An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model

An algebraic deriv ation of the ei genspace s asso ciated with an Ising -lik e sp ec trum of the sup erin tegrable c hiral P ott s mo de l Akinori Nishino ∗ and T e tsuo D eguc hi 1 † Institute of Industrial Science, The Univ ersit y o f T oky o, 4–6–1 Komaba, Meguro-ku, T oky o, 153–8505, Japan 1 Departmen t of Ph ysics, Oc hanomizu Univ ersit y , 2–1–1 Otsuk a, Bunky o-ku, T oky o, 112–8610, Japan No v em b er 13, 2018 Abstract In terms of the sl 2 loop algebra and th e alge braic Be the-ansatz meth od, w e deriv e the inv ariant subspace asso ciated with a giv en I sing-like spectrum consisting of 2 r eigen v alues of the diagonal-to-diagonal transf er matrix of the sup erinteg rable chiral P otts (SCP) model with arbitrary inhomogeneous parameters. W e sho w that ev ery regular Bethe eigenstate of the τ 2 - mod el leads t o an Ising-lik e sp ectrum and is an eigenve ctor of the SCP transfer matrix which is given by the pro duct of tw o diago nal-to-diagonal transfer matrices with a constrain t on th e sp ectral parameters. W e also show in a sector that th e τ 2 -mo del commutes with th e sl 2 loop algebra, L ( sl 2 ), and every regular Bethe state of t he τ 2 -mo del is of highest weigh t. Thus, from physical assumptions suc h as the completeness of the Bethe ansatz, it follo ws in th e sector that ev ery regula r Bethe state of the τ 2 -mo del generates an L ( s l 2 )-degenerate eigenspace and it give s the inv arian t subspace, i.e . the direct sum of the eigenspaces associated w ith the Ising-like spectrum. 1 In tro duction The chiral Potts model [3, 4, 12, 31, 42 ], which is an N -state generalizatio n of the tw o-dimensional Ising mo del, has b een extensively studied from v a rious po ints o f view in recent years. The mo del is solv able in the s ense that its B oltzmann weigh ts satisfy the star-triangle relation to g ive a commutativ e family of transfer matrices [1 2]. In fact, the free energy , int erfacia l tension and o rder parameters of the mo del are exactly calcula ted in the thermo dyna mic limit [2, 8–10]. ∗ E-mail address: n ishino@ii s.u-toky o.ac.jp † E-mail address: d eguc hi@phys.ocha.ac.jp 1 In the sup erintegrable case of the chiral Potts model, all the eigenv alues of the transfer matrix are gro up e d in to s e ts of 2 r eigenv alues simila r to those of free fer mions. W e call it the sup er inte- grable c hiral P otts (SCP) mo del a nd the se t of e ig env alues an Ising- like sp e c trum [1, 2, 6– 8, 41 ]. The Onsager algebra is pow erful to de r ive the spe ctrum of the t w o-dimensiona l Ising mo del [18, 38, 39 ], in which a set of 2 r eigenv alues corresp onds to a 2 r -dimensional irr educible repr esentation o f the algebra. The approa ch is extended to the Z N -symmetric quantum system corresp o nding to the SCP model [17, 1 8 , 42]. How ev er, in con trast to the Ising-cas e, the appr o ach does not w ork enough to der ive a n ex act form of the spectrum for N > 3. A deriv ation of the exact for m is established by the appr oach [1 , 6–8 , 41] us ing functional relations among diag onal-to-dia gonal transfer matrice s of the SCP mo del [11]. There, the Ising-like sp e c trum is des c r ib ed by a p o ly nomial, which we call the SCP p o lynomial. How ev er, it is still nontrivial to define the SCP p olyno mial by an algebra ic metho d. In this pa p e r, w e pr e s ent a metho d for constr ucting basis vectors of the direct sum of the eigenspaces asso cia ted w ith a given Ising-like sp ectrum of the tra ns fer matrix of the SCP mo del in some sector . In sho r t, we constr uct the in v ariant subspace of the Ising-like sp ectr um. Fir s t, by the alg ebraic Bethe-ansatz metho d, w e show that ev ery regula r Bethe state of the τ 2 -mo del is an eigen vector of the SCP transfer matrix. Here it is defined by the pro duct of tw o diago nal-to- diagonal transfer matrices o f the SCP mo del with a constraint on the s pe ctral parameters. W e shall define it in detail in Section 2.1. The τ 2 -mo del is the integrable N -state spin chain corr e s p o nding to a nilp otent case of the cyclic L -op er ator [34, 35]; the transfer matrix constructed from the cyclic L -op era tors commutes with the transfer matrix o f the c hiral Potts mo del [13]. Secondly , we show in a sector that the τ 2 -mo del has the sy mmetry of the sl 2 lo op algebr a, L ( sl 2 ), a nd a lso in the sector that every r egular Bethe state of the τ 2 -mo del is a highest weight vector of the symmetry . Thu s, the degenera te e ig enspaces are generated b y regula r Bethe eig e ns tates [25, 37] in the sector through the symmetry . Her e w e sha ll define r egular Bethe states in Section 2.2. Thirdly , with some physical as sumptions such a s the completeness of the Bethe ansatz, w e show that fo r the diagonal- to -diagona l tra nsfer matrix o f the SCP mo del the in v ariant subspace of the Ising-like sp ectrum asso ciated with a regular Bethe state is giv en by the L ( sl 2 )-degenerate eigenspace of the τ 2 -mo del genera ted by the same regula r Bethe state. W e apply a gener alization of the a lgebraic Bethe ansa tz to the SCP tr a nsfer matrix with arbitrar y inhomogeneous pa rameters, a nd do not use the functional relations a mong the transfer matrices [11]. The algebr aic approa ch treats the SCP mo del and the τ 2 -mo del in a unified way , which might be useful for calculating co rrelation functions for the mo de l. W e r e pro duce the SCP p olyno mial as a k ind of Dr infeld p olynomia l which characterizes the finite-dimensional highes t w eight r epresentation of L ( sl 2 ) g enerated by the regular Bethe s tate. Here it is not necessarily irreducible [24 ]. F o r generic v alues of inhomog eneous pa rameters, how- ever, the zeros of the p olynomia l should b e distinct, and hence the hig hest weigh t re pr esentation should be irreducible. Thu s, the SCP p olyno mial should b e identified with the Drinfeld p olyno- mial [15, 23 , 24, 27]. The algebr aic deriv ation of the in v ariant subspace asso ciated with the Ising - like sp ectr um 2 prov es in the sector a prev ious conjecture that for the Z N -symmetric Hamiltonian the represen- tation space o f the Onsa ger a lgebra asso ciated with a n SCP p olynomia l sho uld co rresp ond to the L ( sl 2 )-degenerate eigenspac e of the τ 2 -mo del asso cia ted with the Drinfeld p olyno mial [37]. F or the τ 2 -mo del w e shall show a Bo r el subalg ebra symmetry of L ( sl 2 ) thro ugh a gauge tra ns- formation o n the L -op erato rs. In fact, it is known that every finite-dimensional irreducible r ep- resentation of the Borel subalgebra is extended to that of the sl 2 lo op algebra [14, 22]. W e shall th us derive the L ( sl 2 ) symmetry of the τ 2 -mo del in the general N -state case with inhomogeneous parameters in the pa p e r . Pr eviously , the s y mmetry has been shown for the o dd N and homoge- neous ca se [37 ]. The present r esult also prov es the L ( sl 2 ) sy mmetr y of the τ 2 -mo del for the ev en N and homog eneous case. It thus proves the conjecture [5] that the pro p o sed set of commuting op erator s forms the L ( sl 2 ) symmetry of the τ 2 -mo del. The L ( sl 2 ) symmetry of the τ 2 -mo del is closely rela ted to that of the spin-1/2 XXZ spin chain at roo ts o f unity [21, 24– 26]. The article consists of the following: in Section 2 , we introduce the SCP mo del and the τ 2 - mo del. W e r eview the algebraic Bethe-ans atz metho d fo r the τ 2 -mo del [41 ] and the Y ang-Ba xter relation betw een the mono dr omy matrices of the tw o models [1 3]. In Section 3 , generalizing the algebraic B ethe a nsatz, w e show that every regular Bethe eigens ta te of the τ 2 -mo del is an eigen- vector of the SCP transfer matrix with a constr aint on the sp ectral par ameters. The expression of eigenv a lues of the pro duct of two diago nal-to-dia g onal transfer matrices suggests the Ising- like sp ectrum to ea ch of the tw o transfer matrices . In Section 4 , we show in a sector the B o rel sub- algebra symmetry of the τ 2 -mo del through a gauge transforma tion o n the L - op erator s. It th us follows fro m [14, 22] that the τ 2 -mo del has the L ( sl 2 ) sy mmetr y . W e also show in the sector that every regular Bethe state of the τ 2 -mo del gener ates an irreducible highest weigh t representation of L ( sl 2 ), which gives the degener a te eigenspa ce asso ciated with the reg ular Bethe state for the τ 2 -mo del. W e then formulate the co njecture that the diagona l-to-diago nal tra ns fer matrix of the SCP mo del has the Ising-like sp ectrum in the L ( sl 2 )-degenerate eigenspac e of the τ 2 -mo del. 2 Mo d els and Y ang-Baxter relations 2.1 The chiral P otts mo del and the sup erintegrabl e conditions W e briefly review the c hiral P otts model [3, 4 , 12] and its super integrable p oint [1, 2, 6–8 ]. The mo del is defined on a tw o-dimensional s quare lattice with N -state lo cal spins in teracting along the edges. F or tw o a dja c ent lo cal spins σ i and σ j which take v alues in Z N , that is, 0, 1, . . . , N − 1 , t wo edge-types of the Boltzmann weights W pq ( σ i − σ j ) and ¯ W pq ( σ i − σ j ) are given as W pq ( n ) = W pq (0) n Y j =1 µ p µ q y q − x p ω j y p − x q ω j , ¯ W pq ( n ) = ¯ W pq (0) n Y j =1 µ p µ q x p ω − x q ω j y q − y p ω j , where ω is an N th ro ot of unit y . Here p = ( x p , y p , µ p ) and q = ( x q , y q , µ q ), whic h we call rapidities, are given on a F ermat curve defined b y k x N p = 1 − k ′ µ − N p , k y N p = 1 − k ′ µ N p , k 2 + k ′ 2 = 1 . (2.1) 3 Note that both W pq ( n ) and ¯ W pq ( n ) ar e functions of v ariable n ∈ Z N . The mo del is integrable in the sense that the Boltzmann weights sa tis fy the star- tr iangle r elations [4, 12]. W e a ls o give the F ourier-tr ansformed Boltzmann w eight: c W pq ( n ) = N − 1 X m =0 ω − nm W pq ( m ) = c W pq (0) n Y j =1 x p µ p ω − x q µ q ω j y q µ p − y p µ q ω j . W e introduce the S -o p erator [13, 16] to construct the mono dromy matrix o f the SCP mo del. Let Z and X b e op er ators which have the a ction Z v σ = ω σ v σ and X v σ = v σ +1 for a s tandard basis { v σ | σ ∈ Z N } of the N - dimensional vector space C N . By using them and combin ing the Boltzmann weigh ts, we define the S -op erator S ( p, p ′ ; q , q ′ ) ∈ E nd  C N ⊗ C N  by S ( p, p ′ ; q , q ′ ) = 1 N 2 P C N ⊗ C N X { n i } w pp ′ qq ′ ( n 1 , n 2 , n 3 , n 4 ) X n 1 Z n 2 X n 4 ⊗ X − n 1 Z n 3 X − n 4 , (2.2) w pp ′ qq ′ ( n 1 , n 2 , n 3 , n 4 ) = c W pq ′ ( n 1 ) W pq ′ (0) ¯ W pq ( n 2 ) ¯ W pq (0) ¯ W p ′ q ′ ( n 3 ) ¯ W p ′ q ′ (0) c W p ′ q ( n 4 ) W p ′ q (0) , where P C N ⊗ C N is the sta nda rd p ermutation ope rator: P C N ⊗ C N : v σ ⊗ v τ 7→ v τ ⊗ v σ . The action of the S -op er a tor is ex tended to a tensor pr o duct ( C N ) ⊗ L ⊗ C N , where the tens or pro duct ( C N ) ⊗ L is the quantum space descr ibing an L - site spin chain and the last space C N is a n auxiliary space. W e denote by S i ( p, p ′ ; q , q ′ ) the S -op erator acting on the i th c omp onent of ( C N ) ⊗ L and auxiliary space C N as the S -op er ator S ( p, p ′ ; q , q ′ ) and other comp onents of ( C N ) ⊗ L as the identit y . Here we use the oper ators Z i and X i on ( C N ) ⊗ L given by Z i = id ⊗ · · · ⊗ i ˇ Z ⊗ · · · ⊗ id , X i = id ⊗ · · · ⊗ i ˇ X ⊗ · · · ⊗ id . W e construct mono dromy ma trix T ( q 1 , q 2 ; { p , p ′ } ) ∈ End  ( C N ) ⊗ L ⊗ C N  and transfer matrix t ( q 1 , q 2 ; { p , p ′ } ) ∈ End  ( C N ) ⊗ L  as T ( q 1 , q 2 ; { p , p ′ } ) = L Y i =1 S i ( p i , p ′ i ; q 1 , q 2 ) , t ( q 1 , q 2 ; { p , p ′ } ) = tr C N  T ( q 1 , q 2 ; { p , p ′ } )  , (2.3) where both p i and p ′ i are rapidities of the i th comp onent of the qua ntu m space ( C N ) ⊗ L and q 1 and q 2 are those of the auxiliary space C N . The pa rameters q 1 and q 2 are called sp ectr a l parameters . Here the symbol { p , p ′ } denotes the set of r apidities p i and p ′ i for i = 1 , 2 , . . . , L . The transfer matrices satisfy the commutativit y t ( q 1 , q 2 ; { p , p ′ } ) t ( r 1 , r 2 ; { p , p ′ } ) = t ( r 1 , r 2 ; { p , p ′ } ) t ( q 1 , q 2 ; { p , p ′ } ) , which is a result o f the star-triangle r e lation [12]. Then the eigen vectors of the tra nsfer matrix t ( q 1 , q 2 ; { p , p ′ } ) are indep endent of the sp ectra l pa rameters q 1 or q 2 . W e ca ll the transfer matrix t ( q 1 , q 2 ; { p , p ′ } ) the row-to-row transfer matrix of the chiral Potts mo del since the Boltzmann weigh t w pp ′ qq ′ ( n 1 , n 2 , n 3 , n 4 ) is co ns idered as that of a vertex model. The r ow-to-row transfer matr ix t ( q 1 , q 2 ; { p , p ′ } ) is given by the product of tw o types of diagonal- 4 to-diagona l tr ansfer matrices T D ( x q 1 , y q 1 ) and ˆ T D ( x q 2 , y q 2 ) which are defined by T D ( x q , y q ) σ ′ σ = L Y i =1 W p ′ i q ( σ i − σ ′ i ) W p ′ i q (0) ¯ W p i +1 q ( σ i +1 − σ ′ i ) ¯ W p i +1 q (0) , ˆ T D ( x q , y q ) σ ′ σ = L Y i =1 ¯ W p ′ i q ( σ i − σ ′ i ) ¯ W p ′ i q (0) W p i +1 q ( σ i − σ ′ i +1 ) W p i +1 q (0) , where the p erio dic b oundary conditions σ L +1 = σ 1 and p L +1 = p 1 are impo sed. The diagona l- to- diagonal transfer ma trices a r e diago nalized by a pair o f inv ertible matrices U and V , which ar e independent of the pa r ameter q , as U − 1 T D ( x q , y q ) V = Λ( x q , y q ) and V − 1 ˆ T D ( x q , y q ) U = ˆ Λ( x q , y q ). Let us now discuss the sup e rintegrable case. When rapidities p and p ′ satisfy the conditions x p = y p ′ , y p = x p ′ , µ p = µ − 1 p ′ , we deno te the rapidity p ′ by ¯ p . Definition 2.1. We c al l the chir al Potts mo del su p erinte gr able, if r apidi ties { p, p ′ } satisfy the c onditio ns p ′ i = ¯ p i for al l i , that is, x p i = y p ′ i , y p i = x p ′ i , µ p i = µ − 1 p ′ i , (1 6 i 6 L ) [6–8 ]. In the sup er int egr a ble case, we deno te by T ( q 1 , q 2 ; { p } ) and t ( q 1 , q 2 ; { p } ) the mo no dromy matrix and the row-to-row transfer matrix of the chiral Potts mo del, resp ectively . That is, w e express T ( q 1 , q 2 ; { p } ) = T ( q 1 , q 2 ; { p , ¯ p } ) and t ( q 1 , q 2 ; { p } ) = t ( q 1 , q 2 ; { p , ¯ p } ). Hereafter w e call the row-to- row transfer ma trix t ( q 1 , q 2 ; { p } ) the SCP transfer matr ix, briefly . W e also call T ( q 1 , q 2 ; { p } ) the SCP mono dromy matr ix. 2.2 The τ 2 -mo del and the algebraic Bethe ansatz Let us now introduce an integrable N -state spin chain whose tra nsfer matr ix commutes with the SCP transfer matr ix . W e introduce the cyclic L -op er ator L ( z ; p, p ′ ) ∈ E nd( C 2 ⊗ C N ) [34, 3 5] b y L ( z ; p, p ′ ) =   − y p y p ′ z + µ p µ p ′ Z − z ( y p − x p ′ µ p µ p ′ Z ) X X − 1 ( y p ′ − x p µ p µ p ′ Z ) 1 − x p x p ′ µ p µ p ′ z ω Z   . (2.4) In the s ame wa y as the S -o pe r ator, the a c tion of the L -op erato r L ( z ; p, p ′ ) is extended to the tensor pro duct C 2 ⊗ ( C N ) ⊗ L where the space C 2 is a nother a uxiliary space ; we denote by L i ( z ; p, p ′ ) the L -op era tor a cting on the auxiliary spa ce and i th comp onent of the q uantum s pace ( C N ) ⊗ L as the L - op erator L ( z ; p, p ′ ) and other comp onents of ( C N ) ⊗ L as the identit y . The following pro pe r ties giv e the reas o n why the chiral Potts mo del is considered as a descendant of the six-vertex mo del [13]: Prop ositi on 2.2. The L -op er ators L i ( z ) = L i ( z ; p, p ′ ) satisfy a Y ang-Baxter re lation R ( z /w )  L i ( z ) ⊗ id C 2  id C 2 ⊗ L i ( w )  =  id C 2 ⊗ L i ( w )  L i ( z ) ⊗ id C 2  R ( z /w ) (2.5) with the R -matrix define d by R ( z ) =         1 − z ω 0 0 0 0 ω (1 − z ) (1 − ω ) z 0 0 1 − ω 1 − z 0 0 0 0 1 − z ω         , (2.6) 5 and another Y ang-Baxter r ela tion S ij ( p, p ′ ; q , q ′ ) L i ( z ; p, p ′ ) L j ( z ; q , q ′ ) = L j ( z ; q , q ′ ) L i ( z ; p, p ′ ) S ij ( p, p ′ ; q , q ′ ) , (2.7) wher e S ij ( p, p ′ ; q , q ′ ) is the S -op er a tor acting on the i th and j th c omp onents of the quantum sp ac e ( C N ) ⊗ L as the S -op er ator S ( p, p ′ ; q , q ′ ) and other c omp o nents of ( C N ) ⊗ L as the identity. A t the sup erintegrable p oint, the cyclic L -op er ator L i ( z ; p i , p ′ i ) is reduced to L i ( z ; p i , ¯ p i ) =   − t p i z + Z i − y p i z (1 − Z i ) X i x p i X − 1 i (1 − Z i ) 1 − t p i z ω Z i   , (2.8) where we hav e defined t p = x p y p . W e int ro duce the mono dromy matrix T ( z ; { p } ) ∈ End  C 2 ⊗ ( C N ) ⊗ L  and the transfer ma trix τ ( z ; { p } ) ∈ End  ( C N ) ⊗ L  by T ( z ; { p } ) = L Y i =1 L i ( z ; p i , ¯ p i ) =:   A ( z ) B ( z ) C ( z ) D ( z )   , τ ( z ; { p } ) = tr C 2  T ( z ; { p } )  . (2.9) Here we ha ve also defined op erato r s A ( z ) , B ( z ) , C ( z ) , D ( z ) ∈ End  ( C N ) ⊗ L  . The spin c hain describ ed by the tra nsfer matrix τ ( z ; { p } ) is called the τ 2 -mo del [5, 10, 40]. W e remark that the original τ 2 -mo del is defined in ter ms of the cy c lic L -op era tor L ( z ; p , p ′ ) (2.4). Prop ositi on 2.3. The m ono dr omy matrix T ( z ; { p } ) s atisfies a Y ang-Baxter r elation R ( z /w )  T ( z ; { p } ) ⊗ id C 2  id C 2 ⊗ T ( w ; { p } )  =  id C 2 ⊗ T ( w ; { p } )  T ( z ; { p } ) ⊗ id C 2  R ( z /w ) , (2.10) wher e R ( z ) is the R -matrix define d in (2.6) . The Y ang-Ba xter r elation (2.1 0) gives the commut ativity τ ( z ; { p } ) τ ( w ; { p } ) = τ ( w ; { p } ) τ ( z ; { p } ). Hence the eige nvectors of the transfer matrix τ ( z ; { p } ) are indep endent of the pa r ameter z . The relation also pro duces relations among op era to rs A ( z ) , B ( z ) , C ( z ) a nd D ( z ) [30]. In the nex t sec- tion, we need more general relations , which ar e collected in Lemma A.1. By using the relations, the algebr aic Bethe-ansatz metho d is readily applicable to the transfer matrix τ ( z ; { p } ) [41]. Let | 0 i b e the reference s tate v 0 ⊗ v 0 ⊗ · · · ⊗ v 0 . It has the following prop erties: A ( z ) | 0 i = a ( z ) | 0 i = L Y n =1 (1 − t p n z ) | 0 i , D ( z ) | 0 i = d ( z ) | 0 i = L Y n =1 (1 − t p n z ω ) | 0 i , C ( z ) | 0 i = 0 , for arbitra ry z . Prop ositi on 2.4. L et { z i | i = 1 , 2 , . . . , M } b e a solution of the Bethe e quations: a ( z i ) M Y j =1 j ( 6 = i ) f ( z i /z j ) = d ( z i ) M Y j =1 j ( 6 = i ) f ( z j /z i ) , (2.11) wher e f ( z ) = ( z − ω ) / ( z − 1) ω . Then, ve ct or | M i = B ( z 1 ) B ( z 2 ) · · · B ( z M ) | 0 i gives an eigenve ctor of the tr ansfer matrix τ ( z ; { p } ) : τ ( z ; { p } ) | M i =  a ( z ) M Y i =1 ω f ( z /z i ) + d ( z ) M Y i =1 ω f ( z i /z )  | M i . (2.12) The ve ctor | M i is r eferr e d to as a Bethe state. 6 If solutions of the Bethe equations (2.11) are non-zero, finite and distinct, w e call them regular [25]. If { z i | i = 1 , 2 , . . . , R } is a regula r so lutio n of the Bethe eq uations, we call the Bethe sta te B ( z 1 ) B ( z 2 ) · · · B ( z R ) | 0 i regula r, and denote it by | R i . 2.3 Comm utativit y of transfer matrices As a co ns equence of the rela tion (2 .7 ), we obtain the Y ang-Baxter relation betw een the mono dromy matrices T ( z ; { p } ) a nd T ( q 1 , q 2 ; { p } ), by whic h we s ha ll generalize the algebra ic Bethe-a nsatz metho d. Prop ositi on 2.5. The m ono dr omy matric es T ( z ; { p } ) and T ( q 1 , q 2 ; { p } ) s atisfy L ( z ; q 1 , q 2 )  T ( z ; { p } ) ⊗ id C N  id C 2 ⊗ T ( q 1 , q 2 ; { p } )  =  id C 2 ⊗ T ( q 1 , q 2 ; { p } )  T ( z ; { p } ) ⊗ id C N  L ( z ; q 1 , q 2 ) . (2.13) Her e the cyclic L - op er ator L ( z ; q 1 , q 2 ) define d in (2.4) is c onsider e d as a 2 N × 2 N m atrix acting on the tensor pr o duct C 2 ⊗ C N of the auxiliary s p ac es. As a c or ol lary, the tr ansfer matrix τ ( z ; { p } ) c ommutes with the tr ansfer m atr ix t ( q 1 , q 2 ; { p } ) . Thanks to the comm utativity o f the tw o transfer matr ic es τ ( z ; { p } ) and t ( q 1 , q 2 ; { p } ), they may hav e a set of common eigen vectors. F or the τ 2 -mo del, w e ha ve obtaine d the eigenstates through the a lgebraic Bethe-a nsatz metho d. If a given Bethe eigenvector o f the τ 2 -mo del has a non-degenera te eigenv a lue o f τ ( z ; { p } ), then it also b eco mes an eigenv ector o f the SCP tra nsfer matrix t ( q 1 , q 2 ; { p } ). How ever, in section 4, we shall show in a sector that the transfer matr ix τ ( z ; { p } ) of the τ 2 -mo del has degenera te eigenv ector s with res p ect to the sl 2 lo op algebr a a nd hence not all the Bethe states of the τ 2 -mo del ar e nec essarily eigenv ectors of t ( q 1 , q 2 ; { p } ). 3 Sp ectrum of the su p erin tegrable c hiral P otts mo del W e shall sho w in this sectio n that, if the sp ectral parameters q 1 and q 2 satisfy the condition q 2 = ¯ q 1 ( s ) = ( y q 1 , x q 1 ω s , µ − 1 q 1 ), every re gular Bethe eigenstate of τ ( z ; { p } ) is an eigenstate of the SCP transfer matr ix t ( q 1 , q 2 ; { p } ). 3.1 Algebraic B ethe-ansatz metho d for the SCP transfer matrix First, we give a fundamental r elation, gener a lizing the standard algebra ic Bethe-ansatz metho d. Prop ositi on 3.1. L et B i , A i and D i denote B ( z i ) , A ( z i ) and D ( z i ) , r esp e ctively. L et T τ ′ τ , ( τ , τ ′ ∈ Z N ) denote the op er ator-value d entries of the S CP mono dr omy matrix T ( q 1 , q 2 ; { p } ) . By setting q 1 = q = ( x q , y q , µ q ) and q 2 = ¯ q ( s ) = ( y q , x q ω s , µ − 1 q ) , ( s = 0 , 1 , . . . , N − 1) we have B 1 · · · B n T τ ′ τ | 0 i = X { i ℓ } , { j ℓ } , { k ℓ } n B + n A + n D = n c τ ′ τ n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) a ( z j 1 ) · · · a ( z j n A ) d ( z k 1 ) · · · d ( z k n D ) T τ ′ − n D τ + n A B i 1 · · · B i n B | 0 i . (3.1) 7 Her e { i ℓ } , { j ℓ } and { k ℓ } ar e such disjoint subsets of the index set Σ n = { 1 , 2 , . . . , n } that the n um- b ers of elements of the subsets denote d by ♯ { i ℓ } = n B , ♯ { j ℓ } = n A and ♯ { k ℓ } = n D , r esp e ctively, satisfy the c o ndition n B + n A + n D = n , and the c o effici ents c τ ′ τ n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) ar e given by c τ ′ τ n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) = n B Y ℓ =1 1 µ τ ′ τ ( z i ℓ ) n A Y ℓ =1 ν τ + ℓ ( z j ℓ ) µ τ ′ ,τ + ℓ − 1 ( z j ℓ ) n D Y ℓ =1 − ν τ ′ − ℓ +1 ( z k ℓ ) µ τ ′ − ℓ +1 ,τ ( z k ℓ ) Y i ∈{ i ℓ } j ∈{ j ℓ } ω f j i Y i ∈{ i ℓ } k ∈{ k ℓ } ω f ik Y j ∈{ j ℓ } k ∈{ k ℓ } ω f j k , with µ τ ′ τ ( z ) , ν τ ( z ) and f ij define d by µ τ ′ τ ( z ) = ( t q z − 1 )( t q z ω s − 1) ω τ ′ ( t q z ω τ ′ − 1)( t q z ω τ +1 − 1) , ν τ ( z ) = y q z (1 − ω τ ) t q z ω τ − 1 , f ij = z i − z j ω ( z i − z j ) ω . (3.2) A pro of of the relation is presen ted in the nex t subsec tion. The po int of the pro o f is to arrange the pro duct B 1 · · · B n T τ ′ τ int o the o rder T B AD C , which is p oss ible by the he lp of the rela tions (A.1) and (3.10). On the re fer ence state | 0 i , the terms with the o p erator C ( z i ) v anish and the op erator s A ( z i ) and D ( z i ) are r e pla ced by the facto rs a ( z i ) and d ( z i ), resp ectively . Next, w e apply the transfer matrix t ( q, ¯ q ( s ); { p } ) to the reference state | 0 i . It is dir e c tly shown that the re fer ence state | 0 i is a n eigenv ector of the transfer matr ix t ( q , ¯ q ( s ); { p } ) [6], t ( q , ¯ q ( s ); { p } ) | 0 i = N − 1 X τ =0 λ τ | 0 i = N L L Y n =1 x p n − x q x N p n − x N q y p n − y q y N p n − y N q ! N − 1 X τ =0 L Y n =1 t N p n − t N q t p n − t q ω τ ! ω τ L | 0 i , (3.3) where t p n = x p n y p n and t q = x q y q . Here we define λ τ by relatio n (3.3). Let { z i | i = 1 , 2 , . . . , R } be a regular solution of the Bethe equa tions (2 .1 1) a nd extract the term T τ ′ τ B 1 · · · B R | 0 i from the right-hand side of the relation (3.1). Then we s ee how the o p er ator T τ ′ τ acts on the r egular Bethe state | R i . By setting τ = τ ′ and ta king the sum on τ , it follows that the Bethe state | R i is an eige nv ector o f the SCP tr a nsfer matrix t ( q , ¯ q ( s ); { p } ). Theorem 3.2. Every r e gular Bethe state | R i is an eigenve ctor of the tr ansfer matrix t ( q , ¯ q ( s ); { p } ) with q = ( x q , y q , µ q ) and ¯ q ( s ) = ( y q , x q ω s , µ − 1 q ) , t ( q , ¯ q ( s ); { p } ) | R i = N − 1 X τ =0 λ τ  R Y i =1 µ τ τ ( z i )  | R i = N L L Y n =1 x p n − x q x N p n − x N q y p n − y q y N p n − y N q ! N − 1 X τ =0 L Y n =1 t N p n − t N q t p n − t q ω τ ! ω τ ( L + R ) F ( t q ) F ( t q ω s ) F ( t q ω τ ) F ( t q ω τ +1 ) | R i , (3.4) wher e F ( t ) is a p olynomi al in t define d by F ( t ) = Q R i =1 (1 − tz i ) and { z i | i = 1 , . . . , R } is a r e gular solution of the Bethe e quations (2.11) . Pr o of. By taking the sum of the left-hand side of the relation (3.1) with τ = τ ′ ov er τ = 0 , 1 , . . . , N − 1 and using the res ult (3.3), we o btain N − 1 X τ =0  R Y i =1 µ τ τ ( z i )  B 1 · · · B R T τ τ | 0 i = N − 1 X τ =0 λ τ  R Y i =1 µ τ τ ( z i )  B 1 · · · B R | 0 i . 8 On the other hand, from the right-hand side of the rela tion (3.1), we have N − 1 X τ =0  R Y i =1 µ τ τ ( z i )  B 1 · · · B R T τ τ | 0 i = N − 1 X τ =0 X { i ℓ } , { j ℓ } , { k ℓ } n B + n A + n D = R n A Y p =1 µ τ τ ( z j p ) ν τ + p ( z j p ) µ τ ,τ + p − 1 ( z j p ) n D Y q =1 − µ τ τ ( z k q ) ν τ − q +1 ( z k q ) µ τ − q +1 ,τ ( z k q ) × Y i ∈{ i ℓ } j ∈{ j ℓ } ω f j i Y i ∈{ i ℓ } k ∈{ k ℓ } ω f ik Y j ∈{ j ℓ } k ∈{ k ℓ } ω f j k ! a j 1 · · · a j n A d k 1 · · · d k n D T τ − n D τ + n A B i 1 · · · B i n B | 0 i = R X m =0 X { i ℓ } n B = m  Y i ∈ Σ R \{ i ℓ } a i  N − 1 X τ =0 X { j ℓ } , { k ℓ } n A + n D = R − m n A Y p =1 µ τ τ ( z j p ) ν τ + p ( z j p ) µ τ ,τ + p − 1 ( z j p ) n D Y q =1 − µ τ τ ( z k q ) ν τ − q +1 ( z k q ) µ τ − q +1 ,τ ( z k q ) × ω m ( R − m )+ n A n D n B Y r =1 n A Y p =1 n D Y q =1 f j p i r f k q i r f k q j p T τ − n D τ + n A B i 1 · · · B i m | 0 i = R X m =0 X { i ℓ } n B = m  Y i ∈ Σ R \{ i ℓ } a i  N − 1 X τ =0 X { j ℓ } , { k ℓ } n A + n D = R − m ω m ( R − m )+ n A n D × n A Y p =1 µ τ + n D ,τ + n D ( z j p ) µ τ + n D ,τ + n D + p − 1 ( z j p ) ν τ + n D + p ( z j p ) n D Y q =1 − µ τ + n D ,τ + n D ( z k q ) µ τ + n D − q +1 ,τ + n D ( z k q ) ν τ + n D − q +1 ( z k q ) × n B Y r =1 n A Y p =1 n D Y q =1 f j p i r f k q i r f k q j p T τ τ + R − m B i 1 · · · B i m | 0 i = N − 1 X τ =0 T τ τ B 1 · · · B R | 0 i = t ( q , ¯ q ( s ); { p } ) | R i . (3.5) In the seco nd equality , we hav e calcula ted as Y i ∈{ i ℓ } Y j ∈{ j ℓ } Y k ∈{ k ℓ } f ik f j k ! d k 1 · · · d k n D = Y k ∈{ k ℓ } d k Y i ∈{ i ℓ }∪{ j ℓ } f ik ! = Y k ∈{ k ℓ } a k Y i ∈{ i ℓ }∪{ j ℓ } f ki ! = Y i ℓ ∈{ i ℓ } Y j ℓ ∈{ j ℓ } Y k ℓ ∈{ k ℓ } f ki f kj ! a k 1 · · · a k n D , where we have employ ed the identit y in Lemma B.3 which is derived fr om the Bethe eq ua- tions (2.11 ). The un wan ted terms m 6 = R in (3.5) hav e b een canceled out b eca use of the iden tity in Lemma B.4. 3.2 The I sing-lik e sp ectrum consisting of 2 r eigen v alues F rom the expression of eigenv alues of the SCP transfer matr ix t ( q , ¯ q ( s ); { p } ) w e can derive eigenv a l- ues of the dia gonal-to- diagonal transfer ma trices T D ( x q , y q ) and ˆ T D ( x q , y q ). F rom the discussio n similar to [8], the set of eigenv a lues in the inv ar ia nt subspace c o ntaining a given re g ular Bethe 9 state | R i a re given in the following forms: Λ( x q , y q ) = N L 2 L Y n =1 x p n − x q x N p n − x N q ! x P a q y P b q µ − N P c q F ( t q ) G ( µ − N q ) , ˆ Λ( x q , y q ) = N L 2 L Y n =1 y p n − x q y N p n − x N q ! x P a q y P b q µ − N P c q F ( t q ) ˆ G ( µ − N q ) . (3.6) Here P a and P b are in tegers satisfying P a + P b ≡ − L − R mo d N , and we reca ll F ( t q ) = Q R i =1 (1 − t q z i ). G ( µ N q ) and ˆ G ( µ N q ) are p olynomials in µ N q satisfying G ( µ N q ) = co ns t . ˆ G ( µ N q ). F rom the r elation t ( q , ¯ q ( s ); { p } ) = T D ( x q , y q ) ˆ T D ( y q , x q ω s ), the pro duct G ( µ − N q ) ˆ G ( µ N q ) is given by G ( µ − N q ) ˆ G ( µ N q ) = P SCP ( t N q ) := N − 1 X τ =0 L Y n =1 t N p n − t N q t p n − t q ω τ ! ω τ ( L + R ) F ( t q ω τ ) F ( t q ω τ +1 ) . (3.7) Here P SCP ( t N q ) is a p olynomial in t N q of degre e at mo st ⌊ L ( N − 1) − 2 R N ⌋ ; the Bethe equa tions (2.11 ) corres p o nd to the p ole-fr ee condition. W e call the p oly nomial P SCP ( ζ ) the SCP p oly nomial. W e remark that, in our result, o nly the cas e P b = 0 app ear s. The relation k 2 t N q = 1 − k ′ ( µ N q + µ − N q )+ k ′ 2 tells us that the po lynomial P SCP ( t N q ) is regarde d as a Laurent polyno mial in µ N q of degree r = deg P SCP ( ζ ) whose zeros o c cur in r ecipro cal pairs. Then, by denoting the 2 r zeros b y { w ± 1 i } , w e hav e 2 r solutions for G ( µ N q ) and ˆ G ( µ N q ) in the fo r ms G ( µ N q ) , ˆ G ( µ N q ) = co ns t. r Y i =1 ( µ N q − w ǫ i i ) , (3.8) where ǫ i = 1 or − 1 is indep endently chosen for the index i . The 2 r solutions for G ( µ N q ) and ˆ G ( µ N q ) ar e similar to the 2 r eigenv alues o f the Ising- like form [18, 38]. W e th us ca ll the set of 2 r eigenv alues of the diago nal-to-diag onal tr a nsfer matr ices asso ciated with a regula r Bethe sta te the Ising-like sp ectrum a sso ciated with the regula r Bethe state. In fact, in the homogeneous ca se o f p 1 = · · · = p L , it follows from the Onsag er-alg ebra structure o f the SCP mo del that each eigenv alue is non-degenerate, that is, the multiplicit y of the eigenv alue sp ecified b y a set o f { ǫ i } is given by one. F or the homogeneous cas e, the Ising -like sp ectrum of the dia gonal-to -diagona l transfer ma trix was shown by a pplying the functional relations amo ng the transfer ma trices [1 , 2, 6–8, 4 1]. Ther e are three t ype s of the functiona l relations [6–8 ]: the first relation is based on the fact that the transfer ma trix of the SCP mo del is exactly a Q -o p e rator for the τ 2 -mo del [13 , 40], and it gives eigenv alues of the tr ansfer matrix o f the τ 2 -mo del. The second rela tion is interpreted as a T - system [32, 33], which r ecursively genera tes the eigenv a lues of the transfer matrices in the fusion hierarch y . The third relation leads to the eigenv alues of the pr o duct of the dia gonal-to -diagona l transfer matrices of the SCP mo del with a constra int on the sp ectral para meter s. The algebra ic Bethe ansatz of the τ 2 -mo del given in the previous section plays the sa me role as the fir st functional relation [41]. The a lgebraic approach formulated in this section plays a similar role a s the second and third functional relations. 10 3.3 Pro of of Prop osition 3.1 The subsection is devoted to a pro of of Prop os ition 3.1 . Our strategy is to derive a recurs ion relation for the coefficients c τ ′ τ n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) in the re la tion (3.1). Lemma 3 .3. The Y ang-Ba xter r elation (2.13) is e quivalent to the fol lowing r elations: α τ ′ ( z ) A ( z ) T τ ′ τ + β τ ′ ( z ) C ( z ) T τ ′ − 1 τ = α τ ( z ) T τ ′ τ A ( z ) + γ τ ( z ) T τ ′ τ − 1 B ( z ) , α τ ′ ( z ) B ( z ) T τ ′ τ + β τ ′ ( z ) D ( z ) T τ ′ − 1 τ = β τ +1 ( z ) T τ ′ τ +1 A ( z ) + δ τ ( z ) T τ ′ τ B ( z ) , γ τ ′ +1 ( z ) A ( z ) T τ ′ +1 τ + δ τ ′ ( z ) C ( z ) T τ ′ τ = α τ ( z ) T τ ′ τ C ( z ) + γ τ ( z ) T τ ′ τ − 1 D ( z ) , γ τ ′ +1 ( z ) B ( z ) T τ ′ +1 τ + δ τ ′ ( z ) D ( z ) T τ ′ τ = β τ +1 ( z ) T τ ′ τ +1 C ( z ) + δ τ ( z ) T τ ′ τ D ( z ) , (3.9) wher e α τ ( z ) = − y q 1 y q 2 z + µ q 1 µ q 2 ω τ , β τ ( z ) = − z ( y q 1 − x q 2 µ q 1 µ q 2 ω τ ) , γ τ ( z ) = y q 2 − x p 1 µ q 1 µ q 2 ω τ , δ τ ( z ) = 1 − x q 1 x q 2 µ q 1 µ q 2 z ω τ +1 . Her e we have omitt e d the dep endenc e of the sp e ctr al p ar ameters q 1 and q 2 in the c o efficients α τ ( z ) , β τ ( z ) , γ τ ( z ) , δ τ ( z ) and the op er ator T τ ′ τ . Lemma 3 .4. F or the op er a tors T τ ′ τ , we have µ τ ′ τ ( z ) B ( z ) T τ ′ τ = T τ ′ τ B ( z ) + ν τ +1 ( z ) T τ ′ τ +1 A ( z ) − ν τ ′ ( z ) T τ ′ − 1 τ D ( z ) − ν τ ′ ( z ) ν τ +1 ( z ) T τ ′ − 1 τ +1 C ( z ) . (3.10) Her e µ τ ′ τ ( z ) and ν τ ( z ) c oincide with those define d in (3.2 ) , r esp e ctively, by sett ing q 1 = ( x q , y q , µ q ) and q 2 = ( y q , x q ω s , µ − 1 q ) . Pr o of. F ro m the second and fourth r elations in Le mma 3.3, we have T τ ′ τ B ( z ) =  α τ ′ ( z ) δ τ ( z ) − β τ ′ ( z ) δ τ ( z ) γ τ ′ ( z ) δ τ ′ − 1 ( z )  B ( z ) T τ ′ τ − β τ +1 ( z ) δ τ ( z ) T τ ′ τ +1 A ( z ) + β τ ′ ( z ) δ τ ′ − 1 ( z ) T τ ′ − 1 τ D ( z ) + β τ ′ ( z ) δ τ ′ − 1 ( z ) β τ +1 ( z ) δ τ ( z ) T τ ′ − 1 τ +1 C ( z ) . By setting q 1 = ( x q , y q , µ q ) and q 2 = ( y q , x q ω s , µ − 1 q ), we prov e the relation. Lemma 3.5. L et I = { i ℓ } , J = { j ℓ } and K = { k ℓ } b e such disjoint subsets of the set Σ n = { 1 , 2 , . . . , n } that ♯I = n B , ♯J = n A , ♯K = n D and n B + n A + n D = n . The c o efficie nts 11 c n ( I ; J ; K ) = c τ ′ τ n ( I ; J ; K ) in the r elation (3.1) satisfy the fol lowing r e cursion r elation on n : c n ( I ; J ; K ) = c n − 1 ( I \ { n } ; J ; K ) 1 µ τ ′ − n D ,τ + n A ( z n ) + c n − 1 ( I ; J \ { n } ; K ) ν τ + n A ( z n ) µ τ ′ − n D ,τ + n A − 1 ( z n ) Y i ∈ I ω f ni − X j ∈ J c n − 1 ( I ∪ { j } \ { n } ; J \ { j } ; K ) ν τ + n A ( z n ) µ τ ′ − n D ,τ + n A − 1 ( z n ) ω  Y i ∈ I \{ n } ω f j i  g nj − c n − 1 ( I ; J ; K \ { n } ) ν τ ′ − n D +1 ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) Y i ∈ I ω f in − X k ∈ K c n − 1 ( I ∪ { k } \ { n } ; J ; K \ { k } ) ν τ ′ − n D +1 ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) ω  Y i ∈ I \{ n } ω f ik  g nk − X k ∈ K c n − 1 ( I ∪ { k } ; J \ { n } ; K \ { k } ) ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω  Y i ∈ I ω f ni ω f ik  g nk + X j ∈ J c n − 1 ( I ∪ { j } ; J \ { j } ; K \ { n } ) ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω  Y i ∈ I ω f j i ω f in  g nj + X j ∈ J k ∈ K c n − 1 ( I ∪ { j } ∪ { k } \ { n } ; J \ { j } ; K \ { k } ) × ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω f j k  Y i ∈ I \{ n } ω f j i ω f ik  g nj g nk . Her e, if the set S 1 is not a subset of S , we set c n ( S \ S 1 ; · ; · ) = c n ( · ; S \ S 1 ; · ) = c n ( · ; · ; S \ S 1 ) = 0 . Pr o of. W e a pply the oper ator B n = B ( z n ) to b oth sides of the equa tion (3 .1) with n − 1 in place of n . Let ˜ I = { ˜ i ℓ } , ˜ J = { ˜ j ℓ } and ˜ K = { ˜ k ℓ } be such disjoin t subsets of the set Σ n − 1 that ♯ ˜ I = m B , ♯ ˜ J = m A , ♯ ˜ K = m D and m B + m A + m D = n − 1. By us ing the relatio n (3.10), we have B 1 · · · B n − 1 B n T τ ′ τ | 0 i = X ˜ I , ˜ J , ˜ K c τ ′ τ n − 1 ( ˜ I ; ˜ J ; ˜ K ) B n T τ ′ − m D τ + m A B ˜ i 1 · · · B ˜ i m B A ˜ j 1 · · · A ˜ j m A D ˜ k 1 · · · D ˜ k m D | 0 i = X ˜ I , ˜ J , ˜ K c τ ′ τ n − 1 ( ˜ I ; ˜ J ; ˜ K )  1 µ τ ′ − m D ,τ + m A ( z n ) T τ ′ − m D τ + m A B n + ν τ + m A +1 ( z n ) µ τ ′ − m D ,τ + m A ( z n ) T τ ′ − m D τ + m A +1 A n − ν τ ′ − m D ( z n ) µ τ ′ − m D ,τ + m A ( z n ) T τ ′ − m D − 1 τ + m A D n − ν τ ′ − m D ( z n ) ν τ + m A +1 ( z n ) µ τ ′ − m D ,τ + m A ( z n ) T τ ′ − m D − 1 τ + m A +1 C n  × B ˜ i 1 · · · B ˜ i m B A ˜ j 1 · · · A ˜ j m A D ˜ k 1 · · · D ˜ k m D | 0 i . By ar ranging the op er ators A ( z ), B ( z ), C ( z ) a nd D ( z ) in the order B AD C with the rela tions in Lemma A.1 and by rewriting the terms in the form of the right-hand side o f (3.1), we obtain the recursion relatio n. W e now pr ov e P rop osition 3.1. F rom the sy mmetry of the relation (3.1), it is enough to so lve the re c ursion r elation in the case i 1 < · · · < i n B < j 1 < · · · < j n A < k 1 < · · · < k n D . First we 12 consider the case n B = n , that is, i ℓ = ℓ for ℓ = 1 , 2 , . . . , n and J = K = φ . The recursion relation is reduced to c n ( I ; φ ; φ ) = c n − 1 ( I \ { n } ; φ ; φ ) 1 µ τ ′ τ ( z n ) . F rom the initial condition c 0 ( φ ; φ ; φ ) = 1, the recur sion relation is solved as c n ( I ; φ ; φ ) = Y i ∈ I 1 µ τ ′ τ ( z i ) , which is cons istent with the form (3.1 ). Second we consider the case n B + n A = n , tha t is, n ∈ { j ℓ } and K = φ . The recursion relatio n is r educed to c n ( I ; J ; φ ) = c n − 1 ( I ; J \ { n } ; φ ) ν τ + n A ( z n ) µ τ ′ ,τ + n A − 1 ( z n ) Y i ∈ I ω f ni . By using the result in the case n B = n , we obtain c n ( I ; J ; φ ) = Y i ∈ I 1 µ τ ′ τ ( z i ) n A Y p =1 ν τ + p ( z j p ) µ τ ′ ,τ + p − 1 ( z j p ) Y i ∈ I j ∈ J ω f j i , which is also consistent with the form (3.1). Thir d we consider the ca se n ∈ { k ℓ } . The recursion relation is reduced to c n ( I ; J ; K ) = − c n − 1 ( I ; J ; K \ { n } ) ν τ ′ − n D +1 ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) Y i ∈ I ω f in + X j ∈ J c n − 1 ( I ∪ { j } ; J \ { j } ; K \ { n } ) ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω  Y i ∈ I ω f j i ω f in  g nj . (3.11) Note that, in the case, the co efficients c n − 1 ( { i ′ ℓ } ; { j ′ ℓ } ; { k ′ ℓ } ) with ge ne r al sets { i ′ ℓ } , { j ′ ℓ } and { k ′ ℓ } , which a re not necessarily in the order i ′ 1 < · · · < i ′ n B < j ′ 1 < · · · < j ′ n A < k ′ 1 < · · · < k ′ n D , a ppe a r. Assume that the co efficients c n − 1 ( I ; J ; K \ { n } ) and c n − 1 ( I ∪ { j } ; J \ { j } ; K \ { n } ) in (3.1 1) are given in the fo r m (3 .1 ). Substituting the form of the co efficien t c n − 1 ( { i ℓ } ; { j ℓ } ; { k ℓ } \ { n } ) and k n D = n into the fir s t term of (3.11), we obtain − c n − 1 ( { i ℓ } ; { j ℓ } ; { k ℓ } \ { n } ) ν τ ′ − n D +1 ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) n B Y r =1 ω f i r n = − ( − ) n D − 1 n B Y r =1 1 µ τ ′ τ ( z i r ) n A Y p =1 ν τ + p ( z j p ) µ τ ′ ,τ + p − 1 ( z j p ) n D − 1 Y q =1 ν τ ′ − q +1 ( z k q ) µ τ ′ − q +1 ,τ ( z k q ) ν τ ′ − n D +1 ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) × Y i ∈{ i ℓ } j ∈{ j ℓ } ω f j i Y i ∈{ i ℓ } k ∈{ k ℓ }\{ n } ω f ik Y j ∈{ j ℓ } k ∈{ k ℓ }\{ n } ω f j k n B Y r =1 ω f i r n = c n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) Y j ∈{ j ℓ } 1 ω f j n . (3.12) In a similar way , substituting the for ms of the co efficients c n − 1 ( { i ℓ } ∪ { j p } ; { j ℓ } \ { j p } ; { k ℓ } \ { n } ) 13 and k n D = n in to the sec ond term of (3.11), w e obtain n A X p =1 j p 6 = n c n − 1 ( { i ℓ } ∪ { j p } ; { j ℓ } \ { j p } ; { k ℓ } \ { n } ) ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω  n B Y r =1 ω f j p i r ω f i r n  g nj p = n A X p =1 j p 6 = n ( − ) n D − 1 n B Y r =1 1 µ τ ′ τ ( z i r ) 1 µ τ ′ τ ( z j p ) p − 1 Y p ′ =1 ν τ + p ′ ( z j p ′ ) µ τ ′ ,τ + p ′ − 1 ( z j p ′ ) n A Y p ′ = p +1 ν τ + p ′ − 1 ( z j p ′ ) µ τ ′ ,τ + p ′ − 2 ( z j p ′ ) n D − 1 Y q =1 ν τ ′ − q +1 ( z k q ) µ τ ′ − q +1 ,τ ( z k q ) × ν τ ′ − n D +1 ( z n ) ν τ + n A ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ω Y i ∈{ i ℓ }∪{ j p } j ∈{ j ℓ }\{ j p } ω f j i Y i ∈{ i ℓ }∪{ j p } k ∈{ k ℓ }\{ n } ω f ik Y j ∈{ j ℓ }\{ j p } k ∈{ k ℓ }\{ n } ω f j k  n B Y r =1 ω f j p i r ω f i r n  g nj p = − c n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) n A X p =1 µ τ ′ ,τ + n A − 1 ( z j p ) µ τ ′ τ ( z j p ) µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ν τ + n A ( z n ) ν τ + n A ( z j p ) g nj p f j p n Y j ∈{ j ℓ }\{ j p } f j j p f j n . (3.13) Hence, b y combining (3.12) a nd (3.13), the co efficient c n ( { i ℓ } ; { j ℓ } ; { k ℓ } ) is s hown to b e in the form (3.1) if the following relation holds : µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) n A Y p =1 1 ω f j p n − n A X p =1 µ τ ′ ,τ + n A − 1 ( z j p ) µ τ ′ τ ( z j p ) µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ν τ + n A ( z n ) ν τ + n A ( z j p ) g nj p f j p n Y j ∈{ j ℓ }\{ j p } f j j p f j n = 1 , which is the iden tity in Lemma B.3. 4 The sl 2 lo op algebra symmetry of the τ 2 -mo del and de- generate eigenspaces 4.1 Gauge transformations on t he L -op erat or W e now intro duce another L -op erator in order to show the sl 2 lo op algebra symmetry of the τ 2 - mo del. The degenerate eige ns pace of the transfer matrix construc ted from the new L -op erator is ident ical to the degenerate eigenspace o f the τ 2 -mo del which we hav e intro duced in Section 2 .2. Let us introduce the L -op erator ˜ L i ( z ) ∈ End( C 2 ⊗ ( C N ) ⊗ L ) , ( i = 1 , 2 , . . . , L ) given by ˜ L i ( z ) =   q − 1 2  z ( k 1 2 ) i − z − 1 ( k − 1 2 ) i  ( q − q − 1 )( f ) i ( q − q − 1 )( e ) i q 1 2  z ( k − 1 2 ) i − z − 1 ( k 1 2 ) i    . (4.1) Here q is not a r apidity on the F er mat curve (2.1 ) but a generic parameter , and { ( k ) i , ( e ) i , ( f ) i } is the N -dimensio nal representation of the quantum algebra U q ( sl 2 ) non-trivia lly a cting only on the i th co mpo nent of the quan tum space ( C N ) ⊗ L as k v σ = εq N − 1 − 2 σ v σ , ev σ = ε α [ N − σ ] v σ − 1 , f v σ = α − 1 [ σ + 1 ] v σ +1 , with α 6 = 0 and [ n ] = q n − q − n q − q − 1 . W e s e t ε = 1 for o dd N a nd ε = − 1 for even N . One sees that the L -op era tor ˜ L i ( z ) is nothing but that of a n XXZ s pin c hain with N -state lo ca l spins and a twist 14 parameter. The L -op era tor ˜ L i ( z ) satisfies the Y ang-Baxter relatio n (2.5) with the R - matrix of the six-vertex model given b y R 6v ( z ) =         1 − z 2 q 2 0 0 0 0 (1 − z 2 ) q z (1 − q 2 ) 0 0 z (1 − q 2 ) (1 − z 2 ) q 0 0 0 0 1 − z 2 q 2         . (4.2) W e int ro duce the mono dro m y matrix ˜ T ( z ; { p } ) ∈ End  C 2 ⊗ ( C N ) ⊗ L  and the transfer matrix ˜ τ ( z ) = ˜ τ ( z ; { p } ) ∈ End  ( C N ) ⊗ L  as ˜ T ( z ; { p } ) = L Y i =1 ˜ L i ( t 1 2 p i z q 1 2 ) =:   ˜ A ( z ) ˜ B ( z ) ˜ C ( z ) ˜ D ( z )   , ˜ τ ( z ; { p } ) = tr C 2  ˜ T ( z ; { p } )  . (4.3) In a wa y similar to Section 2 .2, w e apply the algebraic Bethe-ansatz metho d to the tr ansfer matrix ˜ τ ( z ; { p } ) to obtain Bethe eigenstates. The asso ciated Bethe equatio ns are given by L Y n =1 t p n z 2 i εq N − 1 t p n z 2 i q 2 − εq N = Y j ( 6 = i ) z 2 i q 2 − z 2 j z 2 i − z 2 j q 2 . (4.4) The transfer matr ix τ ( z 2 ; { p } ) of the τ 2 -mo del defined in (2.9) is equiv alen t to the transfer matrix ˜ τ ( z ; { p } ) at εq N = 1. W e set ω = q 2 with the primitive N th ro ot of unity q for o dd N and the primitive 2 N th r o ot of unity q for ev en N , and take α = x 1 2 p i y − 1 2 p i . Then, in terms of the op erator s Z i and X i , the re pr esentation of the quant um algebra U q ( sl 2 ) is express ed as ( k ) i = q − 1 Z − 1 i , ( e ) i = x 1 2 p i y − 1 2 p i q − q − 1 X − 1 i ( Z − 1 2 i − Z 1 2 i ) , ( f ) i = x − 1 2 p i y 1 2 p i q − q − 1 ( Z 1 2 i − Z − 1 2 i ) X i , by which the L -op er a tor ˜ L i ( z ) at εq N = 1 tak es the for m ˜ L i ( z ) =   q − 1 2  − z q − 1 2 Z − 1 2 i + z − 1 q 1 2 Z 1 2 i  x − 1 2 p i y 1 2 p i ( Z 1 2 i − Z − 1 2 i ) X i x 1 2 p i y − 1 2 p i X − 1 i ( Z − 1 2 i − Z 1 2 i ) q 1 2  − z q 1 2 Z 1 2 i + z − 1 q − 1 2 Z − 1 2 i    . The L -o pe rator ˜ L i ( z ) is transformed to the L -op er ator L i ( z 2 ; p i , ¯ p i ) defined in (2 .8) as follows:   1 0 0 z − 1 q 1 2   t 1 2 p i z q 1 2 Z 1 2 i ˜ L i  t 1 2 p i z q 1 2    1 0 0 z q − 1 2   = L i ( z 2 ; p i , ¯ p i ) . Through the gauge transforma tion, the Y ang-Ba x ter rela tion with the R -ma tr ix R 6v ( z ) fo r the L -op era tor ˜ L i ( z ) is transformed to the Y ang-Baxter r elation (2.1 0) with the R -matrix R ( z ) (2.6). In the case of o dd N , the L -op erato r ˜ L i ( z ) sa tisfies the Y ang- Baxter relation (2.7). On the o ther hand, in the case o f even N , the L -o p erator ˜ L i ( z ) do es not satisfy the relation (2.7) due to the m ultiplication by the op er ator Z 1 2 i . How ever, the conserved op erator s deriv ed fro m a n expansio n of the log arithm of the tra nsfer matrix ˜ τ ( z ; { p } ) commute with the transfer ma trix t ( q 1 , q 2 ; { p } ) since the op er ators Z 1 2 i are canceled out in the der iv ation. F urthermor e, the pro duct Z 1 2 1 · · · Z 1 2 L , which a ppea rs in ea ch entry of the mono dro my matrix ˜ T ( z ; { p } ), acts as the cons tant q M on the sector spanned b y the vectors v σ 1 ⊗ · · · ⊗ v σ L satisfying σ 1 + · · · + σ L = M . As w e sha ll see be low, each Bethe eigens ta te and its L ( sl 2 )-descendant state b elong to o ne of the s ectors. Therefor e, transfer matrice s ˜ τ ( z ; { p } ) and τ ( z ; { p } ) thus shar e a set of commo n eig env ectors. 15 4.2 The sl 2 lo op algebra symmetry W e now show the sl 2 lo op algebra L ( sl 2 ) symmetry of the τ 2 -mo del. W e first obtain a represen- tation of the quan tum affine alge br a U ′ q ( sl 2 ) in a limit of the ent ries o f the mono dr omy matrix ˜ T ( z ; { p } ): A := lim z →∞ ˜ A ( z ) m ( z ) q − L 2 = lim z → 0 ˜ D ( z ) m ( z ) q L 2 = k 1 2 ⊗ · · · ⊗ k 1 2 , B ± := lim z ± 1 →∞ ˜ B ( z ) m ( z ) n ± ( z ) = L X i =1 q L +1 2 − i ( t ∓ 1 2 p i q ∓ 1 2 ) k ± 1 2 ⊗ · · · ⊗ k ± 1 2 | {z } i − 1 ⊗ f ⊗ k ∓ 1 2 ⊗ · · · ⊗ k ∓ 1 2 | {z } L − i , C ± := lim z ± 1 →∞ ˜ C ( z ) m ( z ) n ± ( z ) = L X i =1 q − L +1 2 + i ( t ∓ 1 2 p i q ∓ 1 2 ) k ∓ 1 2 ⊗ · · · ⊗ k ∓ 1 2 | {z } i − 1 ⊗ e ⊗ k ± 1 2 ⊗ · · · ⊗ k ± 1 2 | {z } L − i , where m ( z ) = Q L i =1 ( t 1 2 p i z q 1 2 − t − 1 2 p i z − 1 q − 1 2 ) and n ± ( z ) = ± z ∓ 1 ( q − q − 1 ). They indeed give a finite-dimensional repre sentation of U ′ q ( ˆ sl 2 ) through the map π ( L ) : U ′ q ( ˆ sl 2 ) → ( C N ) ⊗ L defined b y π ( L ) : k 0 , 1 , e 0 , e 1 , f 0 , f 1 7→ A ∓ 2 , B + , C + , C − , B − , where { k i , e i , f i | i = 0 , 1 } is a s et of the Chev alley genera tors of U ′ q ( ˆ sl 2 ). Second w e show that, in the limit εq N → 1, the repr esentation π ( L ) of the quan tum affine algebra U ′ q ( ˆ sl 2 ) gives a finite-dimensio nal representation of a Bor el subalg ebra o f L ( sl 2 ). The sl 2 lo op algebr a is rea lized by the Drinfeld generato rs { h n , x + n , x − n | n = 0 , 1 , 2 , . . . } satisfying [ h n , h m ] = 0 , [ h n , x ± m ] = ± 2 x ± n + m , [ x + n , x − m ] = h n + m . The a lgebra has t wo Borel subalgebras b + generated b y { h n , x + n , x − m | n > 0 , m > 0 } and b − generated by { h − n , x + − m , x − − n | n > 0 , m > 0 } . Define the oper ators H ( N ) := 1 N L X i =1 id ⊗ · · · ⊗ id ⊗ h ⊗ id ⊗ · · · ⊗ id , B ( n ) ± := lim εq N → 1 ( B ± ) n [ n ]! , C ( n ) ± := lim εq N → 1 ( C ± ) n [ n ]! for o dd N , where hv σ = ( N − 1 − 2 σ ) v σ and [ n ]! = [ n ][ n − 1 ] · · · [1]. The op er ators B ( N ) ± and C ( N ) ± are well- defined in the limit εq N → 1 since both the op erator s ( B ± ) N and ( C ± ) N include the factor [ N ]!. They satisfy the relations [ B ( N ) + , B ( N ) − ] = [ C ( N ) + , C ( N ) − ] = 0 , [ H ( N ) , B ( N ) ± ] = − 2 B ( N ) ± , [ H ( N ) , C ( N ) ± ] = 2 C ( N ) ± , [ B ( N ) ± , [ B ( N ) ± , [ B ( N ) ± , C ( N ) ± ]]] = 0 , [ C ( N ) ± , [ C ( N ) ± , [ C ( N ) ± , B ( N ) ± ]]] = 0 . Here the las t tw o rela tions ar e o bta ined fro m the limit εq N → 1 of the higher -order q -Serre r elations in U ′ q ( ˆ sl 2 ) [36]. Then w e find that the map ϕ + : b + → End( ( C N ) ⊗ L ) defined by ϕ + ( h 0 ) := H ( N ) , ϕ + ( x + 0 ) := C ( N ) + , ϕ + ( x − 1 ) := B ( N ) + 16 is extended to a finite-dimensional representation of the Borel subalgebr a b + and the map ϕ − : b − → End(( C N ) ⊗ L ) defined by ϕ − ( h 0 ) := H ( N ) , ϕ − ( x + − 1 ) := C ( N ) − , ϕ − ( x − 0 ) := B ( N ) − is also extended to tha t of the Bor el s ubalgebra b − . Prop ositi on 4.1. The τ 2 -mo del in a se ct or sp e cifie d b elow has the Bor el sub algebr a symmetry in the fol lowing sense: the tr ansfer matrix ˜ τ ( z ) = ˜ τ ( z ; { p } ) at εq N = 1 satisfies  ˜ τ (1) − 1 ˜ τ ( z ) , ϕ + ( x )  = 0 for x ∈ b + in the se ct or with A 2 = q L and  ˜ τ (1 ) − 1 ˜ τ ( z ) , ϕ − ( x )  = 0 for x ∈ b − in the se ct or with A 2 = q − L . Pr o of. In the limit εq N → 1, we hav e ˜ A ( z ) B ( N ) ± = εB ( N ) ± ˜ A ( z ) − z ± 1 q − L 2 B ( N − 1) ± ˜ B ( z ) A ± 1 , ˜ D ( z ) B ( N ) ± = ε B ( N ) ± ˜ D ( z ) + z ± 1 q L 2 B ( N − 1) ± ˜ B ( z ) A ∓ 1 , ˜ A ( z ) C ( N ) ± = εC ( N ) ± ˜ A ( z ) + z ± 1 q − L 2 C ( N − 1) ± ˜ C ( z ) A ± 1 , ˜ D ( z ) C ( N ) ± = ε C ( N ) ± ˜ D ( z ) − z ± 1 q L 2 C ( N − 1) ± ˜ C ( z ) A ∓ 1 . By considering them in the sector with A 2 = q ± L , we pr ove the pro p o sition. Let us consider the condition A 2 = q ± L in detail. F r om the r elation A 2 = k ⊗ · · · ⊗ k , we have A 2 = ε L q ( N − 1) L − 2 M = q − L − 2 M in the sector spanned by the vectors v σ 1 ⊗ · · · ⊗ v σ L satisfying σ 1 + · · · + σ L = M . Then the condition A 2 = q L means M + L ≡ 0 mo d N and A 2 = q − L means M ≡ 0 mod N . One notices that the r eference state | 0 i belongs to the sector with A 2 = q − L . W e now show the sl 2 lo op algebra symmetry of the τ 2 -mo del. It is known that every finite- dimensional ir r educible representation of the Borel subalgebra b ± is ex tended to that of the s l 2 lo op algebr a [14, 22 ]. Ther efore, it follows fro m Pr op osition 4.1 tha t the transfer matrix of the τ 2 -mo del has the sl 2 lo op algebr a symmetry . Third we now show that any given regular Bethe state | R i in the sector with A 2 = q ± L is a highest weigh t vector with resp ect to the representation ϕ ± of the Bor el subalgebra b ± and the highest weight r epresentation generated by the Bethe state is irreducible. A v ector Ω is calle d highest weigh t of the Bo rel subalgebra b + if it is annihila ted by x + n , ( n > 0) and is diagona lized by h n , ( n > 0), and is called highest weigh t of b − if it is annihila ted by x + − n , ( n > 0) and is diagonalized by h − n , ( n > 0). The conditions are eq uiv alen t to [21] x + 0 Ω = 0 , h 0 Ω = r Ω , ( x + 0 ) m m ! ( x − 1 ) m m ! Ω = χ + m Ω , ( m ∈ Z > 0 ) for b + x + − 1 Ω = 0 , h 0 Ω = r Ω , ( x + − 1 ) m m ! ( x − 0 ) m m ! Ω = χ − m Ω , ( m ∈ Z > 0 ) for b − , 17 where r ∈ Z > 0 and χ ± m ∈ C . By using the set { χ ± m } for a highest weigh t vector of the Borel subalgebra b ± , we define the highest weigh t p olynomial a s [23 , 24] P ± D ( ζ ) = X m > 0 χ ± m ( − ζ ) m . Prop ositi on 4.2. At εq N = 1 , every r e gular Bethe state | R i in the se ctor with A 2 = q ± L is a highest weight ve ctor with r esp e ct to the r epr esent ation ϕ ± of t he Bor el su b algebr a b ± . The highest weight p ol ynomial is given by P ± D ( ξ N ) = 1 N N − 1 X τ =0 L Y n =1 1 − t ∓ N p n ξ N 1 − t ∓ 1 p n ξ q ± 2 τ ! 1 F ± ( ξ q ± 2 τ ) F ± ( ξ q ± 2( τ +1) ) , (4.5) wher e F ± ( ξ ) = Q R i =1 (1 − ξ z ± 2 i ) and { z i } is a r e gular solution of the Bethe e quations (4.4) . W e shall give a pro of of Pro po sition 4.2 in App endix C. Here we can directly show that every highest weigh t vector of the B orel s ubalgebra beco mes a highest weigh t vector of the sl 2 lo op algebra in a finite-dimensional highest weigh t r epresentation (see, App endix A of [25]). Let us discuss a physical consequence of g eneric inhomogeneous parameters. F o r a given reg ular Bethe s ta te, the zeros o f p olynomial P ± D ( ζ ) (4.5) should b e distinct, if inhomogeneo us parameters { p n } on the F erma t cur ve (2.1) are given by generic v alues. If they ar e distinct, it therefore follows that the hig hest weigh t repr esentation gener ated by the regula r Bethe sta te is ir reducible and the polyno mial P ± D ( ζ ) is identified with the Drinfeld p olynomia l [15, 23, 2 4, 27 ]. As s uming that the ze r os of the Drinfeld polynomia l are distinct, we express the distinct zer os P ± D ( ζ ) by ζ i , ( i = 1 , 2 , . . . , r = deg P ± D ( ζ )). Then, the repr esentation is is o morphic to the tensor pro duct of t wo-dimensional ev aluation repr esentations, V 1 ( ζ 1 ) ⊗ · · · ⊗ V 1 ( ζ r ), and the τ 2 -mo del in the sec to r A 2 = q ± L has the 2 r -dimensional degener ate eigenspace asso ciated with the regular Bethe state. 4.3 Complete N -strings and degenerate eigen v ectors of the sl 2 lo op algebra In P rop ositions 4 .1 and 4.2 of Section 4.2 , it has b een shown in the sector that the τ 2 -mo del has the sl 2 lo op algebr a L ( sl 2 ) symmetry a nd also that every regular Bethe state | R i is a hig hest weigh t vector of L ( sl 2 ). Therefor e, the deg e nerate eigenspace of the τ 2 -mo del asso cia ted with the regular Bethe sta te | R i is g iven by the highest weight representation ge ner ated b y | R i through generator s of L ( sl 2 ). Let us define a complete N -string by the set { e Λ ω − l | l = 1 , 2 , . . . , N } , wher e we call Λ the cen ter of the string [29]. By adding m complete N -strings { e Λ j ω − l | l = 1 , 2 , . . . , N , j = 1 , 2 , . . . , m } to a regular so lution { z i | i = 1 , 2 , . . . , R } of the Bethe equations (2.11) a nd tak ing the limit Λ j → ±∞ , we o btain a formal solution { z i } ∪ { e Λ j ω − l } of the Bethe equations (2.11 ) with M = R + mN . W e call it a non-regular solution. It is clear tha t the tra nsfer-matrix eig env alue (2.12) for a non-regular solution { z i } ∪ { e Λ j ω − l } is the same as that of the or iginal regular solution { z i } . W e now discuss that the SCP tr ansfer matr ix t ( q , ¯ q ( s ); { p } ) with q = ( x q , y q , µ q ) and ¯ q ( s ) = ( y q , x q ω s , µ − 1 q ) sho uld hav e degenera te eigenspa c e s. W e o bserve that the eigenv alue (3 .4) o f the 18 SCP transfer matrix t ( q , ¯ q ( s ); { p } ) with a non-r egular solution { z i } ∪ { e Λ j ω − l } is the same as that with the original regular s olution { z i } . As a consequence, the degenerate eigenspace of the transfer matrix τ ( z ; { p } ), whic h contains a regular Bethe s tate and non-r egular Bethe states, corresp onds to a degene r ate eigenspace of the tra nsfer matrix t ( q , ¯ q ( s ); { p } ). Non-regular Bethe eig enstates with complete N - strings ma y v anish as w e shall see in Sec - tion 4.2. How ever, there ar e several approaches to obtain non-zer o eigenstates corres po nding to non-regula r solutio ns s uch as complete N - strings [19, 20 , 25, 26 , 28]. Thus, from the observ ation that the eigenv alue (3.4 ) do e s not dep end on complete N -strings, w e suggest tha t the SCP trans- fer matrix t ( q , ¯ q ( s ); { p } ) is also degenera te in the L ( sl 2 )-degenerate eigenspace of the τ 2 -mo del generated by a regular Bethe state. W e thus pr op ose a co njecture that the sl 2 lo op algebra symmetr y of the τ 2 -mo del gives a degenerate eigenspace of the SCP trans fer matrix t ( q , ¯ q ( s ); { p } ) in the secto r. In tuitively , in terms of complete N -s trings, we may interpret that every L ( sl 2 )-descendant state of a given reg ular Bethe state should b e expressed as some linear combination o f such no n- regular Bethe states consisting of complete N -strings. F ur thermore, the Drinfeld p oly nomial P ± D ( ζ ) (4.5) is iden tical to the SCP po lynomial P SCP ( ζ ) (3.7) ass o ciated with the regular Bethe state. Th us, the L ( sl 2 )-degenerate eigenspace of the τ 2 -mo del should hav e exa c tly the same dimensio ns as the in v ariant subspace asso ciated with the Is ing-like s p ectr um (3.6) characterized b y the SCP polyno mial P SCP ( ζ ). 4.4 The sl 2 lo op algebra degeneracy and the Ising-lik e sp ectrum W e now discuss an imp or tant consequence of the co mm utativity of the SCP transfer matrix t ( q , ¯ q ( s ); { p } ) with the transfer matr ix τ ( z , { p } ) of the τ 2 -mo del. Here we no te that basis vec- tors diago nalizing co mm uting tr ansfer matrices do not depend o n the sp ectra l parameters . W e define the completeness of the Bethe ans a tz of the τ 2 -mo del at the sup erintegrable point by the following conjecture: Conjecture 4.3. Al l r e gular Bethe states in the se ctor with A 2 = q ± L and their desc endants with r esp e ct t o the sl 2 lo op algebr a give the c omplete s et of the Hilb ert sp ac e in the se ctor on which tr ansfer mat rix τ ( z , { p } ) of the τ 2 -mo del acts. Her e we r e c al l εq N = 1 . F or generic v alues of s p ectr al parameter z , regula r Bethe states in the sector are non-dege nerate with r e s p e ct to the eigenv alue of tra nsfer matrix τ ( z , { p } ). The degeneracy in the eigensp e ctrum of transfer matrix τ ( z , { p } ) should b e given only b y the sl 2 lo op algebra symmetry . Similarly , for generic sp ectral par ameters, reg ular Bethe states a re non-degenerate with r e s p e c t to the eigenv a lue of the SCP transfer matrix t ( q , ¯ q ( s ); { p } ). The eig env alue of t ( q , ¯ q ( s ); { p } ) is also gener ic with resp ect to the spectra l parameters , a s shown in (3.4). Thu s, if Conjecture 4 .3 is v alid, i.e. the completeness o f the Bethe ansa tz for the τ 2 -mo del is v alid, w e hav e the following co rollar y: Corollary 4.4. In the se ctor with A 2 = q ± L , the SCP tr ansfer matrix t ( q 1 , q 2 ; { p } ) is blo ck- diagonal ize d with r esp e ct to t he L ( sl 2 ) -de gener ate eigensp ac es of the τ 2 -mo del asso cia te d with t he r e gula r Bethe states. Her e we r e c al l εq N = 1 . 19 Assuming the arg uments for de r iving the formula of eigenv alues of the diagonal-to- diagonal transfer matrice s T D ( x q , y q ) and ˆ T D ( x q , y q ), we hav e the following conjecture: Conjecture 4.5. In the L ( sl 2 ) -de gener ate eigensp ac e of t he τ 2 -mo del asso ciate d with a re gular Bethe state | R i , the diagonal-to-diag onal tr ansfer matric es T D ( x q , y q ) and ˆ T D ( x q , y q ) of the SCP mo del have the Ising-like sp e ctru m (3.6) asso ciate d with t he r e gula r Bethe state | R i . Let us consider some examples of the inv arian t subspace of the Is ing-like s p ec tr um asso ciated with a regular Bethe state | R i . If the degree of the Drinfeld poly nomial P ± D ( ζ ) is zero, then | R i is an eigen vector of both of the tw o diagonal-to-diag onal transfer matrices T D ( x q , y q ) and ˆ T D ( x q , y q ). The Bethe state should genera te a singlet of the sl 2 lo op algebra, i.e. a one-dimensio na l highest weigh t r epresentation. One notices that, if R = L ( N − 1) / 2, the degree of the Drinfeld polynomia l P ± D ( ζ ) is ze ro. How ev er, if the degr ee of the Drinfeld p olynomial is nonzero and g iven by r , the SCP trans fer matrix t ( q 1 , q 2 ; { p } ) should b e blo ck-diagonalized a t le a st with resp ect to the L ( sl 2 )-degenerate eigenspace of the τ 2 -mo del asso ciated with | R i . F urthermore, the SCP transfer matrix t ( q , ¯ q ( s ); { p } ) should b e dege ner ate in the 2 r -dimensional L ( sl 2 )-degenerate eigenspace of the τ 2 -mo del a sso ciated with | R i , as it was conjectured in Section 4.3. 4.5 N = 2 case W e verify in the cas e o f N = 2 with a set o f homogeneo us parameters that the Ha miltonian of the SCP mo del has the Ising- like sp ectrum in the L ( sl 2 )-degenerate eigenspa ce of the τ 2 -mo del. In the case, the SCP mo de l is the tw o-dimensional Ising model. The Hamiltonians of the SCP mo del and the τ 2 -mo del are given in the forms H SCP = L X i =1 σ z i + λ L X i =1 σ x i σ x i +1 , H τ 2 = L X i =1 ( σ x i σ y i +1 − σ y i σ x i +1 ) , where σ x , σ y and σ z are Pauli’s ma trices. In ter ms o f Jor dan-Wigner’s fermion op erator s: c i = σ + i i − 1 Y j =1 σ z j , ˜ c k = 1 L L X i =1 e − √ − 1( ki + π 4 ) c i , the Hamiltonian H τ 2 in the secto r with S z := 1 2 P L i =1 σ z i ≡ 0 mo d 2 is wr itten as H τ 2 = X k ∈ K sin( k ) ˜ c † k ˜ c k , where K = { π L , 3 π L , . . . , ( L − 1) π L } . F o r e ven L , the sl 2 lo op algebra symmetry describing a degener ate eigenspace of the τ 2 -mo del is given by h n = X k ∈ K cot 2 n  k 2  ( H ) k , x + n = X k ∈ K cot 2 n +1  k 2  ( E ) k , x − n = X k ∈ K cot 2 n − 1  k 2  ( F ) k , where { ( H ) k , ( E ) k , ( F ) k } is a tw o-dimensio na l r epresentation of the sl 2 algebra given b y ( H ) k = 1 − ˜ c † k ˜ c k − ˜ c † − k ˜ c − k , ( E ) k = ˜ c − k ˜ c k , ( F ) k = ˜ c † k ˜ c † − k . 20 Here w e should remark that for the XX mo del under the per io dic boundary conditions the Chev al- ley genera to rs of the sl 2 lo op algebr a s y mmetry were co nstructed in terms of the free fermio n op erator s [2 6]. The reference state | 0 i , which is a highest weigh t vector, i.e. x + n | 0 i = 0 and h n | 0 i = P k cot 2 n ( k / 2) | 0 i , generates a 2 L/ 2 -dimensional irreducible representation corresp o nding to a de- generate eigenspace of the Hamiltonia n H τ 2 . On the o ther hand, in the sector, the Ha miltonian H SCP is expressed as H SCP = 2 X k ∈ K ( H ) k − 2 λ X k ∈ K  cos( k )( H ) k + sin( k )  ( E ) k + ( F ) k  . It is clear that the Hamiltonia n H SCP acts on the 2 L/ 2 -dimensional irreducible representation space. The 2 L/ 2 eigenv alues of H SCP and the co rresp onding eigenstates ar e given by E ( K + ; K − ) = 2 X k ∈ K + p 1 − 2 λ cos( k ) + λ 2 − 2 X k ∈ K − p 1 − 2 λ cos( k ) + λ 2 , | K + ; K − i = Y k ∈ K + (cos θ k + sin θ k ( F ) k ) Y k ∈ K − (sin θ k − cos θ k ( F ) k ) | 0 i , where K + and K − are such dis jo int subsets o f K that K = K + ∪ K − and tan(2 θ k ) = λ sin( k ) λ cos( k ) − 1 . Ac kno wledgmen ts The author s would like to thank P rof. A. Kuniba and Prof. N. Hatano for helpful comments. One of the authors (A.N.) acknowledges supp or t from Gr ant-in-Aid for Y o ung Scientists (B) No. 2074021 7, and a lso fro m Core Research for Ev olutional Scie nc e a nd T ec hnology of Japa n Science and T ec hnology Agency . The present s tudy is par tially s uppo rted b y Grant-in-Aid for Scient ific Res earch (C) No. 175403 5 1. 21 A Relatio ns among the op erators in (2.9 ) Lemma A. 1. L et A i , B i , C i and D i denote A ( z i ) , B ( z i ) , C ( z i ) and D ( z i ) , r esp e ctively. We have A 0 B i 1 · · · B i n = ω n  n Y p =1 f 0 i p  B i 1 · · · B i n A 0 − n X p =1  Y q ( 6 = p ) f i p i q  g 0 i p B 0 B i 1 · · · i p ˇ · · · B i n A i p ! , D 0 B i 1 · · · B i n = ω n  n Y p =1 f i p 0  B i 1 · · · B i n D 0 + n X p =1  Y q ( 6 = p ) f i q i p  g 0 i p B 0 B i 1 · · · i p ˇ · · · B i n D i p ! , C 0 A i 1 · · · A i n = ω n  n Y p =1 f i p 0  A i 1 · · · A i n C 0 + n X p =1  Y q ( 6 = p ) f i q i p  g 0 i p A 0 A i 1 · · · i p ˇ · · · A i n C i p ! , C 0 D i 1 · · · D i n = ω n  n Y p =1 f 0 i p  D i 1 · · · D i n C 0 − n X p =1  Y q ( 6 = p ) f i p i q  g 0 i p D 0 D i 1 · · · i p ˇ · · · D i n C i p ! , D 0 A i 1 · · · A i n = A i 1 · · · A i n D 0 + ω n n X p =1  Y q ( 6 = p ) f i q i p  g 0 i p B 0 A i 1 · · · i p ˇ · · · A i n C i p + g i p 0 B i p A i 1 · · · i p ˇ · · · A i n C 0  , C 0 B i 1 · · · B i n = ω n B i 1 · · · B i n C 0 + ω 2 n − 1 n X p =1 g 0 i p B i 1 · · · i p ˇ · · · B i n  Y q ( 6 = p ) f 0 i q f i q i p  A 0 D i p −  Y q ( 6 = p ) f i p i q f i q 0  A i p D 0  − X p 6 = q g 0 i p g 0 i q B 0 B i 1 · · · i p ˇ · · · i q ˇ · · · B i n f i p i q  Y r ( 6 = p,q ) f i p i r f i r i q  A i p D i q ! , (A.1) wher e f ij = f ( z i /z j ) and g ij = g ( z i /z j ) with f ( z ) = z − ω ( z − 1) ω , g ( z ) = 1 − ω ( z − 1) ω . Pr o of. The r elations with n = 1 are equiv alen t to the Y a ng-Baxter rela tio n (2.10). F or n > 2, we employ induction on n with the identit y in Lemma B.2 . B Iden titie s W e collect here s e veral useful identities. Lemma B . 1. L et S b e a su bset of Σ R = { 1 , 2 , . . . , R } . We then have Y i ∈ S a i Y j ∈ Σ R \ S f ij ! = Y i ∈ S d i Y j ∈ Σ R \ S f j i ! . The following three iden tities of rational functions ar e prov ed b y verifying tha t all the residues in the left-hand side are zero. Lemma B . 2.  n Y i =1 f ik  −  n Y i =1 f il  ! g kl + n X i =1  Y j ( 6 = i ) f ij  g ki g il = 0 . 22 Let { i ℓ } , { j ℓ } and { k ℓ } b e such disjoint subs ets of the set Σ n = { 1 , 2 , . . . , n } that ♯ { i ℓ } = n B , ♯ { j ℓ } = n A , ♯ { k ℓ } = n D and n B + n A + n D = n . W e hav e the following identities: Lemma B . 3. ω n A n A Y p =1 f j p ,n − µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A ( z n ) + n A X p =1 µ τ ′ ,τ + n A − 1 ( z j p ) µ τ ′ τ ( z j p ) µ τ ′ − n D +1 ,τ ( z n ) µ τ ′ − n D +1 ,τ + n A − 1 ( z n ) ν τ + n A ( z n ) ν τ + n A ( z j p ) ω n A g n,j p n A Y r =1 r ( 6 = p ) f j r ,j p = 0 . or, explicitly, n A Y p =1 z n B + p − z n ω z n B + p − z n − tz n ω τ + n A +1 − 1 tz n ω τ +1 − 1 + n A X p =1 tz n B + p ω τ +1 − 1 tz n ω τ +1 − 1 z n (1 − ω ) z n − z n B + p n A Y r =1 r ( 6 = p ) z n B + r − z n B + p ω z n B + r − z n B + p = 0 . Lemma B . 4. X { j ℓ } , { k ℓ } n B + n A + n D = n n A Y p =1 µ τ + n D ,τ + n D ( z j p ) µ τ + n D ,τ + n D + p − 1 ( z j p ) ν τ + n D + p ( z j p ) n D Y q =1 − µ τ + n D ,τ + n D ( z k q ) µ τ + n D − q +1 ,τ + n D ( z k q ) ν τ + n D − q +1 ( z k q ) × Y i ∈{ i ℓ } j ∈{ j ℓ } ω f j p i r Y i ∈{ i ℓ } k ∈{ k ℓ } ω f k q i r Y j ∈{ j ℓ } k ∈{ k ℓ } ω f k q j p = 0 . or, explicitly, X { j ℓ } , { k ℓ } n A + n D = n − n B ( − ) n D n A Y p =1 1 tz j p ω τ + n D +1 − 1 n D Y q =1 ω q − 1 tz k q ω τ + n D − 1 n A Y p =1 n D Y q =1 z k q − z j p ω z k q − z j p = 0 . C Pro of of Prop osition 4.2 W e give a pro of of Pr op osition 4.2. The detailed pro of for the case of the XXZ-Heisenber g spin chain at roo ts o f unity is presented in [25 ]. Here we show only some different po int s from it. F or simplicit y , we consider the r epresentation ϕ + of the Borel subalgebra b + . Let ˜ A i = ˜ A ( z i ), ˜ B i = ˜ B ( z i ), ˜ C i = ˜ C ( z i ) and ˜ D i = ˜ D ( z i ) fo r i ∈ Σ M = { 1 , 2 , . . . , M } . One of the relations in Lemma A.1 is rewr itten as follows: ˜ C 0 ˜ B 1 · · · ˜ B M = ˜ B 1 · · · ˜ B M ˜ C 0 + M X i =1 ˜ g 0 i ˜ B 1 · · · i ˇ · · · ˜ B M  Y j ( 6 = i ) ˜ f 0 j ˜ f j i  ˜ A 0 ˜ D i −  Y j ( 6 = i ) ˜ f j 0 ˜ f ij  ˜ A i ˜ D 0  − X i 6 = j ˜ g 0 i ˜ g 0 j ˜ B 0 ˜ B 1 · · · i ˇ · · · j ˇ · · · ˜ B M ˜ f ij  Y l ( 6 = i,j ) ˜ f il ˜ f lj  ˜ A i ˜ D j , (C.1) where ˜ f ij = ˜ f ( z i /z j ) = z 2 i q − 1 − z 2 j q z 2 i − z 2 j , ˜ g ij = ˜ g ( z i /z j ) = z i z j ( q − 1 − q ) z 2 i − z 2 j . 23 Lemma C . 1. L et S n = { i 1 , i 2 , . . . , i n } b e a subset of the set Σ M . We have ( C + ) n  Y l ∈ Σ M ˜ B l  | 0 i = ∆( S n ; Σ M ) X S n ⊂ Σ M  Y l ∈ Σ M \ S n ˜ B l  | 0 i (C.2) with the c o efficient ∆( S n ; Σ M ) given by ∆( S n ; Σ M ) =  Y i ∈ S n z i  X P ∈ S n n X l =0 ( − ) l h n l i q n ( n − 1) 2 − ( n − 1) l Y 1 6 j 6 n − l α Σ M \ S n i P j Y n − l 0 ˜ χ + m ( − ξ ) m . (C.3) Lemma C . 3. In the limit Λ → ∞ , that is, ǫ 0 → 0 , we have ˆ ∆( Z N ; Σ R + N ) = ˜ χ + N ([ N ]!) 2 + O ( ǫ 0 ) . Pr o of. Put ξ = ǫ 0 q 2 l +1 in the definition of χ + N (C.3). Then N X l =0 ( − ) l  N l  q − ( N − 1) l Q N − 1 j =1 φ + ( ǫ 0 εq 2 j +2 l − N ) F + ( ǫ 0 q 2 l ) F + ( ǫ 0 q 2 l +2 ) = N X l =0 ( − ) l  N l  ∞ X m =0 ˜ χ + m ( − ǫ 0 q ) m q (2 m − N +1) l = ∞ X m =0 ˜ χ + m ( − ǫ 0 q ) m N − 1 Y l =0 (1 − q 2( m − l ) ) = ˜ χ + N ǫ N 0 [ N ]! q N ( N +1) 2 + N ( q − q − 1 ) N + O ( ǫ N +1 0 ) , where we have used the q -binomial theorem and Q N − 1 l =0 (1 − q 2( m − l ) ) = 0 for 0 6 m 6 N − 1. Prop ositi on C. 4. L et q b e the N th primitive r o o t of u nity for o dd N and t he 2 N t h primitive r o ot of unity for o dd N . The r e gu lar Bethe state | R i in the se ctor with A 2 = q L satisfies ϕ + ( x + 0 ) m m ! ϕ + ( x − 1 ) m m ! | R i = χ + mN | R i , wher e χ + m = lim εq N → 1 ˜ χ + m . Pr o of. W e consider only the case m = 1 . The case of gener al m is prov ed in a similar way by setting M = R + mN a nd n = mN in (C.2). F r om the lemma ab ov e, we have ( C + ) N  Y l ∈ Z N 1 m + ( z l ) n ( z l ) ˜ B l  | R i = χ + N ([ N ]!) 2 | R i + O ( ǫ 0 ) + off-diagonal terms , which, in the limit Λ → ∞ , yields ( C + ) N [ N ]! ( B + ) N [ N ]! | R i = ˜ χ + N | R i + off-diagona l terms . In the secto r with A 2 = q L , the off-dia gonal terms v anish in the limit εq N → 1 [21]. By taking the limit εq N → 1 in the definition of ˜ χ + m (C.3), we have Q N − 1 j =1 φ + ( ξ q 2 j − 1 ) F + ( ξ q ) F + ( ξ q − 1 ) = X m > 0 χ + m ( − ξ ) m . The numerator of the left-hand side is r ewritten as N − 1 Y j =1 φ + ( ξ q 2 j − 1 ) = L Y n =1 1 − t − N p n ξ N q − N 1 − t − 1 p n ξ q − 1 . 25 Then we o btain  L Y n =1 1 − t − N p n ξ N q − N 1 − t − 1 p n ξ q − 1  1 F + ( ξ q ) F + ( ξ q − 1 ) = X m > 0 χ + m ( − ξ ) m . By taking the sum o ver τ = 0 , 1 , . . . , N − 1 after the substitution ξ 7→ ξ q 2 τ +1 , w e hav e N − 1 X τ =0  L Y n =1 1 − t − N p n ξ N 1 − t − 1 p n ξ q 2 τ  1 F + ( ξ q 2 τ ) F + ( ξ q 2 τ +2 ) = N − 1 X τ =0 X m > 0 χ + m ( − ξ q ) m q 2 τ m = N X m > 0 χ + mN ( − ξ N ) m = N P + D ( ξ N ) , which pr oves Prop ositio n 4 .2 . W e give a remark. One can der ive P rop osition 4.2 from the pro of of the spin-1/2 inhomogeneous case thr o ugh the fusion metho d [25]. Ho wev er, we hav e presented the direct and straightforw ard approach her e . References [1] G. Albe rtini, B. M. McCoy , and J. H. H. Perk, Eigenvalue sp e ctrum of the sup erinte gr able chir a l Potts mo del , Adv. Stud. Pur e Ma th. 19 (198 9), 1–55. [2] G. Alber tini, B. M. McCoy , J . H. H. Perk, a nd S. T a ng, Excitation sp e ctru m and or der p ar ameter for the inte gr ab le N-state chir al Pott s mo del , Nucl. Ph ys. B 314 (19 89), 741 –763 . [3] H. Au-Y ang, B. M. McCoy , J . H. H. Perk, S. T ang, and M. L. Y an, Commuting tr ansfer matric es in the chir al Potts mo dels: solutions of the star-triangle e quations with genus g > 1, Phys. Lett. A 1 2 3 (1987), 21 9–22 3. [4] H. Au-Y ang and J. H. H. Perk, O n sager’s star-triangle e quation: master key to inte gr ability , Adv. Stud. Pure Ma th. 19 (198 9), 57–94. [5] , Eig enve ctors in the sup erinte gr a ble m o del , J. Ph ys. A: Math. Gen. 41 (2008), 275201 . [6] R. J. Baxter, The sup erinte gr able chir a l Potts mo del , Phys. Lett. A 133 (19 88), 185 –189 . [7] , Su p erinte gr able chir al Potts m o del: thermo dynamic pr op erties, an “inverse” mo del, and a simple asso ciate d Hamiltonian , J. Statist. P hys. 57 (1 989), 1–39. [8] , Chir al Potts mo del with skewe d b oundary c onditions , J. Statist. Phys. 73 (19 93), 461–4 95. [9] , The ”inversion r elation” metho d for obtaining the fr e e ener gy of the chir al Potts mo del , P hysica A 322 (2 003), 40 7 –431 . [10] , D erivation of the or d er p a r ameter of the chir al Potts mo de l , Phys. Rev. Lett. 94 (2006), 11–2 4. 26 [11] R. J. Baxter, V. V. Ba z ha nov, and J . H. H. Perk, F unctional r elations for tr ansfer matric es of t he chir al Potts mo del , Internat. J. Mo dern Phys. B 4 (1990), 803– 870. [12] R. J. Ba xter, J. H. H. Perk, and H. Au-Y ang, New solutions of the st ar-triangle r el ations for the chir al Potts mo del , Phys. L e tt. A 128 (19 88), 138–1 4 2. [13] V. V. Bazhanov and Y u. G. Stroga nov, Chir al Potts mo del as a desc endant of the six - vertex mo del , J . Statist. Phys. 59 (1990), 799–8 1 7. [14] G. Benk a rt and P . T erwilliger, Irr e ducib le mo dules for the quantum affine algebr a U q ( ˆ sl 2 ) and its Bor el sub algebr a , Journal of Algebra 282 (20 04), 172–1 94. [15] V. Chari and A. Pressley , Quantum affine algebr as , Comm. Math. P hys. 1 42 (199 1), 261–283. [16] E. Date, M. Jimbo, K. Miki, and T. Miw a, Gener a lize d chir al Potts mo dels and minimal cyclic r epr esent ations of U q ( b gl ( n, C )), Co mm. Math. P hys. 137 (1991), 133– 1 47. [17] E. Date and S. Roan, The algebr aic s t ructur e of the On s ager algebr a , Czec h. J . Phys. 50 (2000), 37–4 4. [18] B. Davies, Onsager’s algebr a and sup eri nte gr ability , J. P hys. A: Math. Gen. 23 (1990), 224 5– 2261. [19] T. Deguc hi, The 8V CSOS mo del and t he sl 2 lo op algebr a symmetry of t he six-vertex mo del at r o ots of unity , In t. J. Mo d. Ph ys. B 16 (20 02), 189 9–190 5. [20] , Const r u ction of some missing eige nve ctors of the XYZ spin chain at the discr ete c oupling c onstants and the exp onential ly lar ge sp e ctr al de gener acy of t he tr ansfer matrix , J. Phys. A: Math. Gen. 35 (2002), 879– 895. [21] , The sl 2 lo op algebr a symmetry of the twiste d tr ansfer matrix of the six vertex mo del at r o ots of unity , J. Phys. A: Ma th. Gen. 37 (2 004), 347–3 58. [22] , Extension of a Bor el sub algebr a symmetry into the sl (2) lo op algebr a symmetry for the twiste d XXZ spin chain at r o ots of unity and the On s ager algebr a , in “RAQIS’07”, the pro ceedings of the workshop “ Recent Adv a nces in Quantum in tegra ble Systems”, LAPTH, Annecy-le-Vieux, F rance, September 11-14 , 200 7, e dited by L. F rappat a nd E. Ragoucy , (2007) pp. 15-3 4, (ar Xiv:0712.0 066). [23] , Gener alize d Drinfeld p olynomials for highest weight ve ctors of t he Bor el sub algebr a of the sl 2 lo op algebr a , Differential geometry and ph ysics, 169 -178 , Nank ai T racts Ma th., 10, W orld Sci. Publ., Hackensac k, NJ, 2006. [24] , Irr e ducibility criterion for a finite-dimensional highest weight r epr esentation of the sl 2 lo op algebr a and t he dimensions of r e ducible re pr esentations , J. Stat. Mech. (2007 ), P 0 5007 . [25] , R e gular XXZ Bethe states at r o ots of unity as high est weight ve ctors of sl 2 lo op algebr a , J. Phys. A: Math. Gen. 40 (20 07), 7473–7 508. 27 [26] T. Deguch i, K. F abric ius, a nd B. M. McCoy , The sl 2 lo op algebr a symmet ry of the six-vertex mo del at r o ots of unity , J. Statist. Ph ys. 102 (200 1 ), 701–7 36. [27] V. G. Drinfeld, A new r e ali zation of Y angians and quant ize d affine algebr a s , Soviet Math. Doklady 3 6 (1988), 2 12–2 16. [28] K. F abr icius and B. M. McCoy , Evaluation p ar ameters and Bethe r o ots for the six-vertex mo del at r o ots of unity , in Pr o gress in Mathematical Physics V ol. 23 (MathPhys O dyssey 2001), edited by M. Kashiwara and T. Miwa, (Birkh¨ auser, Boston, 20 02) 119–14 4. [29] , Bethe’s e quation is inc omplete for the XX Z mo del at r o ots of unity , J. Statist. Phys. 102 (2001 ), 6 4 7–67 8. [30] L. D. F addeev and L. A. T akhtadzh yan, Sp e c trum and sc attering of excitations in the one- dimensional isotr opic Heisenb er g mo del , J. Sov. Math. 24 (1984), 241 –267 . [31] S. How es, L. P . Kadanoff, and M. den Nijs, Quantu m mo del for c ommensur ate- inc ommensur ate tr ansitions , Nucl. Phys. B 215 (1983), 169– 208. [32] A. N. K irillov and N. Y u. Res hetikhin, Exact solution of the int e gr able X X Z Heisenb er g mo del with arbitr ary s pin. I. the gr ound state and the excitation sp e ctru m , J . Phys. A: Math. Gen. 20 (198 7), 1565–1 585. [33] A. Kl ¨ ump er and P . A. Pearce, Conformal weights of RSOS lattic e mo dels and their fusion hier a r chies , P hysica A 183 (1 992), 304– 3 50. [34] I. G. Ko r epanov, V acuum curves of L -op er ators asso ciate d with the six-vertex mo del , St. Petersburg Math. J . 6 (199 5), 3 49–3 64. [35] , Hidden symmet ries in t he 6-vertex mo del of statistic al physics , J. Math. Sci. 85 (1997), 1661 –1670 . [36] G. Lusztig, Intr o duction to quantu m gr oups , Birkh¨ a user Boston Inc., 1 993. [37] A. Nishino and T. Deguc hi, The L ( sl 2 ) symmetry of the Bazhanov-Str o ganov mo de l asso ciate d with the su p erinte gr able chir al Potts mo del , P hys. Lett. A 356 (200 6), 3 66–3 70. [38] L. Onsager, Crystal statistics. I. A two-dimensional m o del with an or der-diso r der tr ansition , Phys. Rev. 65 (1 944), 117– 149. [39] J. H. H. Perk, Star-triangle e quat ions, quantum lax p airs, and higher genu s curves , Pro c. Sympo sia Pure Math. 49 (1 989), 34 1–354 . [40] S. Roan, The t r ansfer matrix of sup erinte gr able chir al Potts mo del as the Q-op er ator of r o ot- of-unity XXZ chai n with cyclic r epr esent ation of U q ( sl 2 ), J. Stat. Mec h (2007 ), P0 9021 . [41] V. O. T arasov, T r ansfer matrix of t he sup erinte gr able chir al Potts mo del. Bethe ansatz sp e c- trum , Phys. Lett. A 147 (1990), 487– 490. 28 [42] G. von Gehlen and V. Ritten b erg, Z n -symmetric quantum chai ns with an infinite set of c onserve d char ges and Z n zer o mo d es , Nucl. Ph ys. B 257 (19 85), 351–3 7 0. 29