Temperley-Lieb K-matrices

T emp erley-Lieb K- matrices A. Lima-Sa n tos Universidade F e der al de S˜ ao Carlos, Dep artamento de F ´ ısic a Caixa Postal 676, CEP 13569-905 S˜ ao Carlos, B r asil Abstract This w ork concerns to the studies of b oundary integrabilit y of th e v ertex mo dels from represen- tations of the T emp erle y-Lieb algebra associated with the quantum group U q [ X n ] for the affine Lie algebras X n = A (1) 1 , B (1) n , C (1) n and D (1) n . A systematic computation me tho d i s used to constructed solutions of the b oundary Y ang-Baxter equations. W e find a 2 n 2 + 1 free parameter solution fo r A (1) 1 spin-( n − 1 / 2) and C (1) n vertex models. It turns that for A (1) 1 spin- n , B (1) n and D (1) n vertex models, the sol ution has 2 n 2 + 2 n + 1 free parameters. Nov em ber 16, 2018 1 In tro du c ti on The sea rc h for integrable mo dels thro ugh solutions of the Y ang–Ba x ter equation [1, 2 , 3] R 12 ( u − v ) R 13 ( u ) R 23 ( v ) = R 23 ( v ) R 13 ( u ) R 12 ( u − v ) (1.1) has b een p erformed b y the quantum gro up approach in [4]. In this wa y , the R -matrices corre s ponding to vector r epresen tations of all nonexc eptional affine Lie a lgebras have bee n determined in [5]. A similar a pproac h is desirable for finding solutions of the b oundary Y ang – Baxter equation [6, 7] wher e the b oundary weights follo w from K -ma trices whic h satisfy a pair of equations, namely th e r eflection equation R 12 ( u − v ) K − 1 ( u ) R t 1 t 2 12 ( u + v ) K − 2 ( v ) = K − 2 ( v ) R 12 ( u + v ) K − 1 ( u ) R t 1 t 2 12 ( u − v ) (1.2) and the dual reflectio n equation R 12 ( − u + v )  K + 1  t 1 ( u ) M − 1 1 R t 1 t 2 12 ( − u − v − 2 ρ ) M 1  K + 2  t 2 ( v ) =  K + 2  t 2 ( v ) M 1 R 12 ( − u − v − 2 ρ ) M − 1 1  K + 1  t 1 ( u ) R t 1 t 2 12 ( − u + v ) . (1.3) In this case duality supplies a r elation b et ween K − and K + [8] K + ( u ) = K − ( − u − ρ ) t M , M = V t V (1.4) Here t de no tes tra nsposition a nd t i denotes tr ansposition in the i-th s pace. V is the cr ossing matr ix and ρ the cros sing pa r ameter, bo th b eing sp ecific to each mo del [9]. With t his goal in mind, the study of boundar y quantum groups w as initiated in [10]. This s tudy hav e b een us ed to determine A (1) 1 reflection matrices for ar bitrary spin [11], and the A (2) 2 and some A (1) n reflection matrices w ere deriv ed a gain in [12]. Reflection solutions from R -matrices corresp onding to vector representations of Y angians and sup er-Y angians were presented in [13]. How ev er, as obser v ed in [12], an indep e nden t systematic metho d of constructing the b oundary quantum gro up ge ne r ators is not yet a v ailable. In contrast to the bulk cas e [5], one cannot exploit bo unda ry affine T o da field theory , since appropria te classical integrable bo undary conditions are not yet k no wn [14]. How ever, the algebr aic str uctures related to refle c tio n e quations are well-kno wn [15] and a b oundary quantum gr oup appro ac h was recently used in [16] to derive the clas s ification of the cons ta n t K -matrix (without sp ectral para meter) solutions for the T e mperley-Lieb ( TL ) mo dels. The main result, already 1 po in ted in [17], is that for a g iv en consta n t TL R - ma trix, the corr esponding co nstan t K -matrix satisfies a qua dratic r elation: q K 2 + c 1 K + ( q + q − 1 ) − 1 ( c 2 1 + q c 2 ) I = 0 (1.5) with a ppropriate central element s c 1 and c 2 . F rom this result, the Y ang-Ba xterization pro cedure, as us e d in [18, 1 9, 2 0 ], allows to obtain sp ectral- parameter dependent re flection matrices: K ( u ) = u 2 K − 1 u 2 K − 1 + cI (1.6) with a n arbitrary cent ral element c . Independently , there has been an in creasing amoun t of effort to w ards the understanding of tw o- dimensional in tegrable theories with b oundaries via solutions of the functional equation (1.2). In field theory , a tt ent ion is fo cused o n the b o undary S matrix [21, 23]. In statistical mechanics, the emphasis has b een laid on deriving all solutions of (1.2) b ecause different K - matrices lead to different universalit y classes of surface critica l b eha vior [24] and allow the c a lculation o f v arious surface cr itical phenomena, bo th at a nd aw a y fro m criticality [25]. Although b eing a hard tas k, the direct computation has b een used to solve (1.2 ). F or instance, we men tion the s olutions with R matrix based in non-exce ptio nal Lie algebras [22, 26] and sup eralgebra s [27, 28]. The r egular K -ma trices for the exceptiona l U q [ G 2 ] vertex mo del w ere obtained in [29] . Many diagonal so lutions for face a nd vertex mo dels as sociated with affine Lie a lgebras were presented in [2 5]. F or A–D–E interaction-round face ( IRF ) mo dels, diagona l and some non- diagonal solutions were presented in [30]. Reflection matrice s for Andrews– Baxter–F or r ester mo dels in the RS O S/SOS repre s en ta tion were presented in [31]. Apart from these c - num ber solutions of the reflectio n equa tions ther e must also exist non trivia l solutio ns that include b oundary degr ee of freedom as were derived for the s ine-Gordon theory in [32 ] and the pro jected K -matrices [33]. Motived by the results presented in [34] we will a gain touch this issue in o rder to include once more the TL la ttice mo dels [35 ] ar ising from the quantum gr oup U q [ X n ] for X n = A (1) 1 , B (1) n , C (1) n and D (1) n [36]. The TL algebra is very useful in the study of tw o dimensiona l lattice statistical mechanics. It provided an alge br aic fra mework for cons tr ucting a nd ana lyzing different types of integrable la ttice mo dels, such as Q -state Potts model, IR F model, O ( n ) lo op mo del, six-vertex mo del, etc. [37]. 2 W e hav e o rganized this pap er as follows. In Section 2 the mo del is presented, in Section 3 we choose the reflection eq ua tions a nd their s olutions. The Sectio n 4 is rese r v ed for the conclusion. 2 The mo del F rom the r epresen tation of the TL a lgebra, one ca n build solv able vertex mo dels with the R ope rator defined b y R ( u ) = x 1 ( u ) I + x 2 ( u ) U , (2.1) where I is the identit y op erator and U is the TL pro jector. Her e u is the sp ectral para meter and the anisotropic parameter η is choos e so that x 1 ( u ) = sinh( η − u ) sinh η , x 2 ( u ) = sinh u sinh η , 2 cosh η = T r U . (2.2) Setting R j ( u ) = 1 ⊗ · · · 1 ⊗ R ( u ) | {z } j,j +1 ⊗ 1 · · · ⊗ 1 (2.3) one ca n show that the Y a ng-Baxter equation R j +1 ( u ) R j ( u + v ) R j +1 ( v ) = R j ( v ) R j +1 ( u + v ) R j ( u ) (2.4) is v alid due to the de finitio n r e lations o f the TL a lgebra U 2 j = 2 cosh η U j U j U j ± U j = U j U i U j = U j U i | i − j | > 1 (2.5) F or the affine Lie alge br as A (1) 1 , B (1) n , C (1) n and D (1) n i.e., the q -deformations o f the spin- s repr esen tation of sl (2) and the vector repres en ta tion of so (2 n + 1 ), sp (2 n ) and so (2 n ), the corresp onding TL pro jector, using the notatio n a nd r esults of [24], has the form U = N X i,j =1 ε ( i ) ε ( j ) q − <ǫ i + ǫ j , ρ> e i,j ⊗ e i ′ ,j ′ (2.6) 3 where e i,j is the ma trix unit (e i,j v k = δ j,k v i ) a nd w e hav e used the co njuga ted index a ′ = N + 1 − a . Here, one has to take into ac c o un t the s e t of or tho normal vectors < ǫ i , ǫ j > = δ i,j , the sign ε ( i ) and ρ , the ha lf-sum of p ositive ro ots o f the q -deformed affine Lie alg e bras in order to write explicitly the TL pro jecto r for each mo de l: • A (1) 1 : The U q [ sl (2)] spin- s T emp erley-Lieb mo del U = N X i =1 N X j =1 ( − 1) i + j q i + j − N − 1 e i,j ⊗ e i ′ ,j ′ 2 cosh η = [2 s + 1 ] (2.7) where N = 2 s + 1 ( s = 1 2 , 1 , 3 2 , 2 , ... ). W e r e ma rk the us e of the quantum num ber notation [ n ] = ( q n − q − n ) / ( q − q − 1 ) in the trace of the U pro jectors. • B (1) n ( n ≥ 2 ) : The U q [ so (2 n + 1)] T emper ley-Lieb mo del U = n X i =1 n X j =1 q i + j − 2 n − 1 e i,j ⊗ e i ′ ,j ′ − n X i =1 q i − n − 1 / 2 e i,n +1 ⊗ e i ′ ,n +1 + n X i =1 2 n +1 X j = n +2 q i + j − 2 n − 2 e i,j ⊗ e i ′ ,j ′ − n X j =1 q j − n − 1 / 2 e n +1 ,j ⊗ e n +1 ,j ′ +e n +1 ,n +1 ⊗ e n +1 ,n +1 − 2 n +1 X j = n +2 q j − n − 3 / 2 e n +1 ,j ⊗ e n +1 ,j ′ + 2 n +1 X i = n +2 n X j =1 q i + j − 2 n − 2 e i,j ⊗ e i ′ ,j ′ − 2 n +1 X i = n +2 q i − n − 3 / 2 e i,n +1 ⊗ e i ′ ,n +1 + 2 n +1 X i = n +2 2 n +1 X j = n +2 q i + j − 2 n − 3 e i,j ⊗ e i ′ ,j ′ 2 cosh η = [2 n − 1][ n + 1 2 ] [ n − 1 2 ] (2.8) • C (1) n ( n ≥ 1): The U q [ sp (2 n )] T emp erley-Lieb mo del U = n X i =1 , n X j =1 q i + j − 2 n − 2 e i,j ⊗ e i ′ ,j ′ − n X i =1 , 2 n X j = n +1 q i + j − 2 n − 1 e i,j ⊗ e i ′ ,j ′ − N X i = n +1 , n X j =1 q i + j − 2 n − 1 e i,j ⊗ e i ′ ,j ′ + 2 n X i = n +1 , 2 n X j = n +1 q i + j − 2 n e i,j ⊗ e i ′ ,j ′ 2 cosh η = [ n ][2 n + 2] [ n + 1 ] (2.9) 4 • D (1) n ( n ≥ 3 ): T he U q [ so (2 n )] T emp erley-Lieb mo del U = n X i =1 , n X j =1 q i + j − 2 n e i,j ⊗ e i ′ ,j ′ + n X i =1 2 n X j = n +1 q i + j − 2 n − 1 e i,j ⊗ e i ′ ,j ′ + 2 n X i = n +1 n X j =1 q i + j − 2 n − 1 e i,j ⊗ e i ′ ,j ′ + 2 n X i = n +1 2 n X j = n +1 q i + j − 2 n − 2 e i,j ⊗ e i ′ ,j ′ 2 cosh η = [ n ][2 n − 2] [ n − 1 ] (2.10) W e also have to consider the p erm uted op erator R = P R which is regular satisfying PT -symmetry , unitarity and cr ossing symmetry R 12 (0) = P , R t 1 t 2 12 ( u ) = P R 12 ( u ) P = R 21 ( u ) , R 12 ( u ) R 21 ( − u ) = x 1 ( u ) x 1 ( − u ) I , R 21 ( u ) = κ ( V ⊗ 1) R t 2 12 ( − u − ρ )( V ⊗ 1) − 1 (2.11) where ρ = − η is the cr ossing parameter , κ = ( − 1) 2 s for A (1) 1 , κ = − 1 for C (1) n and κ = 1 for B (1) n and D (1) n . P is the p erm utation op erator: P ( a ⊗ b ) = b ⊗ a for a n y vectors a, b . The crossing matrices V for the TL mo dels a r e sp ecified in [2 4 ] b y V i,j = ε ( i ) q − <ǫ i ,ρ> δ i ′ ,j (2.12) How ever, for the iso morphism (1.4) w e only need of the dia gonal matrix M = V t V for each mo del: M i,i = q 2 i − 2 s − 2 i = 1 , ..., 2 s + 1 for A (1) 1 spin − s (2.13) M i,i =    q 2 i − 2 n − 1 i = 1 , ..., n 1 i = n + 1 q 2 i − 2 n − 3 i = n + 2 , ..., 2 n + 1 for B (1) n (2.14) M i,i =  q 2 i − 2 n − 2 i = 1 , ..., n q 2 i − 2 n i = n + 1 , ..., 2 n for C (1) n (2.15) and M i,i =    q 2 i − 2 n i = 1 , ..., n 1 i = n + 1 , n + 2 q 2 i − 2 n − 2 i = n + 3 , ..., 2 n + 1 for D (1) n (2.16) 5 The Hamiltonian limit R ( u ) = I + u ( α − 1 H + β I ) (2.17) with α = s inh η , β = − coth η leads to the quantum s pin chains H = N − 1 X k =1 U k,k +1 + bt (2.18) where, instead of p erio dic b oundary co ndition, w e are taking into acco un t the ex istence of in teg r able bo undary terms bt [7], derived from the K − and K + matrices presented in the next sections. 3 The reflection matrices In the r eflection equation (1.2) we remark the notatio n K − 1 = K − ⊗ I , K − 2 = I ⊗ K − , R 12 = R and R t 1 t 2 12 = P R P . F or a given R -matrix the unknown is the N by N matrix K − ( u ) satisfying the normal condition K − (0) = I . The dimension N is equal to 2 s + 1 , 2 n + 1 , 2 n and 2 n + 1 fo r A (1) 1 , B (1) n , C (1) n and D (1) n , resp ectiv ely . Substituting K − ( u ) = N X i,j =1 k i,j ( u )e i,j (3.1) and R ( u ) = P [ x 1 ( u ) I + x 2 ( u ) U ] into (1.2 ), we will hav e N 4 functional equations for the k i,j elements, many of them not indep enden t equations. In order to solve these functional equations, we shall pro ceed as follo ws. Fir st we consider the ( i, j ) compo nen t of the matrix equa tio n (1.2). By differentiating it with resp ect to v and taking v = 0, we get algebra ic equations inv o lving the single v aria ble u a nd N 2 parameters β i,j = dk i,j ( v ) dv | v =0 , i, j = 1 , 2 , ..., N (3.2) Analyzing the refection equations o ne can see that they p ossess a sp ecial structur e. Several equations exist inv olving only tw o non-dia gonal element s. They ca n b e solved by the relations k i,j ( u ) = β i,j β 1 ,N k 1 ,N ( u ) ( i 6 = j = { 1 , 2 , ..., N } ) (3.3) 6 W e th us left with several equa tions in v olving tw o diagonal elements and k 1 ,N ( u ). Suc h equations a r e solved by the relations k i,i = k 1 , 1 ( u ) + ( β i,i − β 1 , 1 ) k 1 ,N ( u ) β 1 ,N ( i = 2 , 3 , ..., N ) . (3.4) Finally , we c a n use the eq ua tion (1 , N ) in order to find the element k 1 , 1 ( u ): k 1 , 1 ( u ) = k 1 ,N ( u ) β 1 ,N [ x 2 ( u ) cosh η + x 1 ( u )]  x 1 ( u ) x ′ 2 ( u ) − x ′ 1 ( u ) x 2 ( u ) x 2 ( u ) − 1 2 x 1 ( u )( β N ,N − β 1 , 1 + Ψ 1 ,N ) − 1 2 x 2 ( u ) N X j =2 ( β j,j − β 1 , 1 ) M j,j    (3.5) where M j,j are g iv en in (2.13–2 .16) and Ψ 1 ,N belo ngs to the set of new relatio ns of the para meter s β i,j defined b y Ψ i,j = 1 β i,j N − 1 X k =2 β i,k β k,j i 6 = j = 1 , ..., N (3.6) The prime in the Boltzmann w eights x i ( u ) means its first deriv ativ e with resp ect to u . Now, substituting these ex pr essions int o the remained equations ( i , j ), we are left with factor ed equations of the form: F a ( β i,j ) q p x 1 ( u ) x 2 ( u ) k 1 ,N ( u ) = 0 (3.7) where each factor F a ( β i,j ) don’t dep end on the weights x i ( u ) nor o f the corresp onding quantum g roup q -parameter. It means that they are the same in a ll four mo dels b y we conside r ing. Therefore all computation us ed in [34] give us a general pro cedure: First, we c o llect all matrix elemen t ( i, j ) of (1.2) in blo c k s of four equations [38] B [ i, j ] = { ( i, j ) , ( j, i ) , ( i ′′ , j ′′ ) , ( j ′′ , i ′′ )) } i = 1 , ..., N , j = i, ..., i ′′ (3.8) where a ′′ = N 2 + 1 − a . F rom the first equation of the blo cks B [ j, N − 1] , j = 2 , ..., N − 2 one ca n fix N − 1 dia gonal pa r ameters β j,j = β 1 , 1 + Ψ 1 ,N − Ψ j,N j = 2 , 3 , ..., N − 1 (3.9) and the first equation of the blo ck B [ N , N + 1] fixes the parameter β N ,N β N ,N = β 1 , 1 + Ψ 1 ,N − 1 − Ψ N − 1 ,N (3.10) 7 All equations from the blo ck B [1 , k ] to the blo c k B [ N − 1 , k ] are now substituted by N ( N − 1) / 2 sy mmetr ic relations Ψ j,i = Ψ i,j , j > i (3.11) and 2( N − 3) relations involving fo ur Ψ i,j functions Ψ 2 ,j = Ψ 2 , 3 + Ψ 1 ,j − Ψ 1 , 3 , Ψ 3 ,j = Ψ 2 , 3 + Ψ 1 ,j − Ψ 1 , 2 , j = 4 , ..., N , (3.12) The remained equations contained in the blo c k B [ N , k ] are rewritten by ( N − 3)( N − 4) / 2 relations inv o lving six Ψ i,j functions Ψ i,j = Ψ 1 ,i + Ψ 1 ,j + Ψ 2 , 3 − Ψ 1 , 2 − Ψ 1 , 3 , i = 4 , ..., N − 1 , j = i + 1 , ..., N (3.13) and 2 N − 3 relations inv olving the diagona l β k,k parameters , Ψ 1 ,N and a new function Θ j,j , Θ j,j = Θ N ,N + ( β N ,N − β j,j )( β j,j − β 1 , 1 − Ψ 1 ,N ) , j = 2 , 3 , ..., N − 1 , Θ j ′ ,j ′ = Θ 1 , 1 + ( β 1 , 1 − β j ′ ,j ′ )( β j ′ ,j ′ − β N ,N − Ψ 1 ,N ) , j = 2 , 3 , ..., N − 1 , Θ N ,N = Θ 1 , 1 − ( β 1 , 1 − β N ,N )Ψ 1 ,N , (3.14) where j ′ = N + 1 − j and Θ j,j = X k 6 = j β j,k β k,j (3.15) F rom (3.11 ) to (3.14) one can account N 2 − 3 c o nstrain t equations but, a ft er the substitution of the relations (3.9) and (3.10) int o (3 .14), we o nly need to lo ok at the symmetric relations (3.11). How ever, for co mputatio nal co n venience, we added all r elations with four Ψ i,j functions (3 .12). Therefore, our final task is lo ok for solutions of N ( N − 2) constraint equations. F rom thes e relations w e hav e fixed N ( N − 2) / 2 − 1 / 2 parameter s β fo r the TL mo dels with N o dd and N ( N − 2) / 2 − 1 para meters for the TL models with N even. T aking into a ccoun t tha t the para meter β 1 , 1 is determined by the no rmal c ondion, we e nd the calculus with K -matr ix solutions o f (1.2) with N 2 / 2 + 1 / 2 fre e para meters β i,j if N is o dd and with N 2 / 2 + 1 free pa r ameters if N is even. Now, le t us descr ibe a b out the co rrespo nding diagona l K -matr ix so lutio ns. 8 3.1 The diagonal solutions T aking into acc oun t only the diagonal entries, the refle c tion equatio ns ar e so lv ed when we find all matrix elements k j,j ( u ), j = 2 , ..., N as function of k 1 , 1 ( u ), provided that the diagonal par ameters β j,j satisfy ( N − 1)( N − 2 ) / 2 constraint equations of the type ( β N ,N − β i,i ) ( β N ,N − β j,j ) ( β j,j − β i,i ) = 0 ( i 6 = j 6 = N ) (3.16) F rom (3 .16) we find solutions with only tw o type of entries. Let us normalize one of them to be equal to 1 such that the other one has the form k p,p ( u ) = − β p,p x 2 ( u ) [∆ 1 x 2 ( u ) + x 1 ( u )] + 2 [ x 1 ( u ) x ′ 2 ( u ) − x ′ 1 ( u ) x 2 ( u )] β p,p x 2 ( u ) [∆ 2 x 2 ( u ) + x 1 ( u )] − 2 [ x 1 ( u ) x ′ 2 ( u ) − x ′ 1 ( u ) x 2 ( u )] (3.17) with ∆ 1 + ∆ 2 = 2 cosh η . Ident ifying the diagonal p ositions of the K -matrix with the matrix elements of the M -matrix (2.13– 2.16), (1 , 2 , ..., N ) ⊜ ( M 1 , 1 , M 2 , 2 , ..., M N ,N ) one can see that ∆ 1 is the s um of the M j,j corres p onding to the p ositions of the entries 1 and ∆ 2 is the sum of the M j,j corres p onding to the po sitions of the entries k p,p ( u ). Denoting the diagonal so lutions by K [ r ] a where a = ( a 1 , a 2 , ..., a N ) with a i = 0 if k i,i ( u ) = 1 or a i = 1 if k i,i ( u ) = k pp ( u ) a nd r is the num b er of the entries k p,p ( u ) distributed on diago nal p ositions and p b eing the first p osition with the entry different from 1. Th us, we hav e counted Z = N − 1 X r =1 N ! r ! ( N − r )! (3.18) for the num ber of diagonal K − matrix solutions with one free parameter. The dua l equation (1.3 ) is solved by the K + matrices v ia the isomorphism (1 .4) with ρ = − η and the matrix M s pecified by (2.13–2 .16). Here we note that trace of ea c h diag o nal M -matrix is eq ua l to 2 cosh η . Now, we explicitly show these computatio ns for the first mo dels. Before, we can use the identit y x 2 ( u )[ x 2 ( u ) cosh η + x 1 ( u )] x 1 ( u ) x ′ 2 ( x ) − x ′ 1 ( u ) x 2 ( u ) = sinh( u ) cosh( u ) (3.19) in order to simplify our pr esen tation. F rom the s o lution ((3.3) to (3.5)) one can s ee k 1 ,N ( u ) a s an ar bitr ary function satisfying the nor mal conditio n. Ther efore, the choice k 1 ,N ( u ) = 1 2 β 1 ,N sinh(2 u ) (3.20) 9 do esn’t implies any r estriction as c o mpared to the g e neral case. 3.2 The A (1) 1 spin- 1 2 and C (1) 1 T emp erley-Lieb K-matrices F or these mo dels we hav e the w ell- kno wn three free para meter solution for the U q [ sl (2)] spin- 1 2 mo del [21, 22] K − ( u ) =   k 1 , 1 ( u ) 1 2 β 1 , 2 sinh(2 u ) 1 2 β 2 , 1 sinh(2 u ) k 1 , 1 ( u ) + 1 2 ( β 2 , 2 − β 1 , 1 ) sinh(2 u )   (3.21) Using the identit y (3.19 ) a nd (3.20), the expr ession for k 1 , 1 ( u ) (3.5) has a simplified form k 1 , 1 ( u ) = 1 − 1 2 ( β 2 , 2 − β 1 , 1 ) [ x 1 ( u ) + q x 2 ( u )] x 2 ( u ) sinh η (3.22) where β 1 , 2 , β 2 , 1 and β 2 , 2 being the free parameters a nd 2 cosh η = q + q − 1 . Moreover, w e find that the U q [ sp (2)] TL mo del has the sa me K -matrix form but, with k 1 , 1 ( u ) = 1 − 1 2 ( β 2 , 2 − β 1 , 1 )  x 1 ( u ) + q 2 x 2 ( u )  x 2 ( u ) sinh η (3.23) since that now 2 co s h η = q 2 + q − 2 . The entries of the dia gonal so lutions k 1 , 1 ( u ) and k 2 , 2 ( u ) are g iven by (3.17) and we have tw o s o lutions for e a c h mo del K [1] (1 , 0) =  k 1 , 1 ( u ) 0 0 1  , ∆ 1 = M 2 , 2 ∆ 2 = M 1 , 1 K [1] (0 , 1) =  1 0 0 k 2 , 2 ( u )  , ∆ 1 = M 1 , 1 ∆ 2 = M 2 , 2 (3.24) where M 1 , 1 = q − 1 , M 2 , 2 = q for U q [ sl (2)] spin- 1 2 mo del a nd M 1 , 1 = q − 2 , M 2 , 2 = q 2 for U q [ sp (2)] mo del. Of cour se, in bo th mo dels K [1] (1 , 0) and K [1] (0 , 1) are equiv alen t by the exchange q ↔ q − 1 . 3.3 The A (1) 1 spin- 1 T emp er ley- Lieb K matrices F or the biquadratic mo del [39, 4 0 ], it follows from (3.3) and (3.4 ) that K − ( u ) =       k 1 , 1 ( u ) 1 2 β 1 , 2 sinh(2 u ) 1 2 β 1 , 3 sinh(2 u ) 1 2 β 2 , 1 sinh(2 u ) k 1 , 1 ( u ) + 1 2 ( β 2 , 2 − β 1 , 1 ) sinh(2 u ) 1 2 β 2 , 3 sinh(2 u ) 1 2 β 3 , 1 sinh(2 u ) 1 2 β 3 , 2 sinh(2 u ) k 1 , 1 ( u ) + 1 2 ( β 3 , 3 − β 1 , 1 ) sinh(2 u )       (3.25) 10 where k 1 , 1 ( u ) is given by (3.8), k 1 , 1 ( u ) = 1 − 1 2 { ( β 3 , 3 − β 1 , 1 ) [ x 1 ( u ) + M 3 , 3 x 2 ( u )] + x 1 ( u )Ψ 1 , 3 +( β 2 , 2 − β 1 , 1 ) M 2 , 2 x 2 ( u ) } x 2 ( u ) sinh η . (3.26) The diag onal pa rameters are fixed by the co ns train t equa tions (3.9) a nd (3.10) β 2 , 2 = β 1 , 1 + Ψ 1 , 3 − Ψ 2 , 3 = β 1 , 1 + β 1 , 2 β 2 , 3 β 13 − β 21 β 13 β 23 (3.27) β 3 , 3 = β 1 , 1 + Ψ 2 , 1 − Ψ 2 , 3 = β 1 , 1 + β 1 , 3 β 3 , 2 β 1 , 2 − β 2 , 1 β 1 , 3 β 2 , 3 , ( 3.28) and β 11 is fixed b y the nor mal condition. Mor eo v er, all remained constraint equations are solved by the relation β 3 , 1 = β 1 , 3 β 3 , 2 β 2 , 1 β 1 , 2 β 2 , 3 or Ψ 3 , 1 = Ψ 1 , 3 (3.29) and we hav e get a five free par ameter solution. Here 2 cosh η = q − 2 + 1 + q 2 . It means that M 11 = q − 2 , M 22 = 1 and M 33 = q 2 . Among several po ssibilities, we made the choice β 1 , 2 , β 1 , 3 , β 2 , 1 , β 2 , 3 and β 3 , 2 for the free parameter s. The corresp ondig diagonal s olutions are six, ha lf of them w ith one entry different from 1 K [1] (1 , 0 , 0) =   k 1 , 1 ( u ) 0 0 0 1 0 0 0 1   , ∆ 1 = M 2 , 2 + M 3 , 3 , ∆ 2 = M 1 , 1 (3.30) K [1] (0 , 1 , 0) =   1 0 0 0 k 2 , 2 ( u ) 0 0 0 1   , ∆ 1 = M 1 , 1 + M 3 , 3 , ∆ 2 = M 2 , 2 (3.31) K [1] (0 , 0 , 1) =   1 0 0 0 1 0 0 0 k 3 , 3 ( u )   , ∆ 1 = M 1 , 1 + M 2 , 2 , ∆ 2 = M 3 , 3 (3.32) and three further diago nal so lutio ns with tw o equa l e n tr ie s different from unity K [2] (1 , 1 , 0) =   k 1 , 1 ( u ) 0 0 0 k 1 , 1 ( u ) 0 0 0 1   ∆ 1 = M 3 , 3 , ∆ 2 = M 1 , 1 + M 2 , 2 (3.33) K [2] (1 , 0 , 1) =   k 1 , 1 ( u ) 0 0 0 1 0 0 0 k 1 , 1 ( u )   ∆ 1 = M 2 , 2 , ∆ 2 = M 1 , 1 + M 3 , 3 (3.34) 11 K [2] (0 , 1 , 1) =   1 0 0 0 k 2 , 2 ( u ) 0 0 0 k 2 , 2 ( u )   ∆ 1 = M 1 , 1 , ∆ 2 = M 2 , 2 + M 3 , 3 (3.35) where the entries k p,p ( u ) are given by (3.1 7). Here w e notice again that the difference b et ween the diagona l en tries come from the partitions of ∆ 1 + ∆ 2 = q − 2 + 1 + q 2 and the e q uiv ale nc e b et w een them due to the symmetry q ↔ q − 1 . W e also note that if q 2 is replac ed b y q , the eq uiv alence B (1) 1 ≃ A (1) 1 spin-1 is manifested in all expressions ab ov e, since tha t 2 co sh η = q − 1 + 1 + q for B (1) 1 . These diago nal so lutions were recently used in [42] to study the sp ectrum of the spin-1 TL spin chain with integrable o pen bo undary conditions. 3.4 The A (1) 1 spin- 3 2 and C (1) 2 T emp erley-Lieb K matrices F or U q [ sl (2)] spin- 3 2 and U q [ sp (4)] mo dels, we hav e from (3 .3 ) to (3.8) the following non- diagonal entries k i,j ( u ) = 1 2 β i,j sinh(2 u ) , ( i 6 = j = 1 , 2 , 3 , 4 ) (3.36) and the diago nal one k i,i ( u ) = k 1 , 1 ( u ) + 1 2 ( β i,i − β 1 , 1 ) sinh(2 u ) , ( i = 2 , 3 , 4) (3.37) with k 1 , 1 ( u ) = 1 − 1 2 { ( β 4 , 4 − β 1 , 1 ) [ x 1 ( u ) + M 4 , 4 x 2 ( u )] + x 1 ( u )Ψ 1 , 4 +[( β 2 , 2 − β 1 , 1 ) M 2 , 2 + ( β 3 , 3 − β 1 , 1 ) M 3 , 3 ] x 2 ( u ) } x 2 ( u ) sinh η . (3.38) where M 1 , 1 = q − 3 , M 2 , 2 = q − 1 , M 3 , 3 = q and M 4 , 4 = q 3 for U q [ sl (2)] spin- 3 2 mo del and M 1 , 1 = q − 4 , M 2 , 2 = q − 2 , M 3 , 3 = q 2 and M 4 , 4 = q 4 for U q [ sp (4)] mo del. F rom the equations (3.9) and (3.10 ) w e choo se to fix the following diagonal pa rameters: β 2 , 2 = β 1 , 1 + Ψ 1 , 4 − Ψ 2 , 4 = β 1 , 1 + β 1 , 2 β 2 , 4 + β 1 , 3 β 3 , 4 β 1 , 4 − β 2 , 1 β 1 , 4 + β 2 , 3 β 3 , 4 β 2 , 4 , β 3 , 3 = β 1 , 1 + Ψ 1 , 4 − Ψ 3 , 4 = β 1 , 1 + β 1 , 2 β 2 , 4 + β 1 , 3 β 3 , 4 β 1 , 4 − β 3 , 1 β 1 , 4 + β 3 , 2 β 2 , 4 β 3 , 4 , β 4 , 4 = β 1 , 1 + Ψ 1 , 3 − Ψ 3 , 4 = β 1 , 1 + β 1 , 2 β 2 , 3 + β 1 , 4 β 4 , 3 β 1 , 3 − β 3 , 1 β 1 , 4 + β 3 , 2 β 2 , 4 β 3 , 4 . (3.39) 12 All r emained constraint equations are solved by the choice β 4 , 1 = β 4 , 2 β 2 , 1 + β 4 , 3 β 3 , 1 β 1 , 2 β 2 , 4 + β 1 , 3 β 3 , 4 β 1 , 4 or Ψ 4 , 1 = Ψ 1 , 4 β 3 , 2 = − β 3 , 1 β 1 , 2 + β 3 , 4 β 4 , 2 β 1 , 2 β 2 , 4 + β 1 , 3 β 3 , 4 β 1 , 4 or Ψ 3 , 2 = − Ψ 1 , 4 β 2 , 3 = − β 2 , 1 β 1 , 3 + β 2 , 4 β 4 , 3 β 1 , 2 β 2 , 4 + β 1 , 3 β 3 , 4 β 1 , 4 or Ψ 2 , 3 = − Ψ 1 , 4 (3.40) It mea ns that w e hav e found a K − matrix with nine fr ee parameters. There a re four diagonal so lutions with one entry k p,p ( u ) and three igual to unit y K [1] (1 , 0 , 0 , 0) : ∆ 1 = M 1 , 1 , ∆ 2 = M 2 , 2 + M 3 , 3 + M 4 , 4 . . . K [1] (0 , 0 , 0 , 1) : ∆ 1 = M 4 , 4 , ∆ 2 = M 1 , 1 + M 2 , 2 + M 3 , 3 (3.41) There a re s ix diago nal so lutions with tw o entry k p,p ( u ) and tw o igual to unity K [2] (1 , 1 , 0 , 0) : ∆ 1 = M 1 , 1 + M 2 , 2 , ∆ 2 = M 3 , 3 + M 4 , 4 . . . K [2] (0 , 0 , 1 , 1) : ∆ 1 = M 3 , 3 + M 4 , 4 , ∆ 2 = M 1 , 1 + M 2 , 2 (3.42) and mor e four so lutions with three entries k p,p ( u ) and one equal to unit y K [3] (1 , 1 , 1 , 0) : ∆ 1 = M 1 , 1 + M 2 , 2 + M 3 , 3 , ∆ 2 = M 4 , 4 . . . K [3] (0 , 1 , 1 , 1) : ∆ 1 = M 2 , 2 + M 3 , 3 + M 4 , 4 , ∆ 2 = M 1 , 1 (3.43) Remem ber that in these 1 4 diago nal solutions we have ∆ 1 + ∆ 2 = q − 3 + q − 1 + q + q 3 for the A (1) 1 spin- 3 2 mo del and ∆ 1 + ∆ 2 = q − 4 + q − 2 + q 2 + q 4 for the C (1) 2 mo del. 3.5 The A (1) 1 spin- 2 and B (1) 2 T emp erley-Lieb K matrices F or N = 5, the matrix ele men ts ar e k i,j ( u ) = 1 2 β i,j sinh(2 u ) , ( i 6 = j = 1 , ..., 5) (3.44) 13 and k i,i ( u ) = k 1 , 1 ( u ) + 1 2 ( β i,i − β 1 , 1 ) sinh(2 u ) , ( i = 2 , ..., 5) . (3.45) where k 1 , 1 ( u ) = 1 − 1 2 { ( β 5 , 5 − β 1 , 1 ) [ x 1 ( u ) + M 5 , 5 x 2 ( u )] + Ψ 1 , 5 x 1 ( u ) +[( β 2 , 2 − β 1 , 1 ) M 2 , 2 + ( β 3 , 3 − β 1 , 1 ) M 3 , 3 + ( β 4 , 4 − β 1 , 1 ) M 4 , 4 ] x 2 ( u ) } x 2 ( u ) sinh η (3.4 6) The diagonal pa r ameters are given by (3.9) β 2 , 2 = β 1 , 1 + Ψ 1 , 5 − Ψ 2 , 5 , β 3 , 3 = β 1 , 1 + Ψ 1 , 5 − Ψ 3 , 5 β 4 , 4 = β 1 , 1 + Ψ 1 , 5 − Ψ 4 , 5 , (3.47) and by (3.10 ) β 5 , 5 = β 1 , 1 + Ψ 1 , 4 − Ψ 4 , 5 (3.48) Moreov er, we have 5(5 − 1) / 2 = 10 symmetric relations Ψ j,i = Ψ i,j ( j > i ) (3.49) more 2(5 − 3) = 4 relations co m four Ψ i,j Ψ 2 , 4 = Ψ 2 , 3 + Ψ 1 , 4 − Ψ 1 , 3 , Ψ 2 , 5 = Ψ 2 , 3 + Ψ 1 , 5 − Ψ 1 , 3 , Ψ 3 , 4 = Ψ 2 , 3 + Ψ 1 , 4 − Ψ 1 , 2 , Ψ 3 , 5 = Ψ 2 , 3 + Ψ 1 , 5 − Ψ 1 , 2 , (3.50) and (5 − 3)(5 − 4) / 2 = 1 rela tion with six Ψ i,j Ψ 4 , 5 = Ψ 1 , 5 + Ψ 1 , 4 + Ψ 2 , 3 − Ψ 1 , 2 − Ψ 1 , 3 . (3.51) As men tioned ab ov e, theses 5(5 − 2) = 15 relations a re enough to fix seven rema ined pa rameters: β 2 , 1 = − β 1 , 2 β 2 , 3 β 2 , 5 β 1 , 3 β 1 , 5 − β 2 , 3 β 3 , 5 β 1 , 5 − β 2 , 5 β 5 , 3 β 1 , 3 − β 2 , 3 β 4 , 5 ( β 1 , 4 β 2 , 5 − β 1 , 5 β 2 , 4 ) β 1 , 5 ( β 1 , 3 β 2 , 5 − β 1 , 5 β 2 , 3 ) − β 2 , 5 β 4 , 3 ( β 1 , 3 β 2 , 4 − β 1 , 4 β 2 , 3 ) β 1 , 3 ( β 1 , 3 β 2 , 5 − β 1 , 5 β 2 , 3 ) (3.52) 14 from whic h we can find β 3 , 1 , β 4 , 1 and β 5 , 1 , replac ing the indexe s (2 ↔ 3), (2 ↔ 4) and (2 ↔ 5), resp ectiv ely . F rom the pa r ameter β 3 , 2 = β 1 , 2 β 1 , 4 β 4 , 5 β 1 , 3 β 1 , 5 + β 1 , 2 β 3 , 5 β 1 , 5 − β 1 , 4 β 4 , 2 β 1 , 3 + ( β 1 , 3 β 2 , 5 − β 1 , 5 β 2 , 3 )( β 1 , 2 β 4 , 5 − β 1 , 5 β 4 , 2 ) ( β 1 , 3 β 4 , 5 − β 1 , 5 β 4 , 3 )  β 1 , 2 β 1 , 5 + β 1 , 3 β 5 , 4 − β 1 , 4 β 5 , 3 β 1 , 3 β 2 , 4 − β 1 , 4 β 2 , 3  (3.53) we ca n also find β 5 , 2 , repla cing the indexes (3 ↔ 5). The last par ameter is β 3 , 4 = β 1 , 4 β 3 , 5 β 1 , 5 + β 1 , 4 β 2 , 5 − β 1 , 5 β 2 , 4 β 1 , 3 β 1 , 5  β 1 , 4 ( β 1 , 3 β 4 , 5 − β 1 , 5 β 4 , 3 ) β 1 , 3 β 2 , 5 − β 1 , 5 β 2 , 3 + β 1 , 5 ( β 1 , 3 β 5 , 4 − β 1 , 4 β 5 , 3 ) β 1 , 3 β 2 , 4 − β 1 , 4 β 2 , 3 + β 1 , 2  (3.54) These seven para meters plus the five diago nal par ameters β i,i give us a 5 by 5 reflection matrix solution with 13 free parameters. Similarly , the cor responding 30 diagonal solutions can b e written for bo th mo dels. In the seq uence ( N ≥ 6), the expressions of the fixed parameter s ar e too lar g e and cumbers o me. 3.6 A reduced solution An imp o rtan t c haracteristic of the TL b oundary solutions is its large num b er of free par ameters. It means that we have man y different reduced solutions for a given R -matrix. In particular , c ho osing the free pa r ameters one can gets an appro priate K -matrix solution. F or instance, if w e consider a ll β i,j = β ( i 6 = j ) and all β i,i = α , in (3.5), we will get one- pa rameter so lution of the for m K ( u ) = f 1 ( u ) I + f 2 ( u ) G (3.55) where f 1 ( u ) = 1 − N − 2 2 β x 1 ( u ) x 2 ( u ) sinh( η ) f 2 ( u ) = 1 2 β sinh(2 u ) . (3.56) and G is a N by N matrix with en tries G i,j =  0 if i = j 1 if i 6 = j (3.57) 15 Although G satifies a quadratic relation as (1.5 ) G 2 − ( N − 2) G − ( N − 1) I = 0 , (3.58) we don’t know as the so lution (3.55 ) can b e fitted with (1.6). But, certainly , the infinity sp ectral para meter limit o f the solutio ns presented ab o v e will s olv e the constant reflection equations [15]. Many other reduce d solutions can b e derived in a simila r wa y . See, for instance, the cases pre sen ted in [34 ] for the U q [ sl (2)] mo de l. 4 Conclusion In this work we hav e presented s olutions of the reflection equation for the TL vertex mo dels. Our findings can b e s ummarized int o tw o classes o f solutions dep ending on N -parity . A larg e num ber of free par ameters is an imp ortan t ca racteristic o f these solutions. In analog y com the R -matr ices form (2.1), the K -matrices have the sa me for m (3.3 ) and (3.4), with all qua n tum group depe ndenc e in the diago na l entries throug h of matrix elements of M . These results pav e the wa y to construct, so lv e and study physical prop erties of the underlying quantum spin chains with op en b oundaries, generalizing the previo us efforts ma de for the case of p erio dic b oundary conditions [41, 4 3 ]. W e exp ect that the co ordinate Bethe ansatz for all diagonal solutions presented here can b e obtained by a dapting the results of [42, 44] and that its generaliza tion, as in [4 5 ], may b e a p ossibilit y to treat the non-diagona l s o lutions. 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