Integrable boundary conditions for a non-abelian anyon chain with $D(D_3)$ symmetry

In tegrable b oundary conditions for a non-ab elian an y on c hain with D ( D 3 ) symmetry K.A. Dancer, P .E. Finc h, P .S. Isaac and J. Links No vem b er 2, 2018 Abstract A general formulatio n of the Boundary Qu antum In v ers e Scattering Method is giv en whic h is applicable in cases where R - m atrix solutions of the Y ang– Baxter equation do not h a v e the prop ert y of crossing unitarit y . Suitably mo di- fied forms of the reflection equations are pr ese nted w h ic h p ermit the construc- tion of a family of commuting transfer matrices. As an examp le, we apply the formalism to determine the most g en er al solutions of the reflection equations for a solution of the Y ang-Baxter equation with und e r lying symmetry giv en b y the Drinfeld doub le D ( D 3 ) of the dih e d ral group D 3 . This R -mat r ix do es not ha ve the crossin g unitarit y prop ert y . In this mann er we derive inte grable b oundary conditions f o r an op en chain mo del of interact in g n o n - ab elian any ons. 1 In tro duct ion The study o f systems with non-ab elian a n yonic degrees of freedom currently a ttracts high interes t , due to the p ossibilities for exploiting their top ological pro perties to enco de quantum information in a manner whic h is protected from decoherence [1]. An appropriate framew ork in whic h to fo rm ulate systems with any onic symmetries is through the represen tation theory of quasi-triangular Hopf a lgberas [2], whic h in- cludes the class of Drinfeld doubles of finite group algebras [3, 4 ]. This latt e r class of algebras is particularly suited for the description of non-ab elian a n y ons where the conjugacy classes and cen traliser subgroups o f the finite g roup lab el generalised no- tions o f the magnetic and electric c harges [5, 6]. Moreo v er, within the quasi-triangular Hopf algebra framew ork, consisten t bra iding and fusion pro perties for any onic the- ories are naturally obtained. The braiding pro perties are c haracterised b y solutions of the Y ang-Baxter equation without sp ectral parameter, which are realised through the univ ersal R - matrix o f the alg e bra . The fusion prop erties are give n by decompo - sitions of tensor pro duct represen tations of the Hopf alg e bra , which ar e gov erned by the copro duct structure. These fusion rules provide a means to construct in t era cting systems by assigning energies to t he v arious p ossible multiple t structures. This then enables the study of one-dimensional (chain) mo dels with lo cal in tera ctio ns, as has 1 b een recen tly undertak en in [7] using Fib onacci an y o ns . F or this case the in tera ctio n energies w ere chosen in suc h a wa y that the lo cal Hamiltonians pro vided represen- tations o f the T emperley-Lieb algebra, whic h necessarily means that the system is in tegrable and can b e solved exactly . A study of a non- in tegrable non-ab elian any o n c hain can b e found in [8]. The t h eor y of in tegrable c hains has a lo ng history asso ciated with the Quantum In v erse Scattering Metho d (QISM) [9], whic h relies on a solution of the Y ang–Baxter equation with spectral parameter to construct a family of comm ut ing transfer ma- trices. The transfer matrix ma y be used t o generate the conserv ed operat o rs of an in tegrable quan tum system. F ollo wing this pro cedure w e hav e previously show n that , using the Drinfeld double D ( D 3 ) of the dihedral gro up D 3 , there exis ts a sp ectral parameter dep e ndent solution of the Y a ng -Baxter equation whic h can b e used to construct an in tegrable inte ra c ting non- abelian an yon c hain [10]. There the standard approac h o f the QISM w a s used, pro ducing a c hain with p erio dic b oundary condi- tions. F or op en c hain cases inte g rable b oundary conditions are pro vided by solutio ns of the reflection equations, as w a s first elucidated b y Skly anin [11], and is generally kno wn as the Boundary Quantum Inv erse Scattering Metho d (BQISM). Our g oal is to extend the BQISM formalism in a ma nner whic h will enable the construction of a non-ab elian an yon op en c ha in with integrable b oundary conditions. In Skly anin’s original fo rm ula tion of the BQISM sev eral conditions we re imp osed on the R -matr ix including P -symmetry , T -symmetry and crossing symmetry [11 ]. It w as so on realise d that the BQISM can be extended to cases where the P - and T -symmetry prop erties are relaxed to t he more general P T -symme tr y [12], and the crossing symmetry prop ert y can b e replaced b y the more general crossing unitarity condition [13] (see equation (10 ) b elo w for the definition). Later it w as sho wn in [14] that the BQISM can b e formulated for cases without P T -symmetry . Here w e further extend the formulation of the BQISM b y removing the imp osition of the crossing unitarit y pro pert y . This is necess ary to construct in tegrable b oundary conditions for the D ( D 3 ) a ny on chain, as the R -matrix do es not p ossess this prop ert y . In Section 2 w e prese nt the form ulation of the BQISM for R -matrices without crossing unitarity . Using the exp licit example pro vided b y the D ( D 3 ) R -matrix of [10], in Section 3 w e explicitly find the most general solutions of the reflection equations. In Section 4 w e use these results to derive a non- abelian any on c hain with in tegrable b oundary conditions, and concluding remarks a re giv en in Section 5. 2 BQISM for R -matrices without cros s ing un itar- it y Our first ob j e ctive is to reformulate the BQISM with a minimu m num b er of assumed prop erties imp ose d on the R - matrix solution o f t he Y ang–Baxter equation. W e start with inv ertible op erators R ( z ) ∈ End ( V ⊗ V ) and L ( z ) ∈ End ( V ⊗ W ) whic h satisfy 2 the Y ang–Baxter equation on End ( V ⊗ V ⊗ V ): R 12 ( xy − 1 ) R 13 ( x ) R 23 ( y ) = R 23 ( y ) R 13 ( x ) R 12 ( xy − 1 ) , (1) and the in tertwinin g relation o n End ( V ⊗ V ⊗ W ) : R 12 ( xy − 1 ) L 13 ( x ) L 23 ( y ) = L 23 ( y ) L 13 ( x ) R 12 ( xy − 1 ) . (2) Here the subscripts indicate on whic h ve ctor spaces eac h op erator acts, so for example R 23 ( z ) = I ⊗ R ( z ) . Solutions to the Y ang–Baxter equation (1 ) are referred to as R - matrices, while L ( x ) app earing in (2 ) is called an L -op erator. W e impose only the follo wing conditions o n R ( z ): 1. R t 1 ( z ) is in v ertible and 2. R ( z ) ob eys regularity , i.e. R (1) = P . Here t 1 denotes the partial transp ose ov er the first space and P ∈ End ( V ⊗ V ) is the usual p erm utation op erator defined b y P ( v ⊗ w ) = w ⊗ v , v , w ∈ V . The fo llo wing theorem is repro duced from [15]: Theorem 2.1. If R ( z ) is an R -matrix satisfying the r e gularity pr op erty then it also satisfies the unitarity pr op e rt y, i.e. R 12 ( z ) R 21 ( z − 1 ) = f ( z ) I ⊗ I , with f ( z ) a sc alar function satisfying f ( z ) = f ( z − 1 ) . Pr o of. Let R ( z ) b e a n R -matrix satisfying regularity . Then R 12 ( z ) R 13 (1) R 23 ( z − 1 ) = R 23 ( z − 1 ) R 13 (1) R 12 ( z ) ⇒ R 12 ( z ) P 13 R 23 ( z − 1 ) = R 23 ( z − 1 ) P 13 R 12 ( z ) ⇒ R 12 ( z ) R 21 ( z − 1 ) P 13 = R 23 ( z − 1 ) R 32 ( z ) P 13 ⇒ R 12 ( z ) R 21 ( z − 1 ) = R 23 ( z − 1 ) R 32 ( z ) The left hand side a cts trivially on t he the third space while the right hand side acts trivially on the first. Com bining these it follow s that R 12 ( z ) R 21 ( z − 1 ) m ust b e a scalar of the iden tit y , and that the scalar function is inv ariant under z → z − 1 . Utilising the condition that R t 1 ( z ) is in v ertible w e define the op erator R 12 ( z ) = [( R t 1 21 ( z )) − 1 ] t 1 , (3) whic h b y definition implies R t 1 12 ( z ) R t 1 21 ( z ) = R t 2 21 ( z ) R t 2 12 ( z ) = 1 . (4) 3 W e no w in tro duce tw o reflection equations R 12 ( xy − 1 ) K − 1 ( x ) R 21 ( xy ) K − 2 ( y ) = K − 2 ( y ) R 12 ( xy ) K − 1 ( x ) R 21 ( xy − 1 ) , (5) R 12 ( y x − 1 ) K + 1 ( x ) R 21 ( xy ) K + 2 ( y ) = K + 2 ( y ) R 12 ( xy ) K + 1 ( x ) R 21 ( y x − 1 ) , (6) where K + ( z ) , K − ( z ) ∈ End ( V ) are kno wn a s reflection matrices. The ma t r ic es K + ( z ), K − ( z ), R ( z ) and L ( z ) will enable us to construct an integrable mo del on an op en c hain. The transfer matrix is defined as t ( z ) = tr a  K + a ( z ) T ( z )  , where tr a is the trace ov er space a and T ( z ) is the double m o no dr omy matrix T ( z ) = L aN ( z ) ...L a 1 ( z ) K − a ( z ) L − 1 a 1 ( z − 1 ) ...L − 1 aN ( z − 1 ) . It is kno wn and easily v erifiable from (2) and (5) that R 12 ( xy − 1 ) T 13 ( x ) R 21 ( xy ) T 23 ( y ) = T 23 ( y ) R 12 ( xy ) T 13 ( x ) R 21 ( xy − 1 ) . (7) Prop osition 2.2. The tr ans f e r matric es t ( x ) , t ( y ) c ommute for al l x, y ∈ C . Pr o of. f ( xy − 1 ) t ( x ) t ( y ) = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 2 ( y ) K + 1 ( x ) t 1 T t 1 13 ( x ) T 23 ( y )  = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 2 ( y ) K + 1 ( x ) t 1 R t 1 12 ( xy ) R t 1 21 ( xy ) T t 1 13 ( x ) T 23 ( y )  = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 2 ( y ) R 12 ( xy ) K + 1 ( x ) T 13 ( x ) R 21 ( xy ) T 23 ( y )  = (tr 1 ⊗ tr 2 )  K + 2 ( y ) R 12 ( xy ) K + 1 ( x ) R 21 ( y x − 1 )   R 12 ( xy − 1 ) T 13 ( x ) R 21 ( xy ) T 23 ( y )  = (tr 1 ⊗ tr 2 )  R 12 ( y x − 1 ) K + 1 ( x ) R 21 ( xy ) K + 2 ( y )   T 23 ( y ) R 12 ( xy ) T 13 ( x ) R 21 ( xy − 1 )  = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 1 ( x ) R 21 ( xy ) K + 2 ( y ) T 23 ( y ) R 12 ( xy ) T 13 ( x )  = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 1 ( x ) K + 2 ( y ) t 2 R t 2 21 ( xy ) R t 2 12 ( xy ) T t 2 23 ( y ) T 13 ( x )  = (tr 1 ⊗ tr 2 )  f ( xy − 1 ) K + 1 ( x ) K + 2 ( y ) t 2 T t 2 23 ( y ) T 13 ( x )  = f ( xy − 1 ) t ( y ) t ( x ) where we hav e used equations (4,6,7) a nd Theorem 2.1. W e no w imp ose the limit condition K − (1) = I and only consider instances for whic h tr ( K + (1)) 6 = 0 . Then in the case L ( z ) = R ( z ) the global Hamiltonian on an op en c hain with b oundary fields is defined in the follo wing w ay: H = c 2tr ( K + (1))  d dz ( t ( z ))     z =1 − tr  d dz  K + ( z )       z =1  = N − 1 X i =1 H i,i +1 + c 2 d dz  K − 1 ( z )      z =1 + tr a ( cK + a (1) H N ,a ) tr ( K + (1)) , (8) 4 where c ∈ C and the lo cal Hamiltonians are giv en b y H i,i +1 = c P i,i +1 d dz ( R ( z ) i,i +1 )     z =1 . W e will refer to t he first term of (8) a s the bulk Hamiltonian, while the second and third terms describ e b oundary field in teractions. By construction the global Hamilto- nian (8) commutes with the transfer matrix t ( z ), whic h means that the Hamiltonian is necessarily integrable. The conserv ed op era t ors comm uting with the Hamiltonian are obt a ine d as the co-efficien t op erators in the series expansion of t ( z ). 3 Reflection matrices for an R -matrix asso ciated with D ( D 3 ) W e now apply the ab o v e formalism to solv e for the reflection matrices satisfying equations (5) and (6). This R - matrix we use is constructed from the represen tation theory of D ( D 3 ) [1 0], with asso ciated L -op erators g iv en in [16]. Explicitly , w e ha ve R ( z ) =                 1 0 0 0 0 0 0 0 0 0 0 z ( z − 1) z 2 − z +1 z z 2 − z +1 0 0 0 1 − z z 2 − z +1 0 0 z ( z − 1) z 2 − z +1 0 0 0 1 − z z 2 − z +1 z z 2 − z +1 0 0 0 z z 2 − z +1 0 0 0 z ( z − 1) z 2 − z +1 1 − z z 2 − z +1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 − z z 2 − z +1 z ( z − 1) z 2 − z +1 0 0 0 z z 2 − z +1 0 0 0 z z 2 − z +1 1 − z z 2 − z +1 0 0 0 z ( z − 1) z 2 − z +1 0 0 1 − z z 2 − z +1 0 0 0 z z 2 − z +1 z ( z − 1) z 2 − z +1 0 0 0 0 0 0 0 0 0 0 1                 . (9) The prop erties of the R -matrix include regularity , and consequen tly unitarit y . Ho w- ev er it can b e verified that the R -matrix do es not satisfy the crossing unitarit y con- dition, i.e. there do es not exist M ∈ End( V ) and λ ∈ C suc h that R t 1 12 ( λz ) M 1 R t 1 21 ( z − 1 ) M − 1 1 = f ( z ) I ⊗ I . (10) This is in con trast to R -matrices obtained from lo op represen tations of affine quan tum algebras, fo r whic h equation (10) is alw ay s satisfied [17]. 5 W e first calculate R ( z ) defined b y equation (3): R ( z ) = ( z 2 − z + 1) ( z − 1)( z 3 − 1)               z 2 + 1 0 0 0 z 0 0 0 z 0 0 1 − z 0 0 0 z 2 0 0 1 0 0 0 z 2 − z 0 0 0 − z 0 0 0 1 z 2 0 0 z 0 0 0 z 2 + 1 0 0 0 z 0 0 z 2 1 0 0 0 − z 0 0 0 − z z 2 0 0 0 1 0 0 z 2 0 0 0 − z 1 0 0 z 0 0 0 z 0 0 0 z 2 + 1               . These are the t wo op erators required to construct the reflection matrices K − ( z ) and K + ( z ). 3.1 Sp ecial case of the reflection equation T o determine the p ossible mat r ic es, K − ( z ) and K + ( z ), whic h satisfy equations (5 ) and (6) it is first conv enien t to determine all non-diagonal inv ertible matrices K ( z ) whic h satisfy t h e equation K 2 ( y ) ˇ R 12 (0) K 2 ( z 0 ) ˇ R 12 (0) = ˇ R 12 (0) K 2 ( z 0 ) ˇ R 12 (0) K 2 ( y ) , (11) where y , z 0 ∈ C , z 0 is fixed and ˇ R (0) = P R (0). W e scale K ( z ) so tha t the en tries of lim z → z 0 K ( z ) are all finite and that a t least one is non-zero, a s is a lw ays p ossible . Throughout t his section w e write K ( z ) in the form K ( z ) = 3 X i,j =1 h i,j ( z ) E i j where E i j denotes the elemen ta r y matrix with a 1 in the i th ro w and j th column. W e consider the indices of the functions h i,j ( z ) and elemen ta ry matrices E i j mo dulo 3. Using this notation, K ( z ) is a solution to equation (11) if and only if h i,j ( z 0 ) h k ,l ( y ) = h i,j + k +2 l ( z 0 ) h 2 i +2 k, 2 i + 2 l ( y ) for all 1 ≤ i, j, k , l ≤ 3. Prop osition 3.1. If K ( z ) satisfies e quation ( 11 ) a n d h a,a ( z 0 ) = 0 for some 1 ≤ a ≤ 3 then h a,j ( z 0 ) = 0 for al l 1 ≤ j ≤ 3 . Pr o of. Assume there is an in teger b suc h that h a,b ( z 0 ) 6 = 0. As K ( z ) satisfies equation (11), we ha ve h a,a ( z 0 ) h a,l ( y ) = h a, 2 a +2 l ( z 0 ) h a, 2 a +2 l ( y ) ⇒ h a,b ( z 0 ) h a,b ( y ) = 0 . This contradiction prov es the prop osition. 6 Corollary 3.2. If K ( z ) is a solution to equation (11) then K ( z 0 ) has a t least one non-zero dia gonal entry . Pr o of. Assume there is a solutio n where all the diag onal entries of K ( z 0 ) are zero. It follo ws f r o m Prop osition 3.1 that h a,j ( z 0 ) = 0 for all 1 ≤ a, j ≤ 3, whic h contradicts our requiremen t that at least one en try of K ( z 0 ) is no n-ze ro . Prop osition 3.3. If K ( z ) is a non-diagonal matrix satisfying e quation (11) an d h a,a ( z 0 ) 6 = 0 for some 1 ≤ a ≤ 3 then h a,a + b ( z 0 ) 6 = 0 for b ∈ { 1 , 2 } . F urthermor e, h a,a +2 ( z ) = αh a,a +1 ( z ) and α 3 = 1 wh er e α = h a,a +1 ( z 0 ) /h a,a ( z 0 ) . Pr o of. W e no w assume that there exists a b ∈ { 1 , 2 } such tha t h a,a + b ( z 0 ) = 0 . This leads to h a,a + b ( z 0 ) h a,a + b ( y ) = h a,a ( z 0 ) h a,a +2 b ( y ) ⇒ h a,a +2 b ( y ) = 0 . Therefore h a,a +2 b ( y ) = 0, whic h implies through the same argument that h a,a + b ( y ) = 0. Hence h a,a +1 ( y ) = h a,a +2 ( y ) = 0 . But h a,a ( z 0 ) h l + c,l ( y ) = h a,a + c ( z 0 ) h 2 a +2 l +2 c, 2 a +2 l ( y ) ⇒ h l + c,l ( y ) = 0 for 1 ≤ l ≤ 3 and c ∈ { 1 , 2 } . This implies t ha t if h a,a + b ( z 0 ) = 0 for some in teger b ∈ { 1 , 2 } then K ( z ) is diagona l, whic h is a con tradiction. Hence h a,a + b ( z 0 ) 6 = 0 for b ∈ { 1 , 2 } . T o sho w the o the r half of the pro position w e set α = h a,a +1 ( z 0 ) /h a,a ( z 0 ). Then h a,a +1 ( z 0 ) h a,a +1 ( y ) = h a,a ( z 0 ) h a,a +2 ( y ) ⇒ h a,a +2 ( y ) = α h a,a +1 ( y ) and h a,a +2 ( z 0 ) h a,a +2 ( y ) = h a,a ( z 0 ) h a,a +1 ( y ) ⇒ [ α 3 − 1] h a,a +1 ( y ) = 0 . This completes the pro of. Prop osition 3.4. F o r a non-diagonal matrix, K ( z ) , which s a t isfi es e quation (11) and has h a,a ( z 0 ) 6 = 0 and h a +1 ,a +1 ( z 0 ) 6 = 0 for some 1 ≤ a ≤ 3 then h a +2 ,a +2 ( z 0 ) 6 = 0 . Pr o of. W e find h a +1 ,a +1 ( z 0 ) h a +2 ,a +2 ( y ) = h a +1 ,a +1 ( z 0 ) h a,a ( y ) ⇒ h a +2 ,a +2 ( y ) = h a,a ( y ) , whic h pro ve s the prop osition. Prop osition 3.5. Any non-diag onal matrix, K ( z ) , which satisfi e s e quation (11) and has h i,i ( z 0 ) 6 = 0 for al l 1 ≤ i ≤ 3 is o f the form K ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) β 2 B ( z ) A ( z ) β B ( z ) γ B ( z ) γ 2 B ( z ) A ( z )   wher e α 3 = β 3 = γ 3 = αβ γ = 1 and A ( z 0 ) = B ( z 0 ) = 1 . 7 Pr o of. As h 1 , 1 ( z 0 ) 6 = 0 and w e are free to scale by a constan t w e set h 1 , 1 ( z 0 ) = 1. W e find that h i,i ( z 0 )[ h l,l ( y ) − h 2 i +2 l, 2 i +2 l ( y )] = 0 ⇒ h l,l ( y ) = h j,j ( y ) for all 1 ≤ i, j, l ≤ 3 . W e no w define the follo wing v ariables λ a = h a,a +1 ( z 0 ) , for 1 ≤ a ≤ 3. W e see that fro m the previous prop osition t ha t λ 3 a = 1 and h a,a +2 ( y ) = λ a h a,a +1 ( y ) for all a ∈ { 1 , 2 , 3 } . This giv es the pro p erties required for within eac h ro w, but we still need to relate the entries do wn eac h column. W e hav e that h a,a + c ( z 0 ) h a +2 c,a + c ( y ) = h a,a +2 c ( z 0 ) h a + c,a +2 c ( y ) for a ∈ { 1 , 2 , 3 } and c ∈ { 1 , 2 } . Hence h a +2 ,a +1 ( y ) = λ a h a +1 ,a +2 ( y ) ⇒ λ a +2 h a +2 ,a ( y ) = λ a h a +1 ,a +2 ( y ) Expressing this explicitly w e hav e that λ 1 λ 3 h 1 , 2 ( y ) = λ 2 λ 1 h 2 , 3 ( y ) = λ 2 λ 3 h 3 , 1 ( y ) Hence the off-diagonal en tries a re all sc ala r m ultiples of eac h o the r. Moreo ve r, b y considering t he ab o v e equation at y = z 0 w e find λ 2 1 λ 3 = λ 1 λ 2 2 ⇒ λ 1 λ 3 λ 2 = 1 . This prov es the prop osition. Prop osition 3.6. If K ( z ) satisfies e quation (11) and K ( z 0 ) has on ly one non-zer o diagonal entry then K ( z ) c an b e written in the form (after b asis tr ansformation an d sc aling) K ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) αD ( z ) C ( z ) E ( z ) D ( z ) αE ( z ) C ( z )   wher e α 3 = 1 , A ( z 0 ) = B ( z 0 ) = 1 and C ( z 0 ) = D ( z 0 ) = E ( z 0 ) = 0 . Pr o of. First note t ha t ˇ R is in v ariant under relab ellin g of the indices, so without loss of generality w e can set h 1 , 1 ( z 0 ) = 1. W e set A ( z ) = h 1 , 1 ( z ), so A ( z 0 ) = 1. By Prop osition 3.3 w e kno w that h 1 , 2 ( z 0 ) 6 = 0, and th us we set h 1 , 2 ( z ) = αB ( z ) with 8 B ( z 0 ) = 1. By Prop osition 3.3 w e also ha v e that α 3 = 1 and h 1 , 3 ( z ) = α 2 B ( z ). W e no w consider the diagonal entries and see that h 1 , 1 ( z 0 ) h 3 , 3 ( y ) = h 1 , 1 ( z 0 ) h 2 , 2 ( y ) . It fo llows that h 3 , 3 ( z ) = h 2 , 2 ( z ). W e let C ( z ) = h 2 , 2 ( z ). W e no w use h 1 , 1 ( z 0 ) h 2 ,l ( y ) = h 1 , 2 l ( z 0 ) h 3 , 2+2 l ( y ) . This gives h 2 , 1 ( y ) = α h 3 , 1 ( y ) and h 2 , 3 ( y ) = α 2 h 3 , 2 ( y ) . W e let D ( z ) = h 3 , 1 ( z ) and E ( z ) = h 2 , 3 ( z ). Lastly , note that C ( z 0 ) = 0 and hence b y Prop osition 3.1 we hav e D ( z 0 ) = E ( z 0 ) = 0. W e ha v e classified all the p ossible non- diagonal matrix solutions to equation (11). These can b e classified b y the n umber of non-zero diagonal elemen ts of K ( z 0 ). In Corollary 3.2 and Prop osition 3.4 it was sho wn that K ( z ) can only hav e one or three non-zero dia gonal elemen ts. Moreov er, these solutions can all b e written (af t e r basis transformation and scaling) in the for ms given in Propo s itio ns 3.5 and 3.6 resp ectiv ely . 3.2 Reflection matrix K − ( z ) W e no w turn our attention to the reflection equation (5). Setting x = ∞ and using regularit y , we obtain t he equation R 21 (0) K − 1 ( ∞ ) R 12 (0) K − 2 ( y ) = K − 2 ( y ) R 21 (0) K − 1 ( ∞ ) R 12 (0) . This is equiv alen t to equation (11 ) with z 0 = ∞ . Using the pr evious section w e kno w all the p oss ible forms of K − ( z ). W e now o nly need che ck under whic h conditions, if an y , these forms satisfy the reflection equation (5). Prop osition 3.7. The only diagonal matric es satisfying e quation (5) ar e sc alars of the identity. Pr o of. The pro of in volv es a straigh tfo rw a rd calculation so w e omit the details. Prop osition 3.8. Ther e ar e no invertible matric es satisfying e quation (5) of the form K − ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) αD ( z ) C ( z ) E ( z ) D ( z ) αE ( z ) C ( z )   wher e α 3 = 1 , A ( ∞ ) = B ( ∞ ) = 1 and C ( ∞ ) = D ( ∞ ) = E ( ∞ ) = 0 . 9 Pr o of. W e first assume that suc h a K − ( z ) do es exist. Substituting K − ( z ) in to the reflection equation, w e find it m ust satisfy the constraints D ( y ) B ( x ) = B ( y ) D ( x ) and E ( y ) B ( x ) = B ( y ) E ( x ) . This implies that D ( z ) and E ( z ) are b oth scalars of B ( z ). Using the b oundary conditions at z = ∞ w e deduce that D ( z ) = E ( z ) = 0 . After imp osing D ( z ) = E ( z ) = 0 we obtain t he second constrain t equation B ( x ) B ( y ) = 0 . This is a con tradiction, as B ( ∞ ) = 1, and hence there a r e no solutions K − ( z ) of the ab o v e form. Prop osition 3.9. A l l matric es satisfying e quation (5) ar e either sc alars of the iden- tity or o f the form K − ( z ) =   w 2 + bz − z 2 (1 − z 2 ) (1 − z 2 ) w 2 (1 − z 2 ) w 2 + bz − z 2 w (1 − z 2 ) w 2 (1 − z 2 ) w (1 − z 2 ) w 2 + bz − z 2   , for some b ∈ C and w a cub e r o ot of unity, up to sc aling and a change of b asis. Pr o of. Let K − ( z ) b e a solution to equation (5) whic h is not a scalar of the iden tity . Then by the earlier prop ositions w e kno w K − ( z ) can b e written in the fo rm K − ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) β 2 B ( z ) A ( z ) β B ( z ) γ B ( z ) γ 2 B ( z ) A ( z )   where α 3 = β 3 = γ 3 = αβ γ = 1 and A ( ∞ ) = B ( ∞ ) = 1. Substituting this in to equation (5), w e obtain the constrain t equation 0 = − x ( x 2 + α 2 β x 2 + α β 2 ) − x ( x 2 + αβ 2 + α 2 β ) y 2 +( αβ 2 x 2 + β 2 α + x 2 + x 4 + 2 α 2 β x 2 ) y − y (1 − x 2 ) 2 f ( x ) + x ( x 2 − 1)( y 2 − 1) f ( y ) where f ( z ) = A ( z ) /B ( z ). As the co efficien t of f ( x ) is linear in y , w e take the double deriv ative with r esp ect to y , obtaining 2 ( x 2 + α β 2 + α 2 β ) ( x 2 − 1) = 2 f ( y ) + 4 y f ′ ( y ) + ( y 2 − 1) f ( y ) . 10 The rig h t hand side is indep enden t of x , so w e deduce that 0 = 1 + α β 2 + α 2 β = 1 + α 2 β + ( α 2 β ) 2 . This implies that α 2 β is a primitive cub e ro ot of unity . W e set β = w α, where w is a primitiv e cub e ro ot of unity . Our matrix b ecomes K − ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) w 2 α 2 B ( z ) A ( z ) w αB ( z ) w 2 αB ( z ) w α 2 B ( z ) A ( z )   . The differen t c hoices of α a re equiv alent up to a basis tr ans f o rmation under whic h ˇ R ( z ) is in v arian t, so without loss of generalit y w e choose α = 1. No w t he differen tial equation reduces to 2 = 2 f ( y ) + 4 y f ′ ( y ) + [ y 2 − 1] f ′′ ( y ) . This has the general solution f ( y ) = a + by − y 2 1 − y 2 , where a, b ∈ C . Without loss of generalit y w e set A ( z ) = 1 and B ( z ) = 1 − z 2 a + bz − z 2 . Substituting K − ( z ) in to equation (5), w e find that we must hav e a = w 2 . F urthermore if a = w 2 then K − ( z ) satisfies equation (5) for all b ∈ C . 3.3 Reflection matrix K + ( z ) In this section w e construct solutions to the o the r reflection equation (6). Setting x = 0 and using unitarity , we obtain t he equation R 21 (0) K + 1 (0) R 21 (0) K + 2 ( y ) = K + 2 ( y ) R 12 (0) K + 1 (0) R 12 (0) , This is equiv alen t to equation (11) with z 0 = 0 a s R 21 (0) = R 12 (0). Using the previous section w e kno w all the p ossible forms of K + ( z ). It remains to che ck whether these forms pro vide solutio ns to the reflection equation (6), and if so, under what conditions. Prop osition 3.10. The only diagon a l r efle ction matric e s satisfying e quation (6) ar e sc alars of the i d entit y. Pr o of. The pro of in volv es a straigh tfo rw a rd calculation so w e omit the details. 11 Prop osition 3.11. Ther e ar e no invertible matric es satisfying e quation (6) of the form K + ( z ) =   A ( z ) α B ( z ) α 2 B ( z ) αD ( z ) C ( z ) E ( z ) D ( z ) αE ( z ) C ( z )   wher e α 3 = 1 , A (0 ) = B (0) = 1 and C (0) = D (0) = E (0) = 0 . Pr o of. Directly substituting K + ( z ) into equation ( 6 ) w e find the constraints D ( y ) B ( x ) = B ( y ) D ( x ) . This implies that D ( z ) is a scalar of B ( z ). As D (0) = 0 we deduce that D ( z ) = 0 . W e also obta in the constrain t equations. C ( y ) A ( x ) = A ( y ) C ( x ) and E ( y ) B ( x ) = B ( y ) E ( x ) Using similar reasoning, w e find that E ( z ) = C ( z ) = 0 . Hence K + ( z ) is not inv ertible, whic h pro ve s the prop osition. Prop osition 3.12. Al l m a t ric es satisfying e quation (6) ar e either sc alars of the iden- tity or s c alars of K + ( z ) =   1 + bz − w z 2 (1 − w 2 z 2 ) (1 − w 2 z 2 ) w 2 (1 − w 2 z 2 ) 1 + bz − w z 2 w (1 − w 2 z 2 ) w 2 (1 − w 2 z 2 ) w (1 − w 2 z 2 ) 1 + bz − w z 2   , for some b ∈ C and w a primitive cub e r o ot of unity. Pr o of. By the earlier prop ositions, if K + ( z ) is not a scalar o f the iden tit y then it m ust b e of the form K + ( z ) =   A ( z ) αB ( z ) α 2 B ( z ) β 2 B ( z ) A ( z ) β B ( z ) γ B ( z ) γ 2 B ( z ) A ( z )   where α 3 = β 3 = γ 3 = α β γ = 1 and A (0) = B (0) = 1. Using a similar techniq ue to that used to prov e Prop osition 3.9, we find a constrain t equation and differentiate t wice with resp ect to x . W e obtain the following tw o equations: 2 αβ 2 = 2 f ( y ) + 4 y f ′ ( y ) + [ y 2 + 1 + αβ 2 ] f ′′ ( y ) 0 = (1 + β α 2 + ( β α 2 ) 2 ) [2 f ′ ( y ) + y f ′′ ( y )] where f ( y ) = A ( y ) /B ( y ). These equations only ha ve a solution in common when (1 + β α 2 + ( β α 2 ) 2 ) = 0 . 12 Hence w e must ha ve β = w α = w 2 γ where w is a primitiv e cub e ro ot of unity . F ur- thermore, the three differen t c hoices of α are equiv alent up t o a basis transformation under whic h R ( z ) and R ( z ) are b oth in v arian t , so without loss of generalit y w e can c ho ose α = 1. The differen tial equation simplifies to 2 w 2 = 2 f ( y ) + 4 y f ′ ( y ) + [ y 2 − w ] f ′′ ( y ) . This has the general solution f ( z ) = a + bz − w z 2 1 − w 2 z 2 where a, b ∈ C . How eve r f (0) = A (0) /B (0) = 1, so a = 1. W e set A ( z ) = 1 + bz − w z 2 and B ( z ) = 1 − w 2 z 2 . Substituting this in to t he reflection equation (6), w e find t his is a solution for all b ∈ C . This completes the pro of. 4 An in t e grable Hamiltonian w ith op en b ound ary conditi o ns W e fo und in the previous section all p ossible solutions K + ( z ) and K − ( z ) of the reflection equations (5) and (6) for our R -matrix with non-ab elian an yonic symmetry . T o calculate the global Hamiltonian with non-trivial b oundary terms, w e use the reflection ma t r ic es in the follo wing form: K − ( z ) =    1 α (1 − z 2 ) w 2 + az − z 2 α 2 (1 − z 2 ) w 2 + az − z 2 α 2 w 2 (1 − z 2 ) w 2 + az − z 2 1 αw (1 − z 2 ) w 2 + az − z 2 αw 2 (1 − z 2 ) w 2 + az − z 2 α 2 w (1 − z 2 ) w 2 + az − z 2 1    , and K + ( z ) =   1 + bz − w j z 2 β (1 − w 2 j z 2 ) β 2 (1 − w 2 j z 2 ) w 2 j β 2 (1 − w 2 j z 2 ) 1 + bz − w j z 2 w j β (1 − w 2 j z 2 ) w 2 j β (1 − w 2 j z 2 ) w j β 2 (1 − w 2 j z 2 ) 1 + bz − w j z 2   , where α, β ∈ { 1 , w , w 2 } , j ∈ { 1 , 2 } , a, b ∈ C and w is a primitiv e cub e ro ot of unit y . As we are o nly considering the case where tr ( K + (1)) 6 = 0 we imp ose that b 6 = w j − 1 . Cho osing c = i in (8) yields the lo cal Hamiltonian as g iv en in [10]: H i,i +1 = X γ ∈ D 3 i ( E γ (1) γ (2) ⊗ E γ (2) γ (3) − E γ (2) γ (3) ⊗ E γ (1) γ (2) ) where the elemen ts γ ∈ D 3 are written as p erm utations of { 1 , 2 , 3 } . (Recall tha t D 3 is isomorphic to the permutation group S 3 .) The bulk Hamilto nia n, whic h de- scrib e s nearest neighbour in teractions b et w een an y ons with lo cal three-dimensional 13 state spaces, comm utes with the action of D ( D 3 ) algebra [10]. F or the b oundary terms we find i 2 d dz  K − 1 ( z )      z =1 = A   0 α α 2 α 2 w 2 0 αw αw 2 α 2 w 0   and tr a ( iK + a (1) H N ,a ) tr ( K + (1)) = B   0 β β 2 β 2 w j 0 β w 2 j β w j β 2 w 2 j 0   where A = i (1 − w 2 + a ) − 1 ∈ C a nd B = − i (1 − w j + b ) − 1 ∈ C . W e c ho ose a = iω X − 1 + ω − 1 − 1 , b = w j − 1 − iω − j Y − 1 with X , Y ∈ R unconstrained pa ramete rs. This leads to A = w − 1 X , B = w j Y , resulting in a glo bal Hamiltonian whic h is hermitian. Alternativ ely , one can c ho ose the reflection matrices to b e K + ( z ) = K − ( z ) = I . This yields an op en chain Hamiltonian without b oundary interaction terms. It is a ls o p ossible to ha ve one end of the chain with a b oundary interaction term, wh ile the other end is without a b oundary interaction term. 5 Summary W e reform ulated the BQISM through a pair of reflection equations in a fashion whic h do es not rely on the R -matrix solution of the Y ang–Baxter equation satisfying the crossing unita rit y condition. This w as motiv at e d b y the case of the R -matrix (9) as- so ciated with the quasi-tr iangular Hopf algebra D ( D 3 ), f or whic h crossing unitarity do es not hold. W e then pro ceeded to determine the most general solutions of the re- flection equations. With these results we w ere able to determine inte g rable b oundary conditions for an any onic c hain where the bulk Hamiltonian has D ( D 3 ) symmetry . The R -matrix asso ciated with D ( D 3 ) is the simplest example in a heirarch y of solutions asso ciated with D ( D n ) whic h solv e the Y ang–Baxter equation [18]. All of these solutions are c haracterised by an absence of crossing unitarity , meaning that our general formalism is applicable on a wider scale. A future direction in this pro gram of researc h is to compute the Bethe ansatz so- lution of the o pen c hain mo del deriv ed ab o v e. Implemen tation of the algebraic Bethe ansatz app ears problematic in this case, due to the lac k of a suitable pseudo v acuum state. A more promising av enue is offered b y a functional approac h aided b y fusion relations e.g. see [19] and references therein. References [1] C. Na yak, S.H. Simon, A. Stern, M. F reedman, and S. Das Sarma, Non-ab elian anyons and top olo gic al quantum c omputation , Rev. Mod. Ph ys. 80 , 1083- 1159 (2008). 14 [2] V.G. Drinfeld, Quantum g r oups in Pro ceed ing s of the International Congress of Mathematicians, A.M. Gleason (ed.) pp. 79 8 -820 (Pro vidence, RI: American Mathematical So ciet y , 1986). [3] R. Dijkgraaf, V. P asquier, and P . Ro c he, Quasi Hopf algebr as, gr oup c ohomolo gy and orbifold mo dels , Nucl. Ph ys. B ( Pro c. Suppl.) 18 , 60-72 (199 1 ). [4] M.D. Gould, Quantum double finite gr oup algeb r as and their r epr esentations , Bull. Aust. Math. So c. 48 , 275-301 (1 9 93). [5] M. de Wild Pro pitius and F.A. Bais, Discr ete gauge the ories . In G. Semenoff and L. Vinet, (eds.), Particles and Fields, CRM Series in Mathematical Phy sics pp. 353-440 (SpringerV erlag, New Y ork, 19 9 8) [6] A. Y u Kitaev, F ault-toler ant quantum c omputation by anyon s , Ann. Ph ys. 303 , 2-30 (2003 ). [7] A. F eiguin, S. T rebst, A.W.W. Ludwig, M. T ro yer, A. Kitaev, Z . W ang Z, and M. H. F reedman, Inter acting anyons in top olo gic al quantum liquids: the golden chain , Ph ys. Rev. Lett. 98 160409 (2 0 07). [8] N.E. Bonesteel and K. Y ang, Infinite-r andomness fixe d p oin ts for chains of non - ab elian quas i p articles , Ph ys. Rev. Lett. 99 , 140 4 05 (2007) . [9] E.K. Skly anin, L.A. T akh t a dz hy an, and L.D. F addeev, Quantum inv e rse pr o b lem metho d. 1 , Theor. Math. Ph ys. 40 , 688-706 (197 9). [10] K.A. Dancer, P .S. Isaac and J. Links, R epr esentations of the quantum doubles of finite gr oup algeb r as and sp e ctr al p ar ameter dep endent solutions of the Y ang- Baxter e quation , J. Math. Ph ys. 47 , 103511 (2006 ). [11] E.K. Sklyanin, Bounda ry c on ditions for inte g r able quantum systems , J. Ph ys. A: Math. Gen. 21 , 237 5 -2389 (1988 ) . [12] L. M ezincescu and R.I. Nep omec hie, Inte gr a b le op en spin chain s with non- symmetric R -matric es , J. Ph ys. A: Math. Gen. 24 , L15-L23 (1 991). [13] L. Mez incescu and R.I. Nepomechie , A ddendum: Inte gr ability of op en chains with quantum algebr a symmetry , Int. J. Mo d. Phy s. A 7 , 5 6 57-5659 (19 9 2). [14] J. Links and M.D. Gould, Inte gr able systems on op en chains wi th quantum su- p ersymmetry , In t. J. Mo d. Phy s. B 25 , 3461 - 3480 (19 96). [15] M.C. T akizaw a, Th e ladder op er ator appr o ach to c onstructing c on s erve d op er ators in inte gr abl e o ne-dimensional lattic e mo dels , Do ctoral Thesis, The Univ ersit y of Queensland (20 04) 15 [16] K.A. Dancer and J. Links, Universal sp e ctr al p ar ameter-dep endent L a x op er ators for the Drinfe ld double of the dihe dr al gr oup D 3 , arXiv:0810.56 01 . [17] N. Y u Reshetikhin and M.A. Semenov - T ia n-Shansk y , Centr al extensions of quan- tum curr ent gr oups , Lett. Math. Ph ys. 19 , 133-142 (19 90). [18] P .E. Finc h, K.A. D ance r, P .S. Isaac, and J. Links, Solutions o f the Y ang-Baxter e quation: desc endants of the six-vertex mo del fr om the Drinfeld double s of dihe- dr al gr oup algebr as , in prepara t io n. [19] L. F ra ppat, R. Nep omec hie, and E. Ragoucy , Com p lete B e the an s at z so l ution of the op en spin- s X X Z chain with gener al inte gr able b oundary terms , J. Stat. Mec h.: Theor. Exp. 9 , P09008 (20 07). 16