7) the corresponding subspaces P ± C 3 C 3 ) are invariant. Similarly, using 3 Y ), Y U q 3) one can get the decomposition of C 3 C 3 C 3 = C 2 C 3 C 3 4.5) This type of decomposition is valid for any n. The result of applying the co-product, given by 4.5), on the generators of the quantum algebra U q several times can also be presented in the matrix form id N ) L aq = L aqn... L aq2 L aq := T N) 4.6) where N : U q U q ) N, := id, 2 :=, 3 := id ), etc. In general case of the tensor representation of T L N q), with the space C n at each site, the generators of the algebra Ř qk q k+ := Řk,k+ commute with the generators 4.6) of the global diagonal) action of the quantum algebra U q n) in the space H = N C n. This follows from the relation Ř k,k+ L aqk+ L aqk = L aqk+ L aqk Ř k,k+ and the possibility due to the co-associativity of the coproduct to write the product of of L aqj as T N) = L aqn... L aqk+2 k L aqk )L aqk... L aq
98 Petr P. Kulish Hence, Ř k,k+ T N) = T N) Ř k,k+ Thus, the algebras U q n) and T L N q) are each other s centralizers in the space H. The tensor representation of T L N q) in H decomposes into irreducible factors whose multiplicities are given by the dimensions of the irreducible representations of the algebra U q n), corresponding to the same Young diagrams H = N C n = N p k n)w k N) = N ν k N)V k n) 4.7) k k k 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 p k n) = dim V k n), ν k N) = dim W k N) 4.8) of the algebras U q n) and T L N q), respectively. As for the finite dimensional irreducible representations of the Lie algebra sl 2, V 0 n) = C is the one-dimensional scalar) representation and the fundamental representation of the algebra U q n) is n dimensional, V n) C n. The dimensions of other representations follow from the trivial multiplicities of the factors in the decomposition of the tensor product of the V k n) and the fundamental representation V n) into two irreducible factors, as for the sl 2, V n) V k n) = V k+ n) V k n) 4.9) Thus, for the dimensions p k n) = dim V k n) the following recurrence relation is valid n p k n) = p k+ n) + p k n) 4.20) with the initial conditions p n) = 0, p 0 n) =, whose solutions are Chebyshev polynomials of the second kind p k n) = sink + )θ, n = 2 cos θ 4.2) sin θ The multiplicity ν k N), or the dimensions of the subspaces W k N) in 4.7) is the number of paths that go from the top of the Bratteli diagram to the Young diagram corresponding to the representation W k N). If λ N is the partition of N, λ = λ λ 2 λ + λ 2 = N), then k = λ λ 2 and ν k N) = ν k+ N ) + ν k N ) 4.22) The subspaces invariant under the diagonal action of the quantum algebra U q 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 T L N 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] P +) N P N ŘN N +) q N ; q ) P +) N 4.23) Similar construction can be done with the underlying Lie algebra sl3). Then the corresponding q-antisymmetrizer P 4) which defines a quotient of the Hecke algebra is [4, 27] P 4) P 23) Ř 3 4 q 3 ; q)p 23) 4.24)
Models solvable by Bethe Ansatz 99 This form follows from the intertwiner of four monodromy matrices T u )T 2 u 2 )T 3 u 3 )T 4 u 4 ) J 4) = R 2 R 3 R 2 3 R 4 R 2 4 R 3 4 4.25) where R i j := R i j u i /u j ). Multiplying by appropriate product of the permutation operators P k k+ one can get the expression in terms of the baxterized Hecke generators Ř u 3 /u 4 )Ř2 3u 2 /u 4 )Ř 2u 2 /u 3 )Ř3 4u /u 4 )Ř2 3u /u 3 )Ř 2u /u 2 ) The q-antisymmetrizer 4.24) is obtained by fixing shifts of the spectral parameters u k = uq k [4, 27]. Theorem 4.. Consider the quotient of the Hecke algebra H N q) N > 3) by the ideal I generated by the q-antisymmetrizers P 4) / H 4) N q) = H Nq) I P 4) ), 4.26) The tensor product representation of H 4) N q) in the space H N = N Cn n 3) with the q- antisymmetrizers P 3) of rank define the quantum algebra U qsl3); n) as the centralizer algebra of H 4) N q). 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 [2], 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 06-0-0045. References [] E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev. Quantum inverse problem method. Teoret. Mat. Fiz. 40 979), 94 220 in Russian). [2] L. A. Takhtajan and, L. D. Faddeev. The quantum method for the inverse problem and the XY Z Heisenberg model. Uspekhi Mat. Nauk, 34 979) 3 63 in Russian). [3] L. D. Faddeev. How the algebraic Bethe Ansatz works for integrable models. In Quantum symmetries/symetries quantiques, Proceedings of the Les Houches summer school, Session LXIV. Eds. A. Connes, K. Gawedzki and J. Zinn-Justin. North-Holland, 998, 49 29; arxiv:hep-th/960587. [4] P. P. Kulish and E. K. Sklyanin. Quantum spectral transform method. Recent developments. Lecture Notes in Phys. 5 982), 6 9. [5] P. P. Kulish and N. Yu. Reshetikhin. Quantum linear problem for the sine-gordon equation and higher representations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov LOMI), 0 98), 0 0, 207 in Russian). [6] E. K. Sklyanin. On an algebra generated by quadratic relations. Uspekhi Mat. Nauk, 40 No. 2 985) 24 in Russian). [7] V. G. Drinfeld. Quantum groups. In Proc. Intern. Congress Math. Berkeley, CA, AMS, Providence, RI, 987, 798 820. [8] M. Jimbo. A q-difference analogue of UglN + )) and the Yang-Baxter equation. Lett. Math. Phys. 0 985), 63 69.
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