Symmetries of spin systems and Birman-Wenzl-Murakami algebra
aa r X i v : . [ n li n . S I] M a r Symmetries of spin systems andBirman-Wenzl-Murakami algebra
P. P. Kulish, ∗ N. Manojlovi ´c † and Z. Nagy ‡ ∗ St. Petersburg Department of Steklov Mathematical InstituteFontanka 27, 191023, St. Petersburg, Russia ∗ †‡ Grupo de Física Matemática da Universidade de LisboaAv. Prof. Gama Pinto 2, PT-1649-003 Lisboa, Portugal † Departamento de Matemática, F. C. T., Universidade do AlgarveCampus de Gambelas, PT-8005-139 Faro, Portugal
Abstract
We consider integrable open spin chains related to the quantum affinealgebras U q ( d o ( )) and U q ( A ( ) ) . We discuss the symmetry algebras of thesechains with the local C space related to the Birman-Wenzl-Murakami al-gebra. The symmetry algebra and the Birman-Wenzl-Murakami algebracentralize each other in the representation space H = ⊗ N C of the system,and this determines the structure of the spin system spectra. Consequently,the corresponding multiplet structure of the energy spectra is obtained. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected]
INTRODUCTION
I Introduction
The development of the quantum inverse scattering method (QISM) [1, 2, 3] asan approach to construction and exact solution of quantum integrable systemshas lead to the foundations of the theory of quantum groups [4, 5]. The represen-tation theory of quantum groups is naturally connected to the spectral theoryof the integrals of motion of quantum systems. In particular, this connectionappeared in the combinatorial approach to the question of completeness of theeigenvectors of the
XXX
Heisenberg spin chain [6].Important properties of quantum integrable systems are related with theirsymmetry algebra and are defined by a bigger algebra which gives the mainrelations underlining integrability, the so-called RLL-relations [1]. In the case ofmost known isotropic Heisenberg chain of spin 1/2 (XXX-model) the symme-try algebra is sl , the Hamiltonian is an element of the group algebra C [ S N ] ofthe symmetric group S N . The fundamental relations of the auxiliary L-matrixentries generate an infinite dimensional quantum algebra – the Yangian Y ( sl ) .The actions of sl and S N on the state of space H = ⊗ N C are mutually commut-ing (the Schur-Weyl duality). Extension of this scheme to a particular case of theHecke algebra – the Temperley-Lieb algebra, instead of the symmetric groupand corresponding new quantum algebras were proposed in [7, 8]. Here weconsider a further generalization – the case of the Birman-Wenzl-Murakami al-gebra [9] and its specific representations in C ⊗ C given by the spectral param-eter dependent R-matrices. These R-matrices correspond to different quantumaffine algebras U q ( d o ( )) , U q ( A ( ) ) , U q ( \ osp ( | )) and U q ( sl ( | ) ( ) ) . Althoughcorresponding spin systems were analysed in a variety of papers (detailed refer-ences are given below) we point out the connection of the open spin chains withthe Birman-Wenzl-Murakami algebra as a centralizer of the symmetry algebra.For the XXZ-model of spin 1 the appropriate dynamical symmetry algebrais U q ( d o ( )) and its symmetry algebra is U q ( o ( )) [10]. The corresponding R-matrix was found in [11], see also [5], and it can also be obtained by the fusionprocedure starting from the R-matrix of the XXZ-model of spin ½ [3].The R-matrix of U q ( A ( ) ) in C ⊗ C was found in [12] and the correspondingperiodic spin chain was solved by recurrence algebraic Bethe ansatz in [13].These two spectral parameter dependent R-matrices are the two versionsof the Yang-Baxterization procedure for a given representation of the Birman-Wenzl-Murakami algebra W ( q , ν = q − ) in C ⊗ C [14, 15, 16].The two additional R-matrices related to the quantum affine super-algebrascan be obtained by considering the Birman-Wenzl-Murakami algebra W ( − q , ν = − q − ) and taking into account the connection between the solutions of the Yang-Baxter equation and the solutions of the ( Z graded) super-Yang-Baxter equa-tion [17]. In this case the representation of the Birman-Wenzl-Murakami (BMW)algebra is the centralizer of the U q ( osp ( | )) action in the tensor product of itsfundamental representation.We point out the multiplet structure of the energy spectra of the correspond-ing open spin chain Hamiltonians. The quantum determinant of the algebra U q ( A ( ) ) is also given.The symmetry properties of integrable spin chains depend also on the bound-ary conditions, for example, there are soliton preserving versus non-soliton pre-serving boundary conditions, see [18] and the references therein. For the XXZ-chain of spin ½ particular boundary conditions yield the spectrum of the sys-tem which has clear multiplet structure of the irreducible representations of theHecke algebra H N ( q ) and the symmetry algebra U q ( sl ) . However, there arealso K-matrices defining the integrable boundary conditions of the XXZ modelsuch that the whole space of states is just an irreducible representation of thereflection equation algebra.The paper is organised as follows. In Section II the R-matrix of the model XXZ and its properties are reviewed. The emphasis is given to its connectionto the U q ( o ( )) constant R-matrix and the corresponding realisation of the BMWalgebra. In Section III the Izergin-Korepin R-matrix is reviewed along the samelines. It was shown that although the constant R-matrix is the same as in the caseof the XXZ spin-1 however, the corresponding Yang -Baxterization of the BMWalgebra generators yields different spectral parameter dependent R-matrix. InSection IV the definition of the Birman-Wenzl-Murakami algebra is reviewedin general. Also, some properties of the symmetrizers and antisymmetrizers inthe particular case of the BMW algebra W N ( q , q − ) , corresponding to the XXZ and A ( ) R-matricies, are studied. The symmetries of the corresponding openspin chains are discussed in Section V. In particular, the realization of the BMWalgebra as the centralizer of the symmetry algebra of the open spin chain is anal-ysed. The multiplet structure of the energy spectra of the corresponding openspin chain Hamiltonians is the main result of this analysis. Our conclusions anddirections for further research are given in the last Section.3I R-MATRIXOF XXZ SPIN-1CHAIN
II R-matrix of XXZ spin-1chain
Following [11, 17, 3, 5], the 9 × R ( λ , η ) = a a b a b b c a c a b a b c c a c a a , (II.1)where the functions are a = sinh ( λ + η ) sinh ( λ + η ) , b = e λ sinh λ sinh 2 η , a = sinh λ sinh ( λ + η ) , b = e λ sinh η sinh 2 η , a = sinh λ sinh ( λ − η ) , c = e − λ sinh ( λ + η ) sinh 2 η , a = sinh λ sinh ( λ + η ) + sinh η sinh 2 η , c = e − λ sinh λ sinh 2 η , b = e λ sinh ( λ + η ) sinh 2 η , c = e − λ sinh η sinh 2 η .The R-matrix satisfies the Yang-Baxter equation in the space C ⊗ C ⊗ C R ( λ ) R ( λ + µ ) R ( µ ) = R ( µ ) R ( λ + µ ) R ( λ ) , (II.2)where we use the standard notation of the QISM [1, 2, 3].This form of the R-matrix is related with the symmetric one R t ( λ , η ) = R ( λ , η ) by the similarity transformation R ( λ , η ) → Ad exp ( αλ ( h − h )) R ( λ , η ) , (II.3)with α = ½ and h = diag (
1, 0, − ) . The transformed R-matrix still obeys theYang-Baxter equation due to the U ( ) symmetry of the initial R-matrix [ h + h , R ( λ , η )] =
0. (II.4)The R-matrix (II.1) has a few important properties: regularity, unitarity, PT-symmetry and crossing symmetry. The regularity condition at λ = R ( η ) = sinh ( η ) sinh ( η ) P , (II.5)4I R-MATRIXOF XXZ SPIN-1CHAINwhere P is the permutation matrix of C ⊗ C . The unitarity relation is R ( λ ) R ( − λ ) = ρ ( λ ) , (II.6)here R ( λ ) = P R ( λ ) P and ρ is the following function ρ ( λ ) = sinh ( λ + η ) sinh ( λ + η ) sinh ( λ − η ) sinh ( λ − η ) . (II.7)The so-called PT-symmetry states R t ( λ ) = R ( λ ) . (II.8)Finally, it has the following crossing symmetry property R ( λ ) = ( Q ⊗ ) R t ( − λ − η ) ( Q ⊗ ) , (II.9)where t denotes the transpose in the second space and the matrix Q is given by Q = − e − η − e η . (II.10)The R-matrix (II.1) in the braid group formˇ R ( λ , η ) = P R ( λ , η ) , (II.11)admits the spectral decompositionˇ R ( λ , η ) = sinh ( λ + η ) sinh ( λ + η ) P ( η ) − sinh ( λ + η ) sinh ( λ − η ) P ( η )+ sinh ( λ − η ) sinh ( λ − η ) P ( η ) , (II.12)here P ( η ) = − P ( η ) − P ( η ) , (II.13) P ( η ) = e η + e − η e η − ω − − e − η ω ω − ω e η − − − ω − e − η , (II.14)5I R-MATRIXOF XXZ SPIN-1CHAINhere ω ( e η ) = e η − e − η and P ( η ) = e η + + e − η e η − e η − e η − e − η − e − η e − η . (II.15)These are projectors to the five, three and one dimensional eigenspace, respec-tively. Thus the R-matrix (II.1) has four degeneration points λ = ± η , and λ = ± η . Its rank at λ = η is eight, at λ = η is five, at λ = − η is fourand finally at λ = − η is one.The R-matrix (II.11) can also be expressed in the following form, useful forthe asymptoticsˇ R ( λ , η ) = e η (cid:16) e λ − (cid:17) ˇ R ( η ) + ( sinh η sinh 2 η ) + e − η (cid:16) e − λ − (cid:17) ˇ R − ( η ) .(II.16)A relevant observation is that the constant R-matrixˇ R ± ( η ) = lim λ →± ∞ (cid:0) ( ∓ ( λ + η )) ˇ R ( λ , η ) (cid:1) (II.17)being a solution of the Yang-Baxter equation in the braid group formˇ R ˇ R ˇ R = ˇ R ˇ R ˇ R , (II.18)has the spectral decomposition ( q = e η ) ˇ R ( η ) = qP ( η ) − q P ( η ) + q P ( η ) . (II.19)Hence, ˇ R ( η ) satisfies the cubic equation (cid:0) ˇ R ( η ) − q (cid:1) (cid:18) ˇ R ( η ) + q (cid:19) (cid:18) ˇ R ( η ) − q (cid:19) =
0. (II.20)Consequently, its minimal polynomial is ( α − q )( α + q )( α − q ) . (II.21)6I R-MATRIXOF XXZ SPIN-1CHAINIts matrix form isˇ R ( η ) = e η e − η ω e − η ω e − η e − η ω ( − e − η ) ω ω e η , (II.22)here ω ( e η ) = e η − e − η .For the purpose of establishing a relation with the Birman-Wenzl-Murakamialgebra, the one dimensional projector P ( η ) is related to the rank one matrix E ( η ) E ( η ) = µ P ( η ) , (II.23)with µ = q + + q and q = e η . The matrix E ( η ) satisfies E ( η ) = µ E ( η ) , (II.24)ˇ R ( η ) E ( η ) = E ( η ) ˇ R ( η ) = q E ( η ) , (II.25)and also ˇ R ( η ) − ˇ R − ( η ) = ω ( q ) ( − E ( η )) , (II.26)where ω ( q ) = q − q . From these relations we conclude that ˇ R , ˇ R − and E provide a realisation of the Birman-Wenzl-Murakami algebra W N ( q , 1/ q ) [9]in the space H = ⊗ N C .The projector P ( η ) on five dimentional subspace of C ⊗ C correspondsto a symmetrizer of spin 2 irreducible representation of the quantum algebra U q ( o ( )) . It can be used to construct an R-matrix for higher spin R ( ) ( λ , η ) ∈ End ( C ⊗ C ) by the fusion procedure [3] R ( ) ( λ , η ) ≃ ˇ R ( η , η ) R ( λ + η , η ) R ( λ − η , η ) . (II.27)It will be shown in Sec. V that one can use higher symmetrizers of the BMW-algebra W s ( q , 1/ q ) to get R-matrices R ( s ,1 ) ( λ , η ) ∈ End ( C ( s + ) ⊗ C ) , in thisnotation the original R-matrix is R ( ) ( λ , η ) .7II IZERGIN-KOREPINR-MATRIX III Izergin-Korepin R-matrix
Following [13], the Izergin-Korepin R-matrix is expressed as follows R ( λ , η ) = a a b a b b c a c a b a b c c a c a a , (III.1)where the functions are a = sinh ( λ − η ) + sinh ( η ) , b = e η sinh 2 η (cid:0) − e − λ (cid:1) , a = sinh ( λ − η ) + sinh ( η ) , b = − e − λ + η sinh η sinh 2 η − e − η sinh 4 η , a = sinh ( λ − η ) + sinh ( η ) , c = − sinh ( η ) (cid:0) e λ − η + e η (cid:1) , a = sinh ( λ − η ) + sinh ( η ) − sinh ( η ) + sinh ( η ) , c = e − η sinh 2 η (cid:0) − e λ (cid:1) , b = − sinh ( η ) (cid:0) e − λ + η + e − η (cid:1) , c = e λ − η sinh η sinh 2 η − e η sinh 4 η .Like in the case of XXZ spin 1, this R-matrix (III.1) has four important proper-ties: regularity, unitarity, PT-symmetry and crossing symmetry. The regularitycondition at λ = R ( η ) = ( sinh ( η ) − sinh ( η )) P , (III.2)where P is the permutation matrix of C ⊗ C . The unitarity relation is R ( λ ) R ( − λ ) = ρ ( λ ) , (III.3)and ρ is the following function ρ ( λ ) = − ( sinh ( λ + η ) − sinh ( η ))( sinh ( λ − η ) + sinh ( η )) . (III.4)The so-called PT-symmetry states that the transpose of the R-matrix (III.1) isequal to the same R-matrix conjugated by the permutation matrix P , that is R t ( λ ) = R ( λ ) . (III.5)Also, the R-matrix (III.1) has the following crossing symmetry [19] R ( λ ) = ( Q ⊗ ) R t ( − λ + η + ı π ) ( Q ⊗ ) , (III.6)8II IZERGIN-KOREPINR-MATRIXwhere t denotes the transpose in the second space and the matrix Q is given in(II.10).In the braid group form the R-matrix (III.1)ˇ R ( λ , η ) = P R ( λ , η ) , (III.7)admits the spectral decompositionˇ R ( λ , η ) = ( sinh ( λ − η ) + sinh ( η )) P ( η ) − ( sinh ( λ − η ) + sinh ( η )) P ( η )+ ( sinh ( λ + η ) − sinh ( η )) P ( η ) , (III.8)where the projectors P ( η ) , P ( η ) and P ( η ) are the same as in the equation(II.12) and are given in (II.13-15), respectively. Thus the R-matrix (III.1) has fourdegeneration points [19] λ = ± η , and λ = ± ( η + ı π ) . Its rank at λ = − ( η + ı π ) is eight, at λ = − η is six, at λ = η is three and finally at λ = η + ı π isone.The R-matrix (III.7) can also be expressed in the following formˇ R ( λ , η ) = e η (cid:16) − e − λ (cid:17) ˇ R ( η ) − (cid:16) e η + e − η (cid:17) (cid:16) e η − e − η (cid:17) − e − η (cid:16) − e λ (cid:17) ˇ R − ( η ) , (III.9)where the constant R-matrix used here is given in (II.19) and is the same asthe one used in (II.16). This constant R-matrix, as it was pointed out, defines arepresentation of the BMW algebra W N ( q , 1/ q ) in H = ⊗ N C . To confirm this,in the next section, we briefly review basic facts of the Birman-Wenzl-Murakamialgebra.The matrix ˇ R ( λ , η ) (III.8) at degeneration point λ = η is proportional to therank 3 projector P ( η ) (II.14) which is a q-analogue of the antisymmetrizer on C ⊗ C . One can further obtain the antisymmetrizer on C ⊗ C ⊗ C accordingto the fusion procedure [3] A ≃ ˇ R ( η , η ) ˇ R ( η , η ) ˇ R ( η , η ) . (III.10)This matrix A ∈ End (( C ) ⊗ ) has rank one. It can also be used to define aquantum determinant q-det L ( λ ) of operator valued L-matrix L ( λ ) satisfyingthe so-called RLL-relation, a milestone of the QISM,ˇ R ( λ − µ ) L ( λ ) L ( µ ) = L ( µ ) L ( λ ) ˇ R ( λ − µ ) . (III.11)9II IZERGIN-KOREPINR-MATRIXIn this case, the quantum determinantq-det L ( λ ) ≃ ˇ R ( η , η ) ˇ R ( η , η ) ˇ R ( η , η ) L ( λ ) L ( λ − η ) L ( λ − η ) (III.12)is given byq-det L ( λ ) = A ( λ ) C ( λ − η ) C ( λ − η ) − e − η A ( λ ) C ( λ − η ) C ( λ − η ) − e − η C ( λ ) A ( λ − η ) C ( λ − η ) − e − η ω ( e η ) C ( λ ) C ( λ − η ) C ( λ − η )+ e − η C ( λ ) C ( λ − η ) A ( λ − η ) + e − η C ( λ ) A ( λ − η ) C ( λ − η ) − e − η C ( λ ) C ( λ − η ) A ( λ − η ) , (III.13)where A i ( λ ) , B i ( λ ) and C i ( λ ) , i =
1, 2, 3, are the operator entries of the L-matrix L ( λ ) = A B B C A B C C A . (III.14)The vector in ( C ) ⊗ defining the rank one antisymmetrizer (III.10), coinsideswith the quantum completely antisymmetric tensor of [20]. It can be shownthat the quantum determinant (III.12) is central, with respect to the RLL-relation(III.11), due to the proportionality of the R-marix quantum determinant to theidentity matrix q-det R ( λ , η ) ≃ ∈ End ( C ) . (III.15)The q-det L ( λ ) has a group-like propertyq-det ( L ( λ ) L ( λ )) = q-det L ( λ ) · q-det L ( λ ) . (III.16)As the final remark in the discussion of the properties of the XXZ and A ( ) trigonometric R-matrices we point out that these R-matrices have different scal-ing limits. The A ( ) R-matrix in the limit λ → ǫλ , η → ǫη and ǫ → sl ( ) -Yang R-matrix R ( λ , η ) = λ − η P , (III.17)while in the XXZ case the limit yields R ( λ , η ) = λ ( λ + η ) + η ( λ + η ) P + λη K , (III.18)where K is a rank 1 matrix, invariant with respect to the O ( ) transformations.In the quasi-classical limit η → W N ( Q , ν ) IV Birman-Wenzl-Murakami algebra W N ( q , ν ) The defining relations of the BMW algebra W N ( q , ν ) , for the generators 1, σ i , σ − i and e i , i =
1, . . . , N −
1, are recalled for convenience, [9] σ i σ i + σ i = σ i + σ i σ i + , σ i σ j = σ j σ i , for | i − j | >
1, (IV.1) e i σ i = σ i e i = ν e i , (IV.2) e i σ ± i − e i = ν ∓ e i , (IV.3) σ i − σ − i = ω ( q )( − e i ) , (IV.4)where ω ( q ) = q − q . It can be shown that the dimension of the Birman-Wenzl-Murakami algebra W N ( q , ν ) is ( N − ) !! [9].Many useful relations follow from the definition above, for example [16] e i = µ e i , with µ = ω − ν + νω = ( q − ν )( ν + q ) νω . (IV.5)Another important consequence of the relations (IV.2,4) is a cubic relatoin for σ i ( σ i − q )( σ i + q − )( σ i − ν ) =
0. (IV.6)There is the natural inclusion of W M ( q , ν ) ⊂ W N ( q , ν ) , M < N . Namely, the first3 ( M − ) generators { σ ± i , e i ; i =
1, 2 . . . , M − } of W N ( q , ν ) define the algebra W M ( q , ν ) .The Yang-Baxterization procedure yields two spectral parameter dependentelements [14, 15, 16] σ ( ± ) i ( u ) = ω (cid:16) u − σ i − u σ − i (cid:17) + ν ± q ± u ν ± q ± u − e i . (IV.7)These elements satisfy the Yang-Baxter equation in the braid group form σ ( ± ) i ( u ) σ ( ± ) i + ( uv ) σ ( ± ) i ( v ) = σ ( ± ) i + ( v ) σ ( ± ) i ( uv ) σ ( ± ) i + ( u ) . (IV.8)Their unitarity relation is σ ( ± ) i ( u ) σ ( ± ) i ( u − ) = (cid:16) − ω − ( u − u − ) (cid:17) . (IV.9)The regularity property of the Yang-Baxterized elements (IV.7) is important forthe locality of Hamiltonian density of the corresponding spin chains and is validon the algebraic level due to (IV.4). Also, these elements are normalised so that11V BIRMAN-WENZL-MURAKAMIALGEBRA W N ( Q , ν ) σ ( ± ) i ( ± ) = ±
1. In order to see the connection with the previous sections weset ν = q and find that σ ( − ) i ( e − λ ) ≃ ˇ R i , i + ( λ , η ) of (II.16) and σ (+) i ( e λ /2 ) ≃ ˇ R i , i + ( λ , η ) of (III.9).The irreducible representations of the BMW algebra W N ( q , ν ) are more com-plicated than the irreducible representations of the symmetric group S N or theHecke algebra H N ( q ) , although they can be parameterized by the Young di-agrams [9, 14]. The simplest, one-dimensional irreducible representations of W N ( q , ν ) are defined by the symmetrizer and antisymmetrizer, respectively. Thesymmetrizer of the W N ( q , ν ) is given by S N = [ N ] q ! σ ( − ) ( q − ) σ ( − ) ( q − ) · · · σ ( − ) N − ( q − ( N − ) ) S N − , (IV.10)with S = S = [ ] q σ ( − ) ( q − ) . (IV.11)We use the standard notation for the q-factorial [ n ] q ! = [ n ] q [ n − ] q · · · [ ] q [ ] q and the q-numbers [ n ] q = ( q n − q − n ) / ( q − q − ) . The elements S n , n =
1, . . . , N are idempotents, i.e. S n = S n . In addition, the symmetrizer S N is also central.In the realisation on C ⊗ C of the BMW algebra W ( q , q − ) σ = ˇ R ( η ) = qP − q − P + ν P , ν = q , (IV.12)and e is proportional to the rank one projector P e = µ P = ( q + + q − ) P . (IV.13)Thus σ ( − ) ( q − ) = ( q + q − ) P , (IV.14) σ ± P = q ± P , (IV.15) e P =
0. (IV.16)Similarly, the antisymmetrizer of the W N ( q , ν ) is given by A N = [ N ] q ! σ (+) ( q ) σ (+) ( q ) · · · σ (+) N − ( q N − ) A N − , (IV.17)with A = A = [ ] q σ (+) ( q ) . (IV.18)12V BIRMAN-WENZL-MURAKAMIALGEBRA W N ( Q , ν ) The elements A n , n =
1, . . . , N are idempotents and the antisymmetrizer A N isalso central in W N ( q , ν ) . Below we will show how to prove this statement, herewe only notice that σ (+) ( q ) σ (+) ( q ) = [ ] q σ (+) ( q ) . (IV.19)It is straightforward to see that A ≃ σ (+) ( q ) σ (+) ( q ) σ (+) ( q ) = σ (+) ( q ) σ (+) ( q ) σ (+) ( q ) . (IV.20)In the realisation (IV.12,13) σ (+) ( q ) = [ ] q P , (IV.21) σ ± P = − q ∓ P , (IV.22) e P =
0. (IV.23)In addition, in this realisation (with σ (+) i ( e λ /2 ) ≃ ˇ R i , i + ( λ , η ) of (III.9)), the anti-symmetrizer A has rank one, as it was already noticed in (III.10). Furthermore,a straightforward calculation yields A =
0. Consequently all the higher anti-symmetrizers vanish identically, A n ≡
0, for n > W N ( q , ν ) , it can be shown that the following identitiesare valid σ ( − ) i ( q ) S n = S n σ ( − ) i ( q ) =
0, (IV.24) σ (+) i ( q − ) A n = A n σ (+) i ( q − ) =
0, (IV.25)for i =
1, . . . , n − < n N . The relations (IV.24,25) can also be writtenin the following form σ i S n = S n σ i = q S n , (IV.26) e i S n = S n e i =
0, (IV.27) σ i A n = A n σ i = − q A n , (IV.28) e i A n = A n e i =
0, (IV.29)for i =
1, . . . , n − < n N . From these identities it is evident that S N and A N are central in W N ( q , ν ) . Also, using the relations (IV.26-29), it isstraightforward to check that S n and A n are idempotents, i.e. S n = S n and A n = A n , n =
1, . . . , N .In the next section the BMW algebra W N ( q , q − ) will be used to describe themultiplet structure of the spectra of some open quantum spin chains.13 OPEN SPIN CHAIN V Open Spin Chain
According to the quantum inverse scattering method the R-matrix R ( u , q ) canbe used to construct an auxiliary L-operator for an integrable spin system, iden-tifying the two spaces of R ( u , q ) ∈ End ( V ⊗ V ) as auxiliary and quantum space,respectively: L j ( u ) = R j ( u , q ) . (V.1)Notice that in this section we mainly use the multiplicative spectral parame-ter, which in the case of the model XXZ is given by u = exp ( − λ ) . Then themonodromy matrix of a spin chain with N sites is the product of L-matrices inEnd ( V ) whose entries are in End ( V j ) [1] T ( u ) = L N ( u ) L N − ( u ) · · · L ( u ) , (V.2)while the entries of the monodromy matrix T ab ( u ) are operators on the wholespace of states H = ⊗ Nj = V j (in the case under consideration V j = C ). As aconsequence of the Yang-Baxter equation (II.2) for the R-matrix and (V.1) onehas [2, 6, 17] R ′ (cid:16) uw (cid:17) L j ( u ) L ′ j ( w ) = L ′ j ( w ) L j ( u ) R ′ (cid:16) uw (cid:17) (V.3)and R (cid:16) uw (cid:17) T ( u ) T ( w ) = T ( w ) T ( u ) R (cid:16) uw (cid:17) , (V.4)where T ( u ) = T ( u ) ⊗ and T ( u ) = ⊗ T ( u ) are operator valued matrices inthe two auxiliary spaces V ⊗ V , written as elements of End ( V ⊗ V ) . The traceof the monodromy matrix T ( u ) - the transfer matrix t ( u ) = tr T ( u ) , (V.5)is the generating function of the integrals of motion, including the Hamiltonian,of the spin chain with the periodic boundary condition.In order to construct integrable spin chains with non-periodic boundary con-dition one has to use the Sklyanin formalism [22]. The corresponding mon-odromy matrix T ( u ) consists of the two matrices T ( u ) (V.2) and a reflectionmatrix K − ( u ) ∈ End ( V ) T ( u ) = T ( u ) K − ( u ) T − ( u − ) . (V.6)Using the unitarity relation (II.6) ( R − ( u − ) = R ( u ) ) one gets T − ( u − ) = R ( u ) R ( u ) · · · R N ( u ) . (V.7)14 OPEN SPIN CHAINTaking into account the definition R ( u , η ) = P R ( u , η ) P one can trans-form the monodromy matrix T ( u ) into the following form (in order to shortenthe notation in the formulas below the argument η will be dropped) T ( u ) = ˇ R N ( u ) ˇ R N − N ( u ) · · · ˇ R ( u ) K − ( u ) ˇ R ( u ) ˇ R ( u ) · · · ˇ R N ( u ) . (V.8)The generating function τ ( u ) of the integrals of motion [22] is given by the traceof T ( u ) over the auxiliary space with an extra reflection matrix K + ( u ) τ ( u ) = tr (cid:0) K + ( u ) T ( u ) (cid:1) . (V.9)The reflection matrices K ± ( u ) are solutions to the reflection equation with aproperty K − ( ) = ∈ End ( V ) and τ ( ) ≃ . In particular, the Hamiltonian isgiven by H = ddu ln τ ( u ) | u = , H = N − ∑ i = ˇ R ′ i , i + ( ) + tr K + ( ) ˇ R ′ N ( ) tr K + ( ) + dK − ( ) du + K + ( ) d tr K + ( ) du ! .(V.10)The Hamiltonian density h i , i + = ddu ˇ R i , i + ( u ) | u = as one can see from (II.1) is afunction of the generators of W N ( q , q − ) on the space H = ⊗ N C . The two extraboundary terms are contributions from the two reflection matrices K ± ( u ) at thesites 1 and N . In our case we can take the constant K-matrices K − ( u ) = K + ( u ) = Q t Q , where the matrix Q is given by (II.10). It is easy to check that anon-zero contribution at the site N is proportional to the identity, hence it doesnot influence the structure of the spectrum. For general K-matrices the solution,by the algebraic Bethe ansatz, was given in [23].Asymptotic expansion of T ( u ) at u → u → ∞ ) results in some matri-ces which have no spectral parameter dependence in accordance with (II.1) (seealso (II.16)) T ( u ) = u − N L − N L − N − · · · L − + O ( u − N + ) . (V.11)Here the constant L-matrices L − j are upper triangular matrices which coincidewith the asymptotic limit λ → + ∞ (II.17) of the R-matrices (II.1), L − j = R − j = P j ˇ R j . Hence, the Yang-Baxter equation (II.2) for the constant R-matrix can bewritten as follows R − i , i + L − i + L − i = L − i L − i + R − i , i + . (V.12)It follows from the formula (II.1) (and also (II.16)) that R − i , i + = P i , i + ˇ R i , i + . So,multiplying the previous equation by the permutation operator P i , i + from theleft one gets h ˇ R i , i + , L − i + L − i i =
0. (V.13)15 OPEN SPIN CHAINIt is then obvious that ρ W ( σ i ) = ˇ R i , i + ≡ ˇ R i , ρ W ( e i ) = µ ( P ( η )) i , i + as therepresentation ρ W of the generators of the BMW algebra W N ( q , q − ) in the space H = ⊗ N C , commute with the generators T − ab of the global (or diagonal) actionof the quantum algebra U q ( o ( )) on the space H (cid:2) ˇ R i , i + , T − (cid:3) = T − = L − N L − N − · · · L − . (V.14)This product of L − j can be represented as the image of a multiple co-productmap ∆ N : U q ( o ( )) → (cid:0) U q ( o ( )) (cid:1) ⊗ N [4] acting on a universal L-matrix L − withentries in U q ( o ( )) on the representation space H T − = ( id ⊗ ρ W )( id ⊗ ∆ N ) L − . (V.15)Analogously, the asymptotic expansion of T ( u ) at u → ∞ yields the matrix T + = L + N L + N − · · · L + (cf. (V.11)). Similar arguments used to show that T − commutes with ˇ R i i + lead to the conclusion that T + commutes as well. Noticethat the generators of the global action of the quantum algebra U q ( o ( )) areentries of T ± . Analogous arguments are valid in the quantum algebra A ( ) caseas well.It is known that in the space H as a space of representation of U q ( o ( )) and W N ( q , q − ) these algebras are mutual centralizers [21]. According to the cen-tralizer property this induces the decomposition of the representation space H into direct sum of irreducible representations of both algebras, being a general-isation of the Schur-Weyl duality. Similarly to the Hecke algebra case, studiedpreviously in [8], one gets H = N ∑ s = V s ⊗ U s , (V.16)where V s is the ( s + ) -dimensional irreducible representation of U q ( o ( )) while U s is some irreducible representation of W N ( q , q − ) . The dimension of an irre-ducible representation of W N ( q , q − ) is equal to the multiplicity m of the cor-responding irreducible representation of centralizer algebra U q ( o ( )) , and viceversa m ( V s ) = dim U s , m ( U s ) = dim V s . (V.17)The dimension of the irreducible representation V s of U q ( o ( )) and the number n of the inequivalent irreducible representations in the decomposition (V.16) arewell known. It follows from the decomposition of the tensor product of the spin1 representations of o ( ) : dim V s = s + n N = N + m N ( V s ) = ∑ j = s , s ± m N − ( V j ) , s = N − N , (V.18)16 OPEN SPIN CHAINtogether with m N ( V ) = m N − ( V ) , m N ( V N − ) = + m N − ( V N − ) = N − m N ( V N ) =
1. However, the number and the dimensions of representa-tions U s of W N ( q , q − ) can be obtained from its Bratteli diagram [9, 21]. For N =
2, 3 the number of existing irreducible representations of W N ( q , q − ) andthose entering into the decomposition of the space of states are the same 3, 4,respectively, while for N > W N than of U q ( o ( )) , for example n ( W ) = n ( U q ( o ( ))) = W N ( q , q − ) H = N − ∑ i = h i , i + , h i , i + = dd λ ˇ R ( λ , η ) | λ = = f ( ˇ R i ) ∈ W N ( q , q − ) . (V.19)According to the QISM, the R-matrices (II.1) and (III.1) being regular at λ = XXZ -model from (II.16) onegets h XXZ = dd λ ˇ R ( λ , η ) | λ = ≃ q ˇ R ( η ) − ˇ R − ( η )= ( q − ) (cid:18) ( q + + q )( P − P ) + P (cid:19) . (V.20)In order to simplify the expressions of the eigenvalues the factor ( q − ) will bedropped from the Hamiltonian density. In the A ( ) -case from (III.9) it follows h A = dd λ ˇ R ( λ , η ) | λ = ≃ q ˇ R ( η ) + q ˇ R − ( η ) = ( q + q ) P + ( + q )( P − P ) .(V.21)The Hamiltonian of the open spin chain with N-sites is then given by H = N − ∑ i = h i , i + . (V.22)As an example let us consider the case of N = W ( q , 1/ q ) is realised in C ⊗ C ⊗ C and the corresponding Hamiltonians are H = h + h . (V.23)17 OPEN SPIN CHAINFrom the relations (V.20, IV.24, IV.28) it follows H XXZ S = ( q + + q ) S , (V.24) H XXZ A = A (V.25)and similarly for the H A (V.21) H A S = ( q + q ) S , (V.26) H A A = − ( + q ) A . (V.27)In the case N = W :two one-dimensional irreps generated by S and A , respectively, the three-dimensional irrep d (corresponding to the one-box Young diagram) and thetwo-dimensional irrep d (corresponding to the three-box Young diagram withtwo rows). Thus the Hamiltonian being restricted to invariant subspaces canhave up to seven distinct eigenvalues. Their multiplicities are obtained fromthe correspondence between the irreps of W and the irreps of U q ( o ( )) : U ( S ) ∼ V , U ( A ) ∼ V U ( d ) ∼ V U ( d ) ∼ V . (V.28)The degeneracies of corresponding energy values are m ( ǫ ( S )) = m ( ǫ ( A )) = m ( ǫ j ( d )) = m ( ǫ k ( d )) = j =
1, 2, 3; k =
1, 2.(V.29)The exact values of the corresponding energy are obtained by direct calculationsand are given below. For the XXZ-model of spin 1 the corresponding expres-sions are ǫ ( S ) = ( q + + q ) , ǫ ( A ) =
2, (V.30) ǫ ( d ) = ǫ ( d ) = ± s + ( q + + q ) ! , (V.31) ǫ ( d ) = ( q + + q ) , ǫ ( d ) = ( q + + q ) . (V.32)18I CONCLUSIONSIn the A ( ) -case the corresponding expressions are ǫ ( S ) = ( q + q ) , ǫ ( A ) = − ( + q ) , (V.33) ǫ ( d ) = ( q + q ) , ǫ ( d ) = ( q + q ) ± s q + q − q + q − q + q + q ! , (V.34) ǫ ( d ) = ( + q )( q − + q ) , ǫ ( d ) = ( + q )( q − q + − q + q ) .(V.35)Although the Hamiltonian density (V.21) has a common factor ( + q ) it is notconvenient to drop it since the expressions of some eigenvalues become morecumbersome. VI Conclusions
Two integrable spin systems invariant with respect to the quantum algebra U q ( o ( )) were considered. These spin systems are defined in the frameworkof the QISM by trigonometric R-matrices related to the quantum affine algebras U q ( d o ( )) and U q ( A ( ) ) . It was shown that the mutually commuting integrals ofmotion belong to the image of the BMW algerba W N ( q , q − ) in a reducible rep-resentation on the space of states H = ⊗ N C . The symmetry algebra and theBMW algebra centralize each other in the representation space, and this deter-mines the structure of the spin system spectra.We point out that there is a series of quantum super-algebras U q ( osp ( | n )) [21] with corresponding R-matrices in the vector representation defining gener-ators of the Birman-Wenzl-Murakami algebra W N ( − q , − q − n ) similarly to theone considered in our paper. From the representation of the BMW algebra givenby the R-matrix one can get two spectral parameter dependent R-matrices bythe Yang-Baxterization procedure (IV.7). Each of them yields solutions of thestandard and ( Z graded) super-Yang-Baxter equations [17]. This results in apossibility to construct four series of integrable spin chains whose structure ofthe spectrum is similar to the one considered in the previous section.19EFERENCES VII Acknowledgments
We acknowledge useful discussions with V. Tarasov. This work was supportedby RFBR grant 07-02-92166-NZNI_a, 08-01-00638 and the FCT projectPTDC/MAT/69635 /2006.
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