Bethe Equation of $tau^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

Bethe Equation of τ (2) -mo del and Eigenv alues of Finite-siz e T ransfer Matrix of C hiral P otts Mo del with Alternatin g Rapidities Shi-shyr Roan Institute of Mathematics A c ademia Sinic a T aip ei , T aiwan (email: mar o an@gate.sinic a.e du.tw ) Abstract W e establish the Bethe equation of the τ (2) -mo del in the N -state chiral Potts mo del (in- cluding the dege ne r ate selfdual cases) with a lter nating vertical rapidities. The eigenv alues of a finite-size transfer matrix of the c hiral P otts model ar e computed b y use of functional r elations. The sig nificance of the ”a lternating superintegrable” case of the chiral Potts mo del is discussed, and the degeneracy of τ (2) -mo del found as in the homo geneous superintegrable chiral Potts mo del. 2008 P A CS : 05.50.+q, 03.65.F d, 75.10.-b 2000 MSC: 14H50, 39B72 , 82B23 Key wor ds : Bethe equ a tion, τ (2) -mo del, N -state c hiral Pot ts mo del, F unctional relations 1 In tro duction The theory of N -state c h iral P otts mo del is a b eautiful and important, but tec hnically difficult, sub jec t in solv able lattice mo dels. The lac k of difference-prop ert y of rapidities in e vitably causes a 1 tec hnical complexit y in the s t udy of the c hiral P otts mo del, a natur e whic h distinguishes the theory from other known s o lv able lattice mo dels. Neve rtheless, pr o gress has b een made on the c hiral P otts transfer matrix for the p ast t w o decades. F or examples, one can study the maxim um eigen v alue and lo w lying excitations of the homog eneous c hir al P otts model in th e therm o dynamic L → ∞ limit [9, 10, 31], calculate the free energy and the in terfacial tension [4, 12, 14], and also the eigen v alue sp ectrum w as computed in the sup erint egrable case [1, 8 , 11, 12]. F urthermore, the knowledge was culminated in a rece nt pro of of the order parameter by Baxter [16]. Th ese results all rely on th e tec hnique of functional relations [17] ab out the tr an s fer matrix and fusion matrices of the asso ciate d τ (2) -mo del in the extended study of chiral Po tts mo del as a descendent of the six-vertex mo del found in [19]. The stud y is along the line of T Q -r e lation metho d in v ente d b y Baxter in solving the eigen v alue problem of th e eigh t-v ertex mo del [5, 6, 7]. The Q -matrix of the τ (2) -mo del considered is the c hiral P otts tr an s fer matrix. T h rough the functional-relation app roa c h, the exact results ab out Onsager-algebra symmetry app eared in the homogeneous sup erin tegrable c hiral P otts mo del [34] can serve as a useful mo del case in the study of s y m met ry of solv able lattice mo dels, among w h ic h are the r oot-of-unit y six-v ertex, eigh t-v ertex mo del [20 , 21, 22, 23, 24, 35, 36, 38], and XXZ chai ns of h ig her spin [37, 39]. While in th e f unctio nal-relation symmetry study of XXZ chains asso ci ated to the cyclic U q ( sl 2 ) represen tations lab ele d by the parameter ζ ∈ C ∗ with q N = ζ N = 1, these XXZ chains are equiv alen t to certain c hiral P otts mo dels w it h alternating sup erin tegrable rapidities [40]. F urther m o re, the Q -op erator inv estigation of the fi v e-parameter τ (2) -family app earing in [19] has sho wn that the τ (2) -mo del is indeed the same theory as the c hiral P otts m odel of alternating rapidities when the d eg enerate forms are included [41]. This suggests the r esu lt s in [1, 11, 12, 10, 31] ab out the eigen v alue sp ectrum could b e b etter u nderstoo d thr o ugh a general setting of the c hiral P otts mo del with alternating rapidities, includ in g the degenerate cases. Ind ee d, after sorting out the technical d et ails, one find s the fu nctio nal relations in [17] ab out the c h iral Potts mo del with alternating rapidities also v alid in the degenerate cases (for the details, see [41]). Note that in the case of alternating rapidities, Baxter extended the study of c hiral Po tts mo del and fun c tional relations to a general τ (2) -mo del in [15], wh ere the vertica l rapidities are not n ec essarily r equ ired in the same curve. The Baxter’s ”inhomogeneous” model is more general than the τ (2) -mo del in this work where we assume all rapidities in the same rapidity curve . F or conv enience, throughout this pap er by the N -state c hiral Potts mo del (CPM), w e alw a ys mean the c hiral P otts mo del w it h alternating r a pidities and degenerate cases included, unless otherwise stated. The purp ose of this pap er is to compute the eigen v alue sp ectrum of the fin it e(-size) transfer matrix of C PM. Note that the solution of the homogenous CPM, except in the sup erint egrable case, is y et unkno wn to th e b est of the author’s kn owledge. It is a c hallenge to obtain all the eigen v alues as they can b e solv ed completely for the finite-size Ising m o del [7, 32]. In the p resen t pap er, we fi nd an explicit form ula of eigenv alues of the fin ite transfer matrix of CPM by use of functional relat ions in [17] and th e Bethe equation of τ (2) -matrix generalized as in [31 ]. W e first obtain th e τ (2) -eigen v alues using the solutions of Bethe equation established by the Wiener-Hopf splitting metho d in [31]. Th o se Bethe solutions are formed compatibly with the τ (2) T -relation in CPM. W e then find that the sc heme in [10, 31] for the in ve stigation of maximum eige nv alue and excitati ons of large lattice limit in 2 the h o mogeneous CPM can b e extended to our stud y of solving all eigen v alues of the fi nite c hiral P otts trans f e r matrix. Here, aside the conceptual p recisio n, tec h n ic al accuracy is also needed for the correct expression of these eigen v alues and quan tum n umbers as the exact form is requ ir e d to repro duce results in homogeneous sup erin tegrable CPM previously kno wn in [1, 11, 12]. This pap er is organized as follo ws. In section 2 , we r ec all some kno wn results in τ (2) -mo del and CPM. In section 2.1, w e first summarize some basic facts ab out the fusion relat ion of τ (2) -mo del (for more details, see e.g. [37, 41]). Then we form ulate v arious forms of rapidit y parameters in CPM and in tro duce n ecessary notations for later u s e . I n section 2.2, we recall b riefly the main d e finitions of c hiral P otts transfer matrix and its relation with τ (2) -matrix in [1, 17]. In s ection 3, we d iscuss the Bethe equation of τ (2) -matrix in CPM. S u gg ested by the τ (2) T -relation in CPM, w e d e scrib e in section 3.1 a general mec hanism of constructing τ (2) -eigen v alues as solutions of the fu sio n relation. Then in section 3.2, we apply the Wiener-Hopf splitting tec hniqu e in [31] to establish the Bethe equation of τ (2) -mo del. In section 4, we compute the eige nv alues of the fi nite transfer matrix in CPM. Here w e follo w [10] and define the normalized c hiral P otts transfer m a trix with alternating rapidit y parameters. First in section 4.1, we recall and address tec hnical d e tails ab out functional relations in terms of the normalized c hiral P otts tr an s fer matrix. Th o se form ulas could b e kn o wn for sp ecia lists as a straitforw ard extension of the h o mogeneous case [10]. Ho w ev er, the exp licit form ulas of CPM (including the degenerate case) in the alternating-rapidity case ha v e not b een previously pub lished in the literature to the b est of our knowledge . Sin c e the correct expressions will b e n ee ded in later d i scussions, w e p r ese nt a d eriv ation of those relations here in spite of m uch of the r e sult just p a raph rasing the w ork of [10] in a sligh tly general form . In s e ction 4.2, we derive the form ula of eigen v alues of th e finite c hiral Pott s normalized tran s fer matrix by use of fu nctio nal relations and results in section 3.2 ab out Bethe solutions of τ (2) -mo del. In section 4.3, we discuss the ”alternating sup erin tegrable” case, where a simplification in the Bethe equatio n o ccurs, so is the expr essio n of eigen v alues of τ ( j ) and c hiral P otts transfer matrix. The d e generacy of th e τ (2) - matrix is f ou n d, and a comparison of the result with that in th e h o mogeneous CPM is also giv en there. Finally we close in section 5 with some concluding remarks. 2 The τ (2) -mo del and the N -state Chiral Po tts M od e l In th is pap er, C N denotes the v ector sp ac e of N -cyclic ve ctors with the basis | n i , n ∈ Z N (:= Z / N Z ). W e fix the N th ro ot of u nit y ω = e 2 π i N , and X, Z , the W eyl C N -op erato rs : X | n i = | n + 1 i , Z | n i = ω n | n i ( n ∈ Z N ) , satisfying the relations X Z = ω − 1 Z X and X N = Z N = 1. 2.1 The τ (2) -mo d el The τ (2) -mo del (also called the Baxter-Bazhano v-Stogano v mo del [26, 27, 28]) is the fi v e-parameter family asso ci ated to the L -op erat ors of C 2 -auxiliary , C N -quan tum sp a ce with entries expressed 3 b y the W eyl op erators X , Z : L ( t ) = 1 − t c b ′ b X ( 1 b − ω ac b ′ b X ) Z − t ( 1 b ′ − a ′ c b ′ b X ) Z − 1 − t 1 b ′ b + ω a ′ ac b ′ b X ! , (2.1) satisfying the Y ang Baxter equation for th e asymmetry six-v ertex R -matrix. Here t is the sp ectral v ariable, and a , b , a ′ , b ′ , c are non-zero complex parameters. The monodr o my m a trix of the c hain size L is L O ℓ =1 L ℓ ( t ) = A L ( t ) B L ( t ) C L ( t ) D L ( t ) ! , L ℓ ( t ) = L ( t ) at site ℓ, (2.2) The τ (2) -matrix is the ω -t wisted trace of the ab o ve mono drom y matrix: τ (2) ( t ) = A L ( ω t ) + D L ( ω t ) , (2.3) whic h form a comm uting family of L ⊗ C N -op erato rs, co mmuting with th e spin-sh if t op erator, de- noted again b y X (:= Q ℓ X ℓ ) when no confusion could arise. Th e τ (2) -eigenfunction is a p ol ynomial in t of degree L , and its X -eig env alue defi n es the c h arge ω Q with Q ∈ Z N . It is kno wn that the quan tum determinant of (2.2) is equal to z ( t ) X where z ( t ) = ( ω c ( ab − t )( a ′ b ′ − t ) ( b ′ b ) 2 ) L , (2.4) and the ”classica l” mono drom y matrix is h N L ℓ =1 L ℓ i = h A L i h B L i h C L i h D L i ! = h L 1 ih L 2 i · · · h L L i (= h L i L ) , h L i = 1 b ′ N b N b ′ N b N − c N t N b ′ N − a N c N − ( b N − a ′ N c N ) t N a ′ N a N c N − t N ! (2.5) ([26] (88) (45), [37] (2.9) (2.24) , [42]). Here h O i := Q N − 1 i =0 O ( ω i t ) denotes the a v erage of the (comm uting family of ) op erato rs O ( t ) for t ∈ C . F rom the L -op erator (2.1), one can construct τ ( j ) -matrices, with τ (0) = 0 , τ (1) = I and τ (2) in (2.3), so that the f u sion r elation holds: Recursiv e relation : τ (2) ( ω j − 1 t ) τ ( j ) ( t ) = z ( ω j − 1 t ) X τ ( j − 1) ( t ) + τ ( j +1) ( t ) , j ≥ 1; Boundary relation : τ ( N +1) ( t ) = z ( t ) X τ ( N − 1) ( ω t ) + u ( t ) I , (2.6) where z ( t ) is in (2.4), and u ( t ) = h A L i + h D L i (= the trace of h L i L ) (see, [37] Prop osition 2.1, [26] (107), [37] (2.30 ) and [41] (2.25)). By the recur sio n relation in (2.6), τ ( j ) ( t ) can b e exp r essed as a τ (2) -”p olynomia l” of degree ( j − 1) for j ≤ N + 1, τ ( j ) ( t ) = j − 2 Y s =0 τ (2) ( ω s t ) + [ j − 1 2 ] X k =1 ( − X ) k X 1 ≤ i 1 < ′ i 2 < ′ ··· < ′ i k ≤ j − 2 k Y ℓ =1  z ( ω i ℓ t ) τ (2) ( ω i ℓ − 1 t ) τ (2) ( ω i ℓ t ) j − 2 Y s =0 τ (2) ( ω s t )  where the notion i ℓ < ′ i ℓ +1 means i ℓ + 1 < i ℓ +1 . Th e b oundary relation in (2.6 ) then giv es rise to the f unctional equation of τ (2) ( t ): N − 1 Y s =0 τ (2) ( ω s t ) + [ N 2 ] X k =1 ( − X ) k X I k ∈I k Y i ∈ I k  z ( ω i t ) τ (2) ( ω i − 1 t ) τ (2) ( ω i t ) N − 1 Y s =0 τ (2) ( ω s t )  = u ( t ) I , (2.7) 4 where the in dex sets I k run through all sub set s I k of Z N with k d istinct elements suc h that i 6≡ i ′ + 1 (mo d N ) for i, i ′ ∈ I k . F or examples, the expressions of (2.7) for N = 3 , 4 are N = 3 : Q 2 j =0 τ (2) ( ω j t ) − ( P 2 j =0 z ( ω j t ) τ (2) ( ω j +1 t )) X = u ( t ) I ; N = 4 : Q 3 j =0 τ (2) ( ω j t ) − ( P 3 j =0 z ( ω j t ) τ (2) ( ω j +1 t )) X + ( z ( t ) z ( ω 2 t ) + z ( ω t ) z ( ω 3 t )) X 2 = u ( t ) I , (see [33] (13)-(15 )). There is a three-parameter sub family of τ (2) -mo dels, denoted by τ (2) p,p ′ , app eared in the N -state c hiral P otts mo del with alternating vertical rapidities p, p ′ , whose co ordinates ( x, y , µ ) ∈ C 3 satisfy either an algebraic curve r el ation f o r the p a rameter k ′ : W k ′ : kx N = 1 − k ′ µ − N , k y N = 1 − k ′ µ N , ( k ′ 6 = ± 1 , 0 , k 2 + k ′ 2 = 1) , (2.8) (see, e. g. [17]) or the v arious degenerate k ′ = ± 1 v ersions of the ab o ve curves, defined by one of the f o llo wing equations ([41] (3.22) (3.25) (4.3)): W ′ 1 : x N = 1 − µ − N , y N = 1 − µ N ; W ′′ 1 : x N + y N = 1 , µ N = 1; W ′′′ 1 : x N + y N = 0 , µ N = ± 1 . (2.9) The genus of a rapidit y curv e in (2.8) (2.9) is ( N 3 − 2 N 2 + 1) , N 3 − 3 N 2 +2 2 , N 2 − 3 N +2 2 , 0 resp ectiv ely , and eac h is in v ariant und er the automorphisms: R : ( x, y , µ ) 7→ ( y , ω x, µ − 1 ) , T : ( x, y , µ ) 7→ ( ω x, ω − 1 y , ω − 1 µ ) , U : ( x, y , µ ) 7→ ( ω x, y , µ ) , U ′ (= R 2 U − 1 ) : ( x, y , µ ) 7→ ( x, ω y , µ ) . (2.10) W e shall use t, λ, x to d e note the v ariables t = xy , λ = µ N , x = x N . (2.11) By eliminating λ , a curv e in (2.8 ) or (2.9) is reduced to a xy -curve corresp ondingly: x N + y N = k ( 1 + x N y N ) , x N + y N = x N y N , x N + y N = 1 , x N + y N = 0 . (2.12) The qu o tien t of the ab ov e xy -curv e b y the automorphism T in (2.10) dep ends on the v ariables t and x , whic h defines a hyperelliptic cur v e W of gen us N − 1 , [ N − 1 2 ] , [ N − 1 2 ] and 0 resp ectiv ely: W = W k ′ : t N = (1 − k ′ λ )(1 − k ′ λ − 1 ) k 2 , W ′ 1 : t N = (1 − λ )(1 − λ − 1 ) , W ′′ 1 : t N = x (1 − x ) , W ′′′ 1 : t N = − x 2 . (2.13) Hereafter in the case W ′′′ 1 , w e shall consider only the o dd N as for ev en N it consists of t wo rational irreducible comp onen ts. W e will u se η to den ote the complex num b er η := ( 1 − k ′ 1 + k ′ ) 1 N , 4 1 N , 4 − 1 N , 0 (2.14) 5 resp ectiv ely for W k ′ , W ′ 1 , W ′′ 1 , W ′′′ 1 in (2.13). The branc hed lo cus in the cases of W k ′ and W ′′′ 1 is defined b y t N = η ± N . F or W ′ 1 and W ′′ 1 , the br a nched v alue is: t N = η N , wh ile for o dd N , there is another branc hed v alue with t = 0 , ∞ resp ectiv ely . W e denote t + 0 , t − 0 the follo wing branched v alues: ( t + 0 , t − 0 ) := ( η − 1 , η ) for W k ′ and W ′′′ 1 , ( η , η ω ) for W ′ 1 and W ′′ 1 . (2.15) F or later con venience, w e shall use ( t, σ ) to denote co o rdin a tes of W in (2.13) together with the conjugate v ariable σ † of σ : σ = λ, σ † = λ − 1 , ( t, λ ) ∈ W k ′ or W ′ 1 ; σ = x , σ † = 1 − x , − x , ( t, x ) ∈ W ′′ 1 or W ′′′ 1 resp ectiv ely . (2.16) The interc h an ge of x N and y N of a xy -curve in (2.12) ind uces the hyp e relliptic inv olution of W in (2.13): ( t, σ ) 7→ ( t, σ † ). Hereafter, the letters p, q , . . . will d enot e rapidities in a curv e in (2.8) or (2.9), and write their co ordinates b y x p , y p , µ p , t p , λ p whenev er it will b e n ec essary to sp ecify the elemen t p . T he L -op erator of τ (2) p,p ′ ( t ) in (2.1) is defin ed by the affine co o rdin a tes of p, p ′ in (2.8 ) or (2.9) with the parameters ( a , b , a ′ , b ′ , c ) = ( x p , y p , x p ′ , y p ′ , µ p µ p ′ ) , (2.17) (see [1 7] (3.44a), [41] (2.18) (3.23 ) (4.4 )), and the sp ectral parameter t is related to th e rapidity v ariable q in (2.8) or (2.9) b y (2.11). T h e z ( t ) in (2.6) b ecomes z ( t ) = ( ω µ p µ p ′ ( t p − t )( t p ′ − t ) y 2 p y 2 p ′ ) L , (2.18) and u ( t ) in (2.6) is now expr essed b y u ( t ) = α q + α q , where α q , α q are the eigen v alues of h L i L in (2.5) ([17] (4.28) (4.29), [41] (2.26) an d section 5): α q = ( ( t N p − t N )( y N p ′ − x N ) y N p y N p ′ ( x N p − x N ) ) L = ( ( t N p ′ − t N )( y N p − x N ) y N p y N p ′ ( x N p ′ − x N ) ) L = α ( σ ) , α q = ( ( t N p − t N )( y N p ′ − y N ) y N p y N p ′ ( x N p − y N ) ) L = ( ( t N p ′ − t N )( y N p − y N ) y N p y N p ′ ( x N p ′ − y N ) ) L = α ( σ † ) , (2.19) with the v ariables σ, σ † in (2.16). Note that α q , α q are related to z ( t ) by the equalit y α q α q = z ( t ) z ( ωt ) · · · z ( ω N − 1 t ) , (2.20) whic h is the connection b et ween the determinan t of h N L ℓ =1 L ℓ i and quant um d e terminant of N L ℓ =1 L ℓ : det h N L ℓ =1 L ℓ i = h det q N L ℓ =1 L ℓ i . It is s ho wn in [41 ] that one can alw a ys reduce a τ (2) -mo del to a c hiral Pot ts τ (2) p,p ′ through a pro ce dur e of a s pecial gauge transf o rm and the rescaling of sp ectral parameters ([41] (2.21) and (2.22)): ( a , b , a ′ , b ′ , c ) 7→ ( λν − 1 a , ν b , ν a ′ , λν − 1 b ′ , c ) , ν , λ ∈ C ∗ . Indeed, a τ (2) -mo del wh o se the parameters satisfy ( c N a ′ N − b N )( c N a N − b ′ N )( a ′ N − b N )( a N − b ′ N ) 6 = 0 , 6 is equiv alen t to a τ (2) p,p ′ . The rest τ (2) -mo dels are either w it h the pseud o v acuum state where the algebraic Bet he tec hnique app lie s, or the ”zero- temp erature” ( k ′ = 0) limit of CPM (see, [41] (3. 20) and section 4.2). F or the rest of this p ap er, w e shall only consider the τ (2) -mo del with the ab o v e constrain t, and write τ (2) ( t ) = τ (2) p,p ′ ( t ) , t ∈ C , with rapidities p, p ′ in a curve in (2.8) or (2.9). 2.2 Th e c hiral P otts mo del and t h e degenerate selfdual mo del Using the co o rdin a tes ( x, y , µ ) of rapid it ies p, q in (2.8 ) or (2.9), one defines the Boltzmann weigh ts of th e CPM by W p,q ( n ) W p,q (0) = ( µ p µ q ) n n Y j =1 y q − ω j x p y p − ω j x q , W p,q ( n ) W p,q (0) = ( µ p µ q ) n n Y j =1 ω x p − ω j x q y q − ω j y p . (2.21) The rapidit y condition ensures the N -p erio dic p roperty of th e ab o v e Boltzmann w eigh ts for in tegers n . F or conv enience, w e shall assume W p,q (0) = W p,q (0) = 1 without loss of generalit y . Let W f pq ( n ) b e the F ourier transform of W pq ( n ) ([17] (2.22)-(2 .24)): W f pq ( n ) = P N − 1 k =0 ω nk W pq ( k ) , W f pq ( n ) W f pq (0) = Q n j =1 y q − ω j µ p µ q x p y p − ω j µ p µ q x q . (2.22) Denote g p ( q ) := Q N − 1 n =0 W pq ( n ) = ( µ p µ q ) ( N − 1) N 2 Q N − 1 j =1 ( x p − ω j y q x q − ω j y p ) j , g p ( q ) := det N ( W pq ( i − j )) (= Q N − 1 n =0 W f pq ( n )) = ( W f pq (0)) N Q N − 1 j =1 ( µ p µ q x p − ω j y q µ p µ q x q − ω j y p ) j . (2.23) Using (2.22 ) , one finds the id e ntit y ([17] (2.44)) 1 : g p ( q ) = N N 2 e i π ( N − 1)( N − 2) / 1 2 N − 1 Y j =1 ( t p − ω j t q ) j ( x p − ω j x q ) j ( y p − ω j y q ) j . (2.24) It is known [2, 3, 18, 25, 29, 30] that the Boltz mann w eigh ts in (2.21) satisfy the star-triangle relation N − 1 X n =0 W q r ( j ′ − n ) W pr ( j − n ) W pq ( n − j ′′ ) = R pq r W pq ( j − j ′ ) W pr ( j ′ − j ′′ ) W q r ( j − j ′′ ) (2.25) where R pq r = f pq f qr f pr with f pq = g p ( q ) 1 / N g p ( q ) 1 / N . On a lattice of the horizon tal size L , the combined w eigh ts of inte rsections with ve rtical rapidities p, p ′ b et ween tw o consecutiv e ro ws defi ne the L ⊗ C N - op erato r, calle d the c hiral Pot ts transfer m atrix, comm uting with the sp in-shift op erator X : T p,p ′ ( q ) { j } , { j ′ } = Q L ℓ =1 W p,q ( j ℓ − j ′ ℓ ) W p ′ ,q ( j ℓ +1 − j ′ ℓ ) , b T p,p ′ ( q ) { j } , { j ′ } = Q L ℓ =1 W p,q ( j ℓ − j ′ ℓ ) W p ′ ,q ( j ℓ − j ′ ℓ +1 ) . (2.26) 1 The identity (2.24) is formula (2.44) in [17] where rapidities are in ( 2 .8). The same equality h olds also in the degenerate model with rapidities in (2.9). 7 Here q is an arb itrary r ap id it y , and j ℓ , j ′ ℓ ∈ Z N with the p eriod ic condition imp osed b y defining L + 1 = 1. As in [17] (2.39)-(2.4 2), the op erators in (2.26) are indeed s in g le-v alued in x q and y q : T p,p ′ ( q ) = T p,p ′ ( x q , y q ), b T p,p ′ ( q ) = b T p,p ′ ( x q , y q ), satisfying the relations T p,p ′ ( ω x q , ω − 1 y q ) = ( ( y p − ω x q )( y p ′ − ω − 1 y q ) µ p µ p ′ ( ωx p − y q )( x p ′ − x q ) ) L X − 1 T p,p ′ ( x q , y q ) , b T p,p ′ ( ω x q , ω − 1 y q ) = ( ( y p ′ − ω x q )( y p − ω − 1 y q ) µ p µ p ′ ( ωx p ′ − y q )( x p − x q ) ) L X − 1 b T p,p ′ ( x q , y q ) . (2.27) The star-triangle relation (2.25) yields the commutati ve relation for rap id it ies q an d r : T p,p ′ ( q ) b T p,p ′ ( r ) = ( f p ′ q f pr f pq f p ′ r ) L T p,p ′ ( r ) b T p,p ′ ( q ) , b T p,p ′ ( q ) T p,p ′ ( r ) = ( f pq f p ′ r f p ′ q f pr ) L b T p,p ′ ( r ) T p,p ′ ( q ) . (2.28) The commuting family of Q -op erators is d efined b y Q p,p ′ ( q ) = b T p,p ′ ( q 0 ) − 1 b T p,p ′ ( q ) = ( f pq f p ′ q 0 f p ′ q f pq 0 ) L T p,p ′ ( q ) T p,p ′ ( q 0 ) − 1 ( q ∈ W k ′ ) where q 0 is an elemen t in W k ′ with non-singular b T p,p ′ ( q 0 ) and T p,p ′ ( q 0 ). Note that for p 6 = p ′ , the comm utativit y of the T p,p ′ - and b T p,p ′ -family fails in general. Nev ertheless, one can still d ia gonal the matrices T p,p ′ ( q ) , b T p,p ′ ( q ) by usin g t wo inv ertib le q -indep endent matrices P B , P W so that b oth P − 1 W T p,p ′ ( q ) P B , P − 1 B b T p,p ′ ( q ) P W are diagonalized ([17 ] (2.34)). F or conv enience, w e shall refer those diagonal entries as the ”eig env alues” of T p,p ′ , b T p,p ′ . By (2.28), the diagonal forms of T p,p ′ , b T p,p ′ are related b y ([17] (4.46)): b T p,p ′ ( q ) = ( f pq f p ′ q ) L T p,p ′ ( q ) D (2.29) where D is a q -indep enden t diagonal matrix. The C P M transfer matrices in (2.26) are the Q R , Q L -op erato r of the τ (2) -matrix, satisfying the τ (2) T -relations ([17] (4.20) (4.21), [41]): τ (2) ( t q ) T p,p ′ ( U q ) = { ( y p − ω x q )( t p ′ − t q ) y p y p ′ ( x p ′ − x q ) } L T p,p ′ ( q ) + { ( y p ′ − y q )( t p − ω t q ) y p y p ′ ( x p − y q ) } L T p,p ′ ( R 2 q ) , b T p,p ′ ( U q ) τ (2) ( t q ) = { ( y p ′ − ω x q )( t p − t q ) y p y p ′ ( x p − x q ) } L b T p,p ′ ( q ) + { ( y p − y q )( t p ′ − ω t q ) y p y p ′ ( x p ′ − y q ) } L b T p,p ′ ( R 2 q ) , (2.30) where U, R are in (2.10). Using (2.27), one can write the first τ (2) T -relation in (2.30) in an equiv alen t form in terms of automorphisms U or U ′ ([17] (4.31)) : τ (2) ( t q ) T p,p ′ ( U q ) = ϕ q T p,p ′ ( q ) + ϕ U q X T p,p ′ ( U 2 q ) , τ (2) ( t q ) T p,p ′ ( U ′ q ) = ϕ ′ q X T p,p ′ ( q ) + ϕ ′ U ′ q T p,p ′ ( U ′ 2 q ) , (2.31) where ϕ q (= ϕ p,p ′ ; q ) = { ( t p ′ − t q )( y p − ω x q ) y p y p ′ ( x p ′ − x q ) } L , ϕ q (= ϕ p,p ′ ; q ) = { ω µ p ′ µ p ( t p − t q )( x p ′ − x q ) y p y p ′ ( y p − ω x q ) } L , ϕ ′ q (= ϕ ′ p,p ′ ; q ) = { ω µ p µ p ′ ( t p ′ − t q )( x p − y q ) y p y p ′ ( y p ′ − y q ) } L , ϕ ′ q (= ϕ ′ p,p ′ ; q ) = { ( t p − t q )( y p ′ − y q ) y p y p ′ ( x p − y q ) } L . (2.32) Similarly , the second τ (2) T -relation in (2.30 ) can b e written equiv alently as b T p,p ′ ( U q ) τ (2) ( t q ) = ϕ p ′ ,p ; q b T p,p ′ ( q ) + ϕ p ′ ,p ; U q X b T p,p ′ ( U 2 q ) , b T p,p ′ ( U ′ q ) τ (2) ( t q ) = ϕ ′ p ′ ,p ; q X b T p,p ′ ( q ) + ϕ ′ p ′ ,p ; U ′ q b T p,p ′ ( U ′ 2 y q ) . (2.33) 8 Using τ (2) T -relation (2.31) and the recursiv e f u sion relation (2.6), one fi nds τ ( j ) T -relation ([17 ] (4 . 34) k =0 ): τ ( j ) ( t q ) = P j − 1 m =0 ϕ q ϕ U q · · · ϕ U m − 1 q ϕ U m +1 q ϕ U m +2 q · · · ϕ U j − 1 q × T p,p ′ ( x q , y q ) T p,p ′ ( ω m x q , y q ) − 1 T p,p ′ ( ω j x q , y q ) T p,p ′ ( ω m +1 x q , y q ) − 1 X j − m − 1 . (2.34) 3 Bethe equation of τ (2) -mo del In this section, we establish the Bethe equation of τ (2) -mo del b y the Wiener-Hopf sp l itting metho d in [31]. 3.1 A general description of τ (2) -eigen v alues First w e describ e a mec hanism of constru ct ing τ (2) ( t ) as a general solution of equ a tion (2.7 ) . Let s b e a (system of ) v ariable (or v ariables) algebraically d e p endent on t , s 0 a base p oint corresp o nd i ng to t = 0, and s 7→ φ ( s ) an order- N automorphism so that t can b e exp ressed as a r e gular fu nctio n of s with φ ( s ) co rresp onding to ω t . Ass u me z ( t ) in (2 .18) can b e factored as the prod uct of t w o s -functions, H + ( s ) an d H − ( s ), with resp ect to α q , α q in (2. 19 ), su c h that the follo wing relati ons hold: z ( t ) = H + ( s ) H − ( s ) , H + ( s 0 ) = 1 , α q = H + ( s ) H + ( φ ( s )) · · · H + ( φ N − 1 ( s )) , α q = H − ( s ) H − ( φ ( s )) · · · H − ( φ N − 1 ( s )) . (3.1) The abov e relations are co nsistent with (2.2 0 ). Note that (3.1) r ema ins v alid when replacing H ± ( s ) b y γ ± 1 H ± ( s ) with γ N = 1. Using H ± ( s ) in (3.1 ), on e can v erify that an exact solution of (2.7) is giv en by τ (2) ( t ) = H + ( s ) Λ( s ) Λ( φ ( s )) + ω Q H − ( φ ( s )) Λ( φ 2 ( s )) Λ( φ ( s )) , (3.2) where Λ( s ) is a function with the constraint that the function in ab o ve righ t-hand side defin es a t -function, (a condition automatically true in ca se s = t and φ ( s ) = ω t ). Using (3.2) and the recursiv e fusion relation in (2.6), one finds the follo w in g exp r essio n of τ ( j ) -eigen v alue: τ ( j ) ( t ) = ω j Q Λ( s )Λ( φ j ( s )) P j − 1 m =0  H + ( s ) H + ( φ ( s )) · · · H + ( φ m − 1 s ) × H − ( φ m +1 ( s )) H − ( φ m +2 ( s )) · · · H − ( φ j − 1 ( s ))Λ( φ m ( s )) − 1 Λ( φ m +1 ( s )) − 1 ω − ( m +1) Q  . (3.3) In p artic ular, b y setting s = ( x q , y q , µ q ) f o r q b eing the r a pidity in (2.8) or (2.9), and s 0 b eing the elemen t w hose x -v alue equals to 0, the r e lations, (2.18) and (2.19), imp ly (3.1) holds for ( H + ( s ) , H − ( s )) = ( ϕ q , ϕ q ) w it h ϕ q , ϕ q in (2.32). The formula (3.2) (with φ = U in (2.10)) is an equiv alent form of τ (2) T -relation (2. 31 ), where Λ( s ) is a T p,p ′ -eigen v alue. The r e lation (3.3) is equiv alen t to τ ( j ) T -relation (2. 34 ). Similarly , the first relation in (2 .33) is the same as (3.2) with ( H + ( s ) , H − ( s )) = ( ϕ p ′ ,p ; q , ϕ p ′ p ; q ) and φ = U . One may consider another pair of functions, H ′− ( s ) and H ′ + ( s ), an order N automorphism φ ′ , and a base p oint s ′ 0 with the relation z ( t ) = H ′− ( s ) H ′ + ( s ) , H ′ + ( s ′ 0 ) = 1 , α q = H ′− ( s ) H ′− ( φ ′ ( s )) · · · H ′− ( φ ′ N − 1 ( s )) , α q = H ′ + ( s ) H ′ + ( φ ′ ( s )) · · · H ′ + ( φ ′ N − 1 ( s )) . (3.4) 9 W rite the solutions of (2.7) in the f o rm with the factor ω Q in front : τ (2) ( t ) = ω Q H ′ − ( s ) Λ ′ ( s ) Λ ′ ( φ ′ ( s )) + H ′ + ( φ ( s )) Λ ′ ( φ 2 ( s )) Λ ′ ( φ ′ ( s )) . (3.5) By setting ( H ′ − ( s ) , H ′ + ( s )) = ( ϕ ′ q , ϕ ′ q ) or ( ϕ ′ p ′ ,p ; q , ϕ ′ p ′ p ; q ), one finds the other τ (2) T -relation in (2.31) or (2.33) with s ′ 0 the elemen t with zero y -coord inat e, φ ′ = U ′ , an d Λ( s ) b eing an eigenv alue of T p,p ′ or b T p,p ′ . Note that one may interc hange the f unctions H ± in (3.2), or H ′± in (3.5), to form τ (2) ( t ). The resulting ones can b e obtained fr om the previous ones b y c hanging Q, τ 2 ( t ) to − Q, ω − Q τ 2 ( t ) and φ − 1 , Λ( φ 2 ( s )) to φ, Λ( s ). How eve r, in ord er to identify Q w it h c harge of the eigen v alue τ (2) ( t ), w e need only to consider the cases (3.2 ) and (3.5). 3.2 Bethe equa tion of chiral P otts mo del with alternating v ertical rapidities In solving the eigen v alue problem of the τ (2) -matrix, it will b e conv enient to c ho ose the v ariable s = t w it h φ ( t ) = φ ′ ( t ) = ω t in (3.2 ) and (3.5 ). In this section, w e follo w the metho d in [31] to construct the t -functions h ± ( t ) as H ± and h ′ ∓ ( t ) as H ′ ∓ through the Wiener-Hopf splitting of α q , α q . With Λ( s ) defin ed by Λ( t ) = t P a F ( t ) , F ( t ) = Q J j =1 (1 + ω v j t ) ( v N j 6 = v N i for j 6 = i ) , (3.6) where P a is an intege r, (3.2) is expr e ssed by τ (2) ( t ) = ω − P a h + ( t ) F ( t ) F ( ω t ) + ω Q + P a h − ( ω t ) F ( ω 2 t ) F ( ω t ) . (3.7) The r egular-fu n ct ion condition of τ (2) ( t ) requires the ro ots of F ( t ) satisfy the Bethe equ a tion: J Y j =1 v i − ω − 1 v j v i − ω v j = − ω Q +2 P a h − ( − ω − 1 v − 1 i ) h + ( − ω − 2 v − 1 i ) , i = 1 , . . . , J. (3.8) By (3.3 ), the functions τ ( j ) ( t ) ( j ≥ 2) are expressed b y τ ( j ) ( t ) = ω ( j − 1)( Q + P a ) F ( t ) F ( ω j t ) P j − 1 n =0 h + ( t ) ··· h + ( ω n − 1 t ) h − ( ω n +1 t ) ··· h − ( ω j − 1 t ) ω − n ( Q +2 P a ) F ( ω n t ) F ( ω n +1 t ) ; τ ( N ) ( t ) = ω − Q − P a F ( t ) 2 P N − 1 n =0 h + ( t ) ··· h + ( ω n − 1 t ) h − ( ω n +1 t ) ··· h − ( ω N − 1 t ) ω − n ( Q +2 P a ) F ( ω n t ) F ( ω n +1 t ) . (3.9) The Bethe equation (3.8) can also b e c haracterized as the regular-function criterion of an y one of the ab o ve τ ( j ) ( t )s. Similarly , w hen u s ing th e expression (3.5) with H ′∓ ( s ) = h ′∓ ( t ), and Λ ′ ( s ) = t P b F ′ ( t ) with F ′ ( t ) = Q J ′ j =1 (1 + ω v ′ j t ) as in (3.6), τ (2) ( t ) = ω Q − P b h ′− ( t ) F ′ ( t ) F ′ ( ωt ) + ω P b h ′ + ( ω t ) F ′ ( ω 2 t ) F ′ ( ωt ) , (3.10) the Bethe equation tak es the form J ′ Y j =1 v ′ i − ω − 1 v ′ j v ′ i − ω v ′ j = − ω − Q +2 P b h ′ + ( − ω − 1 v ′− 1 i ) h ′− ( − ω − 2 v ′− 1 i ) , i = 1 , . . . , J ′ , (3.11) 10 with the τ ( j ) -expression τ ( j ) ( t ) = ω ( j − 1) P b F ′ ( t ) F ′ ( ω j t ) P j − 1 n =0 h ′− ( t ) ··· h ′− ( ω n − 1 t ) h ′ + ( ω n +1 t ) ··· h ′ + ( ω j − 1 t ) ω n ( Q − 2 P b ) F ′ ( ω n t ) F ′ ( ω n +1 t ) ; τ ( N ) ( t ) = ω − P b F ′ ( t ) 2 P N − 1 n =0 h ′− ( t ) ··· h ′− ( ω n − 1 t ) h ′ + ( ω n +1 t ) ··· h ′ + ( ω N − 1 t ) ω n ( Q − 2 P b ) F ′ ( ω n t ) F ′ ( ω n +1 t ) . (3.12) Next w e constru c t th e functions h ± ( t ) , h ′∓ ( t ) in the Bethe equation (3.8 ) , (3.11 ) for a r a pidity curv e in (2.8) or (2.9). W r it e the functions in (2.19) in the f orm α q = ( t N p ′ − t N y N p y N p ′ ) L β ( σ q ) L , α q = ( µ N p µ N p ′ ( t N p − t N ) y N p y N p ′ ) L β ( σ q ) − L (3.13) where β ( σ ) is the follo wing σ -function for the v ariable σ in (2.16): β ( σ ) = 1 − λ p λ 1 − λ − 1 p ′ λ for W k ′ and W ′ 1 ; 1 − x p − x x p ′ − x for W ′′ 1 ; − x p − x x p ′ − x for W ′′′ 1 . (3.14) Let C t b e a conto ur in the t -p lane circled clo c k-wise around the in terv al b et ween th e elemen ts, t + 0 and t − 0 , defined in (2.15). With t outside the con tour, we defin e the fun c tion e + ( t ) = exp  L 2 N π i Z C t dt ′ t ′ − t ln β ( σ ′ )  (3.15) where σ , σ ′ are in th e ” p ositive ” sheet W + of W in (2.13): | λ | < 1 for W k ′ and W ′ 1 ; ℜ ( x ) < 1 2 for W ′′ 1 , and ℜ ( x ) < 0 for W ′′′ 1 . Note that outsid e the con tour C t , the t -domain can b e lifted to the h yp erelliptic curv e W w it h the assigned σ -v alue, by whic h σ can b e regarded as a function of t , hence e + ( t ) coincides with the algebraic-function β ( σ ) L N = ( y N p − x N x N p ′ − x N ) L N = ω L µ L p µ L p ′ ( x N p − y N y N p ′ − y N ) L N . (3.16) More pr ec isely , e + ( t ) is expressed b y (3 .16) with the v alues of x and y r e stricted on the p ositiv e region W + of W in (2.13): W + := W + k ′ : | 1 − k x N | ≥ | k ′ | , | 1 − k y N | ≤ | k ′ | ; W ′ + 1 : | 1 − x N | ≥ 1 , | 1 − y N | ≤ 1; W ′′ + 1 : ℜ (1 − x N ) = ℜ ( y N ) ≥ 1 2 ; W ′′′ + 1 : −ℜ ( x N ) = ℜ ( y N ) ≥ 0 . (3.17) In particular, when t = 0, one tak es x = 0 with the corresp onding σ -v alue b eing k ′ , 1 , 1 , 0 resp ec- tiv ely . F or later u se , we denote the negat iv e r e gion W − of W b y rev ersing inequalit y signs in (3.17). Let e − ( t ) b e the function defined b y form ula (3.15) where σ ′ is in the ne gative sheet W − of W , hence e − ( t ) aga in expr e ssed b y (3.16). W e ma y assume e ± ( ω N t ) = e ± ( t ) by a suitable c hoice of phase-factors. Note that e ± ( t ) are indeed fu ncti ons defined on the region W ± , and w e shall also write e ± ( t ) = e ± ( t, σ ) for ( t, σ ) ∈ W ± , whenev er a clearer expression w ill b e needed. Using the function in (3.15), w e defi n e the function h ± ( t ) in (3.7) through the Wiener-Hopf splitting of α q , α q as f ormula (24a,b) in [31]: h + ( t ) = [ t p ′ − t y p y p ′ ] L e + ( t ) γ , h − ( t ) = [ ω µ p µ p ′ ( t p − t ) y p y p ′ ] L 1 e + ( t ) γ (3.18) 11 where γ is a constan t factor d e termined by the condition h + (0) = 1. Indeed, the ev aluation of the in tegral in (3.15) shows that γ is a N th r oot of unity , γ = ω P γ , with lim t →∞ h + ( t ) t L = γ ( − 1 y p y p ′ ) L , lim t →∞ h − ( t ) t L = γ − 1 ( − ω µ p µ p ′ y p y p ′ ) L . (3.19) The expr e ssion (3.18) yields the condition (3.1) with H ± = h ± ( t ): z ( t ) = h + ( t ) h − ( t ) , α q = Q N − 1 j =0 h + ( ω j t ) , α q = Q N − 1 j =0 h − ( ω j t ) , h + (0) = 1 . (3.20) Hence w e can express τ ( j ) ( t ) by (3.9) using h ± ( t ) in (3.18) and the Bethe solution F ( t ) for (3.8). One ma y use e − ( t ) to define the Wiener-Hopf sp l itting h ′∓ of α q , α q : h ′− ( t ) = [ t p ′ − t y p y p ′ ] L e − ( t ) γ ′ , h ′ + ( t ) = [ ω µ p µ p ′ ( t p − t ) y p y p ′ ] L 1 e − ( t ) γ ′ (3.21) where γ ′ = ω P γ ′ is determined by the cond i tion h ′ + (0) = 1. Then one has lim t →∞ h ′− ( t ) t L = γ ′ ( − 1 y p y p ′ ) L ( µ N p µ N p ′ ) L N , lim t →∞ h ′ + ( t ) t L = γ ′− 1 ( − ω µ p µ p ′ y p y p ′ ) L ( µ N p µ N p ′ ) − L N . (3.22) Using (3.16 ) , one finds the algebraic-function-expression of h ± , h ′∓ : ( ( t p ′ − t ) L y L p y L p ′ ( y N p − x N x N p ′ − x N ) L N , ( t p − t ) L y L p y L p ′ ( y N p ′ − y N x N p − y N ) L N ) = ( ( γ − 1 h + ( t ) , γ h − ( t )) for ( t, σ ) ∈ W + ; ( γ ′− 1 h ′− ( t ) , γ ′ h ′ + ( t )) for ( t, σ ) ∈ W − . (3.23) Note that for the general p and p ′ , th e ab o v e expr e ssion holds only on a ”half” of Riemann sur fac e W , i.e. W + (or W − ) where t can b e considered as an algebraic fu nctio n of σ in the restricted domain, bu t not on th e ent ire W except th e sp ecial alternating sup erin tegrable case discus s e d later in section 4.3. Accordingly , the relation (3.7) (or (3.10)) is v alid by regarding t as a function of th e v ariable σ , and it b ecomes the t -polynomial r el ation only in the alternating sup erin tegrable case. The r elation (3.4) holds with H ′∓ = h ′∓ ( t ): z ( t ) = h ′− ( t ) h ′ + ( t ) , α q = Q N − 1 j =0 h ′− ( ω j t ) , α q = Q N − 1 j =0 h ′ + ( ω j t ) , h ′ + (0) = 1 , (3.24) with the τ ( j ) ( t )-expression (3.12) by usin g h ′∓ ( t )in (3.22) and the Bethe solution F ′ ( t ) of (3.11). Remark : The Wiener-Hopf splittings, (3.20) and (3.24), of α q , α q are relate d as follo w s. In the previous discuss ion, it is u nderstoo d that the t -fu nctio ns, h ± ( t ) and h ′∓ ( t ), are indeed functions on one sh ee t of th e hyp erelliptic curve W : h ± ( t ) = h ± ( t, σ ) , τ ( j ) ( t ) = τ ( j ) ( t, σ ) ( t, σ ) ∈ W + ; h ′∓ ( t ) = h ′∓ ( t, σ ) , τ ( j ) ( t ) = τ ( j ) ( t, σ ) ( t, σ ) ∈ W − . F or con v enience, w e define e + ( t ) ∗ to b e the fun c tion on W + b y c hanging x N in e + ( t ) to y N : e + ( t ) ∗ (= e + ( t, σ ) ∗ ) = ( y N p − y N x N p ′ − y N ) L N , ( t, σ ) ∈ W + , 12 and h ± ( t ) ∗ , τ ( j ) ( t ) ∗ are the functions of W + b y r e placing e + ( t ) by e + ( t ) ∗ in (3.18), (3.3) resp ectiv ely . Using (3.16 ) , one finds e + ( t ) ∗ = e + ( t, σ )( α q α q ) 1 N = e + ( t )( α q α q ) 1 N (= e − ( t, σ † )) = e − ( t ) for ( t, σ ) ∈ W + . Hence one obtains h + ( t ) ∗ = h + ( t )( α q α q ) 1 N = γ γ ′− 1 h ′− ( t ) , h − ( t ) ∗ = h − ( t )( α q α q ) − 1 N = γ ′ γ − 1 h ′ + ( t ) . F or generic p and p ′ , τ ( j ) ( t ) ∗ is not a regular fun c tion, hence not equ a l to the t -p olynomial τ ( j ) ( t ). Indeed b y (3.9), τ ( j ) ( t ) ∗ , ( j ≥ 2) , is exp ressed by τ ( j ) ( t ) ∗ = ω ( j − 1)( Q + P a ) F ( t ) F ( ω j t ) P j − 1 n =0 h + ( t ) ∗ ··· h + ( ω n − 1 t ) ∗ h − ( ω n +1 t ) ∗ ··· h − ( ω j − 1 t ) ∗ ω − n ( Q +2 P a ) F ( ω n t ) F ( ω n +1 t ) = ω ( j − 1)( Q + P a ) F ( t ) F ( ω j t ) P j − 1 n =0 h + ( t ) ··· h + ( ω n − 1 t ) h − ( ω n +1 t ) ··· h − ( ω j − 1 t )( ω − ( Q +2 P a ) ( α q α q ) 2 N ) n F ( ω n t ) F ( ω n +1 t ) ( α q α q ) − j +1 N . By (3.8 ), the condition of τ ( j ) ( t ) ∗ b eing regular at zeros of F ( ω n +1 t ) for 0 ≤ n ≤ j − 2 is the requirement of ( α q α q ) 2 N = 1 wh en t = − ω − 1 v − 1 i (1 ≤ j ≤ J ). If the functions α q and α q are not equal, τ ( j ) ( t ) ∗ is not finite f or t = t p , t p ′ . Using the relation b et wee n h ± ( t ) ∗ and h ′± ( t ), one concludes that the p olynomials F ( t ) , F ′ ( t ) in (3.9 ), (3.12) are not the same when α q 6 = α q . Indeed b y (3.8) and (3.11), F ( t ) = F ′ ( t ) is pro vided by h − ( ωt ) N h + ( t ) N = h ′ + ( ωt ) N h ′− ( t ) N , equ iv alen t to α q = ± α q , whic h is the same conclusion for the p olynomial condition of τ ( j ) ( t ) ∗ . In p a rticular, w hen α q = α q , i.e. th e alternating sup erin tegrable case in sec tion 4.3, b y (3.12), one fin ds τ ( j ) ( t ) ∗ = τ ( j ) ( t ) with P b ≡ Q + P a + P γ ′ − P γ . 4 Eigen v alues of Chiral P otts T ransfer Matrix with Alternating Rapidities In th i s section, we discus s the eigen v alue problem of th e trans fer matrices T p,p ′ , b T p,p ′ in (2.26) for a rapidit y curve in (2.8) or (2.9) by employi ng th e f unctio ns h ± ( t ) , h ′∓ ( t ) in (3.18) (3.21). As in the homogeneous CPM case [10], we normalize the transfer matrices by multi plying a p o wer of g p ( q ) and g p ( q ) in (2.23) to eliminate of the f actor app eared in form ula (2.27). Note that µ N ( N − 1) 2 q g p ( q ) is a function of x q and y q only , in depend en t of µ q , wh ose v alue und er the automorphism T in (2.10) c hanges by µ N ( N − 1) 2 T q g p ( T q ) = x N p − y N q y N p − x N q ( y p − ω x q ω x p − y q ) N µ N ( N − 1) 2 q g p ( q ) . (4.1) The function g p ( q ) also dep ends o n x q and y q only , i.e. g p ( q ) = g p ( q ) with ( x q , y q , µ q ) = ( x q , y q , , ω µ q ), whic h follo ws f r o m W f p, q (0) = W f p,q (1) and W f p q ( n ) W f p q (0) = W f p,q ( n +1) W f p,q (1) . By (2.23), The relations, W f p T q (0) = − ( y p − ω − 1 y q ) µ p µ q ( x p − x q ) W f pq (0) and ( µ p µ q ) N ( x N p − x N q ) = y N q − y N p , in turn yield g p ( T q ) = x N p − x N q y N p − y N q ( y p − ω − 1 y q x p − x q ) N g p ( q ) . (4.2) 13 By µ N p µ N p ′ ( x N p − y N q )( x N p ′ − x N q ) = ( y N p − x N q )( y N p ′ − y N q ), the relations, (4.1) and (4.2), imply µ N ( N − 1) 2 T q g p ( T q ) g p ′ ( T q ) = ( ( y p − ω x q )( y p ′ − ω − 1 y q ) µ p µ p ′ ( ω x p − y q )( x p ′ − x q ) ) N µ N ( N − 1) 2 q g p ( q ) g p ′ ( q ) ω N ( N +1) . (4.3) Comparing the factors in (2.27) and (4.3), we consider the normalized transfer matrices as in [10] Sect. 2 : V ( q ) (= V ( x q , y q )) := T p,p ′ ( q )  µ N ( N − 1) 2 q g p ( q ) g p ′ ( q )  − L N , b V ( q ) (= b V ( x q , y q )) := b T p,p ′ ( q )  µ N ( N − 1) 2 q g p ′ ( q ) g p ( q )  − L N . (4.4) By (2.29), the eigen v alues of V ( q ) and b V ( q ) are related by b V ( q ) = V ( q ) D (4.5) where D is a q -indep enden t scale. By (2.27), the follo wing r elations hold (see, [10] page 111): V ( ω x q , ω − 1 y q ) = ω − L X − 1 V ( x q , y q ) , b V ( ω x q , ω − 1 y q ) = ω − L X − 1 b V ( x q , y q ) , (4.6) b y whic h V ( q ) , b V ( q ) dep end on th e v alues of ( t q , σ q ) in (2.16), u p to a N th r o ot of u nit y 2 . In particular, V ( x q , y q ) N is a meromorphic fun ctions of ( t q , σ q ). 4.1 F unctional relations of the c hiral P otts mo del The transfer c hiral Pott s matrix and th e τ ( j ) -matrix for rapidities in (2.8) are kn o wn to satisfy a set of fu n ct ional relations [17], whic h again hold in the degenerate mo dels w it h rapidities in (2.9) ([41] section 5). Among the fun c tional relations are the fusion relation (2.6), τ (2) T -relation (2.31), τ ( j ) T -relation (2.34), and the T ˆ T -relation ([13] (13), [34] (15)): T p,p ′ ( x q , y q ) b T p,p ′ ( y q , ω j x q ) r p ′ ( q ) h j ; p,p ′ ( q ) = τ ( j ) ( t q ) + z ( t q ) z ( ω t q ) · · · z ( ω j − 1 t q ) α q τ ( N − j ) ( ω j t q ) X j (4.7) where r p ′ ( q ) = ( N ( x p ′ − x q )( y p ′ − y q )( t N p ′ − t N q ) ( x N p ′ − x N q )( y N p ′ − y N q )( t p ′ − t q ) ) L , h j ; p,p ′ ( q ) = ( Q j − 1 m =1 y p y p ′ ( x p ′ − ω m x q ) ( y p − ω m x q )( t p ′ − ω m t q ) ) L , z ( t ) and α q in (2.18), (2.19) resp ectiv ely . By (2.3 4 ) and (4.7 ), follo ws the functional relati on of the transfer matrix ([17](4.40 )): b T p,p ′ ( y q , x q ) = N − 1 X m =0 C m ( q ) T p,p ′ ( ω m x q , y q ) − 1 T p,p ′ ( x q , y q ) T p,p ′ ( ω m +1 x q , y q ) − 1 X − m − 1 where C m ( q ) = ϕ ( q ) · · · ϕ ( U m − 1 q ) ϕ ( U m +1 q ) · · · ϕ ( U N − 1 q )( N ( y p y p ′ ) N − 1 ( y p − x q )( y p ′ − y q ) ( y N p − x N q )( y N p ′ − y N q ) ) L . F or the eigenv alue problem of the c hiral Pott s transfer matrix, it is conv enient to express the functional relations in terms of th e norm a lized op erators (4.4). By (2.23), (2.24 ) and g p ( U q ) g p ′ ( U q ) g p ( q ) g p ′ ( q ) = ( y p − ω x q ) N ( t p ′ − t q ) N ( x N p ′ − x N q ) ( x p ′ − x q ) N ( y N p − x N q )( t N p ′ − t N q ) , g p ( U ′ q ) g p ′ ( U ′ q ) g p ( q ) g p ′ ( q ) = ( x p − y q ) N ( t p ′ − t q ) N ( y N p ′ − y N q ) ( y p ′ − y q ) N ( x N p − y N q )( t N p ′ − t N q ) , 2 More precisely , V ( t q , σ q ) , b V ( t q , σ q ) are indeed meromorphic sections of a N th torsion line bundle asso cia ted to the N -fold co veri ng xy -curve ov er W in (2.13). 14 the τ (2) T -relation (2.31) takes the f o rm ([33] (35)): τ (2) ( t q ) V ( U q ) = ( ( y N p − x N q )( t N p ′ − t N q ) y N p y N p ′ ( x N p ′ − x N q ) ) L N V ( q ) + z ( ω t q )( y N p y N p ′ ( x N p ′ − x N q ) ( y N p − x N q )( t N p ′ − t N q ) ) L N X V ( U 2 q ) . (4.8 ) Note th at one can also deriv e the ab o v e r e lation u sing (2.33) and (4.5). Denote r j ( t q , σ q ) := α q X − j τ ( j ) ( t q ) + z ( t q ) z ( ω t q ) · · · z ( ω j − 1 t q ) τ ( N − j ) ( ω j t q ) , 0 ≤ j ≤ N , (4.9) where the v ariables t, σ are in (2.16 ). By u sing (2.27), the T T -relation (4.7 ) is the same as Γ ( j ) q T p,p ′ ( x q , y q ) b T p,p ′ ( ω j y q , x q ) = r j ( t q , σ q ) (4.10) where Γ ( j ) q is defined by Γ ( j ) q = { µ j p µ j p ′ ( y N p − x N q )( y N p ′ − y N q ) N y N + j − 1 p y N + j − 1 p ′ ( y p ′ − y q )( y p − x q ) j − 1 Y m =0 ( t p ′ − ω m t q ) j Y k =1 ω x p − ω k y q y p ′ − ω k y q } L . Define ζ ( t ) = ζ − 2 L N 0 N − 1 Y k =1 z ( ω k t ) k N , ζ 0 := e π i ( N − 1)( N +4) 12 . By (2.23) (2.24), the T T -relation (4.10) b ecomes ([10] (8)): α j N q ζ ( ω j t q ) V ( x q , y q ) b V ( ω j y q , x q ) = r j ( t q , σ q ) , ζ ( t q ) V ( x q , y q ) b V ( y q , x q ) = τ ( N ) ( t q ) , (4.11) where the j N th-p o w er of α q in (2.19) carries the ph a se-factor ω Lj . Hence w e obtain V N ( x q , y q ) Q N j =1 b V ( ω j y q , x q ) = ω − N ( N +1) L 2 ζ 2 L 0 α ( σ q ) − N α ( σ † q ) − N +1 2 Q N j =1 r j ( t q , σ q ) , ( Q N j =1 V ( ω j x q , y q ))( Q N j =1 b V ( y q , ω j x q )) = ζ 2 L 0 α ( σ q ) − N +1 2 α ( σ † q ) − N +1 2 Q N j =1 τ ( N ) ( ω j t q ) . (4.12) By f ormula (3) of [10], one fin ds N − 1 Y j =0 T p,p ′ ( ω j x q , y q ) = C s ( y q )  N − 1 Y j =1 ( y q − ω j +1 x p ) j ( y p ′ − ω j y q ) j  − L for some q -indep enden t constan t C , and a matrix s ( y q ) of degree N ( N − 1) L with y q -rational- function en tries, fin it e when b oth x q , y q are finite. Hence N − 1 Y j =0 V ( ω j x q , y q ) = ζ L 0 α ( σ q ) − N +1 2 S ( σ q ) (4.13) where the en tries of matrix S ( σ q ) are σ q -rational fu nctio ns with only p ossible p ole at σ = 0 ( [10] (4) ) 3 . Using (4.5), (4.6), and (4.11)-(4.1 3 ), one can express V ( x q , y q ) in terms of S, r j and τ ( N ) : V ( x q , y q ) N = ζ L 0 X N ( N +1) 2 α ( σ q ) − N S ( σ q ) Q N j =1 r j ( t q ,σ q ) τ ( N ) ( ω j t q ) = ζ L 0 ( Q N − 1 k =1 z ( ω k t ) − k ) α ( σ † ) N S ( σ q ) Q N j =1 τ ( N ) ( t q ) r j ( t q ,σ † ) . (4.14) 3 F or the rapidities in W k ′ (2.8), σ q = λ q by (2.16), and form ula (4.13) here is the same as [10] (4) except the factor λ − ( N − 1) L/ 2 q . The d i fference is due to the extra factor µ N ( N − 1) / 2 q of V ( q ) in (4.4) w e add in this paper. 15 4.2 Eigen v alues of a finite-size transfer matrix V ( q ) of the c hiral Potts model In this sub se ction, w e compu ter the eigen v alue V ( x q , y q ) using the Bethe solutions, F ( t q ) and F ′ ( t q ), of equation (3.8) (3.11 ) , where h ± , h ′∓ are defined in (3.18 ), (3.21) resp ectiv ely . By (3.23), the τ (2) T -relation (4.8) can b e written as τ (2) ( t q ) V ( U q ) = h + ( t q ) ( t N p ′ − t N q ) L N ( t p ′ − t q ) L V ( q ) + h − ( ω t q ) ( t p ′ − ω t q ) L ( t N p ′ − t N q ) L N X V ( U 2 q ) , τ (2) ( t q ) V ( U ′ q ) = h ′− ( t q ) ( t N p ′ − t N q ) L N ( t p ′ − t q ) L X V ( q ) + h ′ + ( ω t q ) ( t p ′ − ω t q ) L ( t N p ′ − t N q ) L N V ( U ′ 2 q ) . (4.15) Here the N th ro ot-o f-unity γ , γ ′ in h + ( t q ), h ′− ( t q ) are absorb ed in the L N th p o w er of t N p ′ − t N q . By (4.6) and (4.15), the τ (2) T -relation can also b e expressed by τ (2) ( t q ) V ( U ′ q ) = h + ( t q ) ( t N p ′ − t N q ) L N ( t p ′ − t q ) L ω P X V ( q ) + h − ( ω t q ) ( t p ′ − ω t q ) L ( t N p ′ − t N q ) L N ω − P V ( U ′ 2 q ) , τ (2) ( t q ) V ( U q ) = h ′− ( t q ) ( t N p ′ − t N q ) L N ( t p ′ − t q ) L ω − P V ( q ) + h ′ + ( ω t q ) ( t p ′ − ω t q ) L ( t N p ′ − t N q ) L N ω P X V ( U 2 q ) , (4.16) where ω P is some N th r oot of u nit y . Note that for generic p and p ′ , th e v ariable q are restricted is ( t q , σ q ) ∈ W + for the first relations of (4.15)and (4.16), with but ( t q , σ q ) ∈ W − for the second ones there. By (3.9), r j ( t, σ ) in (4.9) and τ ( N ) ( t ) are related by ([31] (30ab c)): r j ( t, σ ) = ω − j ( Q + P a ) h + ( t ) · · · h + ( ω j − 1 t ) F ( t ) F ( ω j t ) τ ( N ) ( ω j t ) , r j ( t, σ † ) = ω j P a h − ( t ) h − ( ω t ) · · · h − ( ω j − 1 t ) F ( ω j t ) F ( t ) τ ( N ) ( t ) (4.17) for j = 1 , . . . , N , where ( t, σ ) ∈ W + . The pro duct of formulas in (4.17), together with (3. 20 ), yields Q N j =1 r j ( t, σ ) = ω − N ( N +1)( Q + P a ) 2 α N q F ( t ) N ( Q N − 1 k =1 h + ( ω k t ) k )( Q N j =1 F ( ω j t )) Q N j =1 τ ( N ) ( ω j t ) , Q N j =1 r j ( t, σ † ) = ω N ( N +1) P a 2 α N q Q N j =1 F ( ω j t ) ( Q N − 1 k =1 h − ( ω k t ) k ) F ( t ) N τ ( N ) ( t ) N . Substituting either one of the ab o v e relatio ns in to (4.14), one fi nds the expression of V ( x q , y q ) : V ( x q , y q ) = ζ L N 0 F ( t q ) Q N − 1 k =1 ( t p ′ − ω k t q ) kL N  S ( σ q ) Q N − 1 k =1 ( t p ′ − ω k t q ) k L ( Q N − 1 k =1 h + ( ω k t q ) k )( Q N j =1 F ( ω j t q ))  1 N ω − ( N +1) P a 2 . By the pr o p erties of S ( σ ) in (4.1 3 ) and F ( t ) in (3.6), the analysis of p o w er orders near p oint s where t = 0 or t = ∞ with a finite y -v alue in tur n yields the expr e ssion  S ( σ q ) Q N − 1 k =1 ( t p ′ − ω k t q ) k L ( Q N − 1 k =1 h + ( ω k t q ) k )( Q N j =1 F ( ω j t q ))  1 N ω − ( N +1) P a 2 = x P x q y P y q µ − P µ q G ( σ q ) (4.18) where P x , P y and P µ are in tegers with P µ ≡ 0 (mo d N ), and G ( σ q ) is an algebraic fu ncti on of σ q with ( t q , σ q ) ∈ W + and a finite v alue except w h ere F ( t q ) = 0 or σ = σ † p . T o a vo id the am biguity of assigning integ ers P x , P y in case W ′′′ 1 where x q differs fr o m y q b y a constant factor, for the rest of 16 this sub sec tion, w e sh a ll assume the rapidit y curv e n o t equal to W ′′′ 1 , i.e. W = W k ′ , W ′ 1 , W ′′ 1 . The in tegers P x , P y , P µ are c hosen so that G (0) 6 = 0 with G ( σ q ) taking non-zero v alue wh en t q = 0, ( P µ is defined to b e zero in the case W = W ′′ 1 ). Hence V ( x q , y q ) = ζ L N 0 x P x q y P y q µ − P µ q F ( t q ) Q N − 1 k =1 ( t p ′ − ω k t q ) kL N G ( σ q ) , ( t q , σ q ) ∈ W + . (4.19) By the consistency of Bethe relation (3.7) and τ (2) V -relations (4.15) (4.16), the int egers P x , P y satisfy P x ≡ P a , P y ≡ P a + Q + P (mo d N ) wh er e P is in (4.16). Similarly , by (3.12) one finds r j ( t, σ ) = ω − j P b h ′− ( t ) · · · h ′− ( ω j − 1 t ) F ′ ( t ) F ′ ( ω j t ) τ ( N ) ( ω j t ) , r j ( t, σ † ) = ω j ( P b − Q ) ( Q j − 1 k =0 h ′ + ( ω k t )) F ′ ( ω j t ) F ′ ( t ) τ ( N ) ( t ) , (4.20) for ( t, σ ) ∈ W − , w hic h in tu r n yields V ( x q , y q ) = ζ L N 0 x P ′ x q y P ′ y q µ − P ′ µ q F ′ ( t q ) Q N − 1 k =1 ( t p ′ − ω k t q ) kL N G ′ ( σ q ) , ( t q , σ q ) ∈ W − , (4.21) where P ′ x ≡ P b − Q − P , P ′ y ≡ P b (mo d N ), and G ′ ( σ ) is defined b y  S ( σ q ) Q N − 1 k =1 ( t p ′ − ω k t q ) k L ( Q N − 1 k =1 h ′− ( ω k t q ) k )( Q N j =1 F ′ ( ω j t q ))  1 N ω − ( N +1)( P b − Q ) 2 = x P ′ x q y P ′ y q µ − P ′ µ q G ′ ( σ q ) (4.22) suc h that G ′ (0 † ) 6 = 0 and G ′ ( σ q ) is non-zero when t q = 0. Ther efore one obtains P x + P ′ y ≡ P y + P ′ x ≡ P a + P b . Note that (4.19) and (4.21) are v alid on W + and W − resp ectiv ely , but the form ulas are not true on the whole Riemann surface W in general when the v ertical rap id it ies p, p ′ are arb it rary . O n e needs b oth relations, (4.19) and (4.21), f o r an equ iv alen t expr e ssion of the N th T T -relation in (4.11): D y N m µ − N d q G ( σ q ) G ′ ( σ † q ) = t − ( P x + P ′ y ) q τ ( N ) ( t q ) F ( t q ) F ′ ( t q ) Q N − 1 k =1 ( y 2 p y 2 p ′ ( t p ′ − ω k t q ) ω µ p µ p ′ ( t p − ω k t q ) ) kL N , D x − N m µ − N d q G ( σ q ) G ′ ( σ † q ) = t − ( P ′ x + P y ) q τ ( N ) ( t q ) F ( t q ) F ′ ( t q ) Q N − 1 k =1 ( y 2 p y 2 p ′ ( t p ′ − ω k t q ) ω µ p µ p ′ ( t p − ω k t q ) ) kL N , (4.23) where ( t q , σ q ) ∈ W + , and d, m are in tegers with N d := P µ − P ′ µ , and N m = P ′ x + P y − ( P x + P ′ y ) ≡ 0 (mo d N ). On e can also u s e th e v ariable ( t q , σ q ) ∈ W − to the express (4.23) by in terc hanging σ q , x q , µ q with σ † q , y q , µ − 1 q resp ectiv ely . T he right hand sides of (4.23) are alg ebraic fu ncti ons of t , in v ariant when replacing t by ω t , and they defin e the same f u nctio n wh e n m = 0. It is kno wn that y N q is a nev er-v an ish ing fun ct ion on W + when W = W k ′ ( | k ′ | < 1), W ′ 1 , W ′′ 1 ; while x N q is n o n-zero on W + when W = W k ′ ( | k ′ | > 1). Corresp ondingly , w e denote the t -function of the righ t hand side in (4.23) by P ( t ). Then P (0) 6 = 0, and P ( t ) is ind eed a t N -function. In the case W ′ 1 , one find s m = 0 in (4.23), wh ere t w o relations defi ne the same function P ( t ). By adjusting the int egers P a and P b , one ma y assume P x = P a , P ′ y = P b in the firs t relation of (4.23), or P y = P a + Q + P , P ′ x = P b − Q − P in second on e. Hence P ( t ) is expr e ssed by P ( t ) = t − P a − P b q τ ( N ) ( t ) F ( t ) F ′ ( t ) N − 1 Y k =1 ( y 2 p y 2 p ′ ( t p ′ − ω k t ) ω µ p µ p ′ ( t p − ω k t ) ) kL N (4.24) 17 with P (0) 6 = 0. The ab o v e P ( t ) is related to the f u nctio ns, G and G ′ , by D y N m µ − N d q G ( σ q ) G ′ ( σ † q ) = P ( t q ) , ( t q , σ q ) ∈ W + k ′ ( | k ′ | < 1) , W ′ + 1 , W ′′ + 1 ; D x − N m µ − N d q G ( σ q ) G ′ ( σ † q ) = P ( t q ) , ( t q , σ q ) ∈ W + k ′ ( | k ′ | > 1) , W ′ + 1 . (4.25) Note that d = 0 in the case W ′′ 1 where P µ , P ′ µ are defined to b e zero. In the case W = W k ′ or W ′ 1 , σ q = µ N q , and the ∞ -limit of t q in W + corresp onds to ∞ -limit of x N q with a fin it e y N q -v alue. When t q → ∞ , one finds d ≥ 0 in the fi rst relation of (4.25), and d − m ≥ 0 in the second one. Note that as f unctio ns of W , the t N -function P ( t ) in (4.24) can b e decomp osed as a pro duct f u nctio n : P ( t ) = G ( σ ) G ( σ † ), where G ( σ ) is a σ -function for σ ∈ C . How eve r, the functions G ( σ ) , µ − N d q G ′ ( σ † ) in (4.23) can not b e expressed as G ( σ ), G ( σ † ) in general (for arbitrary p ′ , p ′ ). Indeed, sa y when m = 0, the relation (4. 25 ) implies that there exists s ome factorizat ion of P ( t ) = G ( σ ) G ( σ † ) suc h that G ( σ ) = G ( σ ) g ( t, σ ) , µ N d G ′ ( σ ) = G ( σ ) g ′ ( t, σ ), wh ere g ( t, σ ) and g ′ ( t, σ ) are n e ve r-v anishing functions on W + , W − resp ectiv ely suc h that D g ( t, σ ) g ′ ( t, σ † ) = 1 for ( t, σ ) ∈ W + . F or the rest of this subs ection, we determine the quantum n um b ers P a , P b in the case of generic p and p ′ . W e lea v e the sp ec ial, i.e. the alternating sup erin tegrable (4.2 9 ), case in th e n e xt sub- section. By the L -op erat or exp ressio n (2.1) with parameters in (2.17), the eigen v alue τ (2) ( t ) is a t -p olynomial of degree L with the leading and constan t terms giv en by lim t →∞ τ (2) ( t ) t L = ( − ω y p y p ′ ) L (1 + ω Q µ L p µ L p ′ ) , τ (2) (0) = 1 + ω Q z (0) . (4.26) On the other hand, τ (2) ( t ) is expressed by (3.7) and(3.10 ), wh ere F ( t ) , F ′ ( t ) are the Bethe solution of (3.8 ), (3.11) resp ectiv ely . Using (3.19), (3.22), (3.20) and (3.24), one obtains another expression of th e leading and constant terms of τ (2) ( t ): lim t →∞ τ (2) ( t ) t L = ( − ω y p y p ′ ) L ( ω − P a + P γ − J − L + ω Q + P a − P γ + L + J µ L p µ L p ′ ) = ( − ω y p y p ′ ) L ( ω Q − P b + P γ ′ − J ′ − L ( µ N p µ N p ′ ) L N + ω P b − P γ ′ + L + J ′ µ L p µ L p ′ ( µ N p µ N p ′ ) − L N ) , τ (2) (0) = ω − P a + ω Q + P a z (0) = ω P b + ω Q − P b z (0) . (4.27) F or generic p, p ′ in (2.17) when b ot h z (0 ) and µ L p µ L p ′ not b eing N th ro o ts of u nit y , the relations, (4.26) an d (4.27), imply the follo wing constraints of the quantum num b ers in (3.8) and (3.11): P a ≡ P b ≡ 0 , J ≡ P γ − L, J ′ ≡ P γ ′ − L + r . (4.2 8) where r is the in tege r determined b y µ L p µ L p ′ = ( µ N p µ N p ′ ) L N ω − r . In th is situation, h ± ( t ) in (3.18) is defined only on W + , while h ′∓ ( t ) in (3.21) defin e d only on W − . By the discuss io n in section 3.2, F ( t ) in (3.8) and F ′ ( t ) in (3.11) are different p olynomials. The eigen v alues of chiral P otts transfer matrix defined by (4.18) and (4.19) are in one-to-one corresp ondence w ith eigen v alues (3.7) of τ (2) -matrix obtained b y the Bethe equation (3.8). Hence the τ (2) -matrix is non-singular wh en th e rapidit y parameters p, p ′ in (2.17) of th e L -op er ator (2.1) are generic. The corresp ondin g c hiral P otts eigen v alues are expressed by (4.19) and (4.21 ), whic h satisfy the relation (4.15). 18 4.3 Eigen v alues V ( q ) for the alternating sup erin tegr a ble c hiral Potts model The alternating sup erinte grable case 4 is when the v ertical r ap id it ies p, p ′ in (2.17) satisfy one of the follo wing equiv alent r el ations: x N p = y N p ′ ⇐ ⇒ y N p = x N p ′ ⇐ ⇒ σ p = σ † p ′ ⇐ ⇒ α 1 L q = α 1 L q (4.29) where the v ariables σ, σ † are in (2.16), and α q , α q ( q ∈ W ) are fu nctio ns in (2.19). When the rapidit y curv e is W k ′ in (2.8) or W ′ 1 in (2.9), the ab o v e cond itions are equiv alen t to λ p λ p ′ = 1. In the alternating sup erinteg rable case, the function τ ( N ) ( t ) F ( t ) 2 with the Bethe solution F ( t ) of (3.8) is a t -p olynomial: τ ( N ) ( t ) F ( t ) 2 ∈ C [ t ]. Indeed by (3.9 ), τ ( N ) ( t ) F ( t ) 2 is regular except th e zeros t 0 ’s of F ( t ). By (3.20), the finite-v alued condition of τ ( N ) ( t ) F ( t ) 2 at t 0 ’s is equiv alen t to the v anish ing of α q α q h + ( ω − 1 t ) F ( ω − 1 t ) + ω ( Q +2 P a ) h − ( t ) F ( ω t ) at t 0 ’s, wh ic h is the same as the Bethe condition (3.8 ) when α q = α q . In this section, we discu ss the alternating sup erinteg rable case. W rite x p = ω m y p ′ , x p ′ = ω m ′ y p , µ p µ p ′ = ω n (m , m ′ , n ∈ Z ) , (4.30) and d e fine the v ariable t := t y p y p ′ . Since e ± ( t ) = 1, th e fun c tions h ± ( t ) , h ′∓ ( t ) in (3.18), (3.21) are h + ( t ) = h + ( t ) = (1 − ω − m ′ t ) L , h − ( t ) = h − ( t ) = ω (1+m+m ′ +n) L (1 − ω − m t ) L ; h ′− ( t ) = h ′− ( t ) = ω (1+m+m ′ +n) L (1 − ω − m ′ t ) L , h ′ + ( t ) = h ′ + ( t ) = (1 − ω − m t ) L (4.31) with P γ = − m ′ L and P γ ′ = (1 + m + n) L . The ab o v e t -p olynomials are considered as functions on W . Using the relation b et ween h ± and h ′∓ , the expr e ssions (3.7) and (3.10) for τ (2) ( t ) are the same with F ( t ) = F ′ ( t ) and P b − P a ≡ Q + (1 + m + m ′ + n) L (mo d N ). Th e ω P in (4.16) is giv en b y P = P γ ′ − P γ . W rite F ( t ) in (3.6) as F ( t ) = F ( t ) = Q J j =1 (1 + ω v j t ) where v j = y p y p ′ v j . Both (3.8) and (3.11) define the same Bethe equation: ( v i + ω − m − 1 v i + ω − m ′ − 2 ) L = − ω − P a − P b Q J j =1 v i − ω − 1 v j v i − ω v j , i = 1 , . . . , J. (4.32) Hence τ j ( t ) in (3.9) and (3.12) is exp r essed by τ ( j ) ( t ) = ω ( j − 1) P b F ( t ) F ( ω j t ) P j − 1 n =0 { (1 − ω − m ′ t ) ··· (1 − ω − m ′ + n − 1 t )(1 − ω − m+ n +1 t ) ··· (1 − ω − m+ j − 1 t ) } L ω − n ( P a + P b ) F ( ω n t ) F ( ω n +1 t ) . In p a rticular, one obtains the t -p olynomial τ ( N ) ( t ) F ( t ) 2 = ω − P b P N − 1 n =0 { (1 − ω − m ′ t ) ··· (1 − ω − m ′ + n − 1 t )(1 − ω − m+ n +1 t ) ··· (1 − ω − m+ N − 1 t ) } L ω − n ( P a + P b ) F ( ω n t ) F ( ω n +1 t ) . (4.33) No w (4.18 ) and (4.22) are defined on W with P x = P ′ x , P y = P ′ y . The functions G, G ′ (b y a prop er c hoice of phase-factors) are r el ated by G ( σ ) = µ N d G ′ ( σ ) , N d := P µ − P ′ µ . 4 Here we use a sligh tly general notion of alternating sup erin tegrabilit y than t h at in [11] (6.1), [12] (14) or [17] section 6, where the alternating sup erin tegrable CPM is when x p ′ = y p , y p ′ = x p . Our general setting includes one sp ecial case of X XZ chains associated to cyclic U q ( sl 2 ) representations with b oth q and the representatio n parameter ς b eing N th roots of unity [40]. 19 The eigen v alue in (4.19), equiv alen tly (4.21), b ecomes V ( x q , y q ) = ζ L N 0 x P a q y P b q µ − P µ q F ( t q ) Q N − 1 k =1 ( t p ′ − ω k t q ) kL N G ( σ q ) , ( t q , σ q ) ∈ W . (4.34) where G ( σ ) in (4.25) and the t N -function P ( t ) in (4.24) are expressed in the follo wing form: D G ( σ ) G ( σ † ) = P ( t ) , ( t, σ ) ∈ W , P ( t ) := C t − P a − P b Q N − 1 k =1 ( (1 − ω − m ′ + k t ) (1 − ω − m+ k t ) ) kL N τ ( N ) ( t ) F ( t ) 2 , P (0) 6 = 0 , (4.35) with C = ω (m ′ − m − n − 1)( N − 1) L 2 ( y p y p ′ ) − P a − P b +( N − 1) L . The non-v anishing of G ( σ ) and G ( σ † ) w hen t q = 0 implies the in tegers P a , P b satisfying 0 ≤ P a + P b ≤ N − 1 , P b − P a ≡ Q + (1 + m + m ′ + n) L (mo d N ) . (4.36) Note th a t the relations, (4.34) and (4.35), are still v alid when the rapidit y curve is W ′′′ 1 in the alternating sup erinte grable case. By (4.26), (4.27 ) and z (0) = ω (1+n+m+m ′ ) L , one finds 1 + ω P b − P a = ω P b + ω − P a , 1 + ω Q +n L = ω − P a − m ′ L − J − L + ω Q + P a +m ′ L + L + J +n L , whic h imply P a ≡ 0 or P b ≡ 0 , J + P b ≡ (m + n) L + Q, (m + 2m ′ ) L (mo d N ) . (4.37) The parameters in the L - op erator (2.1) of τ (2) -mo dels equiv alen t to the alternating s u perin - tegrable case sati sfy th e relations: c N = 1, a N = b ′ N , a ′ N = b N ([41] (2.23) ). Th e se mo dels a re c haracterized by ω m , ω m ′ , ω n in (4.30) with the L -op erator L ( t ) = 1 − ω n t X (1 − ω m+n+1 X ) Z − t (1 − ω m ′ +n X ) Z − 1 − t + ω m+m ′ +n+1 X ! . (4.38) The c hiral Pot ts transfer matrix T p,p ′ ( q ) with p, p ′ in (2.8) or (2.9) can b e constructed from the τ (2) -matrix as a Baxter’s Q -op erato r (see [17, 19] or [41] section 3.1). F or eac h τ (2) -eigen v alue, one asso cia tes the t N -function P ( t ) in (4.25). Eac h factor function G ( σ ) of P ( t ) giv es rise to an eigen v alue (4.19) of T p,p ′ ( q ). Hence the d e generacy of τ (2) -mo del o ccur s, with a p o w er of 2 degenerate states for eac h τ (2) -eigen v alue. Note that the Q -op e rators of the same τ (2) -matrix in this cont ext are distinct f or differen t c h oi ces of r a pidity curves. It is kn own [40] when N is od d: N = 2 M + 1, the τ (2) -mo del with ω m+m ′ +1 = ω n = 1 is identi fied with XXZ c h a ins asso ciat ed to cyclic U q ( sl 2 ) rep resen tations with b oth q and the representat ion parameter ς b eing N th ro ots of unit y . In these cases, the τ (2) -degeneracy carries the s l 2 -lo op -algebra symmetry . In particular, when m = m ′ = M , the τ (2) -mo del is the spin- N − 1 2 XXZ c hain, with the Q -op erato r identi fied w i th the homogeneous c h i ral P otts tran s fer matrix for p = p ′ : ( x p , y p , µ p ) = ( ω M η 1 2 , η 1 2 , 1), where η is in (2.14). In this case, th e s l 2 -lo op -algebra s y m met ry of XXZ c hain inherits the On sa ger-algebra- symmetry induced from the Hamiltonian chain of the chiral Pott s mo del [34]. 20 The homogeneous sup erint egrable c hiral Pot ts mo del is the case (4.29) wh e n p = p ′ 6∈ W ′′′ with the co ordinates p = p ′ : ( x p , y p , µ p ) = ( ω m+ k η ± 1 2 , ω k η ± 1 2 , ω n 2 ) , where η is in (2.14). Th e τ ( N ) ( t ) , P ( t ) in (4.33) and (4.25) resp ectiv ely are p olynomial s expressed b y τ ( N ) ( t ) = C − 1 t P a + P b F ( t ) 2 P ( t ) , P ( t ) = C ω − P b P N − 1 n =0 (1 − t N ) L ( ω n t ) − ( P a + P b ) (1 − ω n − m t ) L F ( ω n t ) F ( ω n +1 t ) , (4.39) where C = ω − (n+1)( N − 1) L 2 ( ω 2 k η ± 1 ) − P a − P b +( N − 1) L , and P a , P b are inte gers in (4.36) w i th m ′ = m. Indeed, P ( t ) is a t N -p olynomial wh ich defines the ev aluation p olynomial of the Onsager-algebra symmetry in the homoge neous sup erin tegrable chiral Pott s mo del [34]. T he homogeneous s up er - in tegrable CP M with rapidities in W k ′ and m = k = n = 0 has b een discu ssed extensively in literature, and the eigen v alues of c hiral P otts tr a nsfer matrix are giv en in [1] [8, 9, 10, 11, 12]: T ( q ) = N L ( η − 1 2 x q − 1) L ( η − N/ 2 x N q − 1) L ( η − 1 2 x q ) P a ( η − 1 2 y q ) P b µ − P µ q F ( η − 1 t q ) ω P b F ( ω η ) G ( λ q ) where G ( λ q ) satisfies G ( λ q ) G ( λ − 1 q ) = P ( t ) P ( η ) . By (4.4), one fin ds V ( x q , y q ) = e π i ( N − 1)( N − 2) L 12 N N L 2 η − P a − P b 2 ω P b F ( ω η ) x P a q y P b q µ − P µ q F ( η − 1 t q ) Q N − 1 k =1 (1 − ω k η − 1 t q ) kL N G ( λ q ) . Then the eigen v alue S ( λ q ) in (4.13) is expressed by ( S ( λ q ) Q N k =1 F ( ω k η − 1 t q ) ) 1 N = N L 2 η − P a − P b 2 ω P b + ( N − 1) L 4 F ( ω η ) x P a q y P b q µ − P µ q G ( λ q ) ω P a ( N − 1) 2 . Hence G ( λ q ) r e lates to G ( λ q ) by G ( λ q ) = e − π i ( N − 1) L 2 N N L 2 η 1 2 ( − P a − P b + L ( N − 1)) ω P b F ( ω η ) G ( λ q ) so that (4.18), (4.19) and (4.34) hold. The ev aluation p olynomial P ( t ) is defined by (4.39) with m = k = n = 0. Th e r e lation (4.2 5 ) holds for D = ω P b F ( ωη ) F ( η ) (= e i P ), where P is the total momen tum ([17] (4.49) , [1 ] (2.24)) 5 . The quan tum num b ers, P a , P b and J , are the in tegers in (4.36) (4.37), no w expressed by ( i ) 0 ≤ P a + P b ≤ N − 1 , P b − P a ≡ Q + L (mo d N ) , ( ii ) P a ≡ 0 or P b ≡ 0 , J + P b ≡ 0 , Q (mo d N ) . (4.40) The ab o ve ( i ) is the condition [11] (6.16), [12] (24) for r = 0, and [1] (C.4) in the case N = 3. The condition ( ii ) could b e kno wn for the sp eciali sts as it is sup ported by th e n umerical studies for N = 3 in T able I of [1 ], b ut it seems not app eared in literature b efore to the b est of the author’s kno wledge. This no v el constraint is consisten t with the total momen tum e i P L = 1 condition: LP b ≡ J ( Q − 2 P b − J ) (mo d N ) ([1] (C.3), [34] (63) ), and it also agrees with the conclusion, P a ≡ 0 or P b ≡ 0, in the algebraic Bethe ansatz discussion of τ (2) -mo del ([37] section 3.2). 5 Here we use th e conv ention in [17], where the total m omentum d iffers from th a t in [1] (2.24) or [34] (62) by a minus sign. 21 5 Concluding Remarks In this w ork, w e study the eigen v alue sp ectrum of a fin it e-size transfer matrix of the c hir a l Potts mo del with alternating rapidities b y use of the fun ct ional relations in [17]. Here the rapidity cur v e is either W k ′ in (2.8) [18], or a curve in (2.9) for the degenerate mo dels, whic h include the selfdual solutions of the s tar-triangle relati on (2 .25) [2, 3 , 25, 30]. W e first establish the Bethe equations, (3.8) and (3.11), of τ (2) -mo del thr o ugh the Wiener-Ho pf splitting (3.18) (3.21) of α q and α q as in [31]. W e express th e τ ( j ) -eigen v alues by usin g the Bethe solution, then obtain the expression, (4.19), (4.21) together with (4.25), of eigen v alues V ( x q , y q ) of the fi nite transfer matrix in CPM. T he pro cedure enables one to solv e the eigen v alue problem of CPM of a fin it e size. In the alternating sup erin tegrable case (4.29), the τ (2) -mo dels for all k ′ are the same, p rodu c ed by the L -op erato r (4.38) with a simp le form of Bethe equation (4.32). As in the homogeneous su p erinteg rable C P M [34], the d eg eneracy of the τ (2) -matrix o c curs, and the c hiral P otts transfer matrix serv es a Q - op erato r of th e τ (2) -matrix. Th e eigen v alues, (4.34) and (4.25), of the chiral Pot ts transfer matrix share a similar str ucture as those of homogeneous sup erin tegrable CPM in [1, 11, 12]. W e fin d the deriv ation in this pap er has enhanced our structural un derstanding ab out the natur e s of eig env alues in CPM. The exp li cit calculation w ill lead to some physical imp lications as compared with previous w orks in homogeneous c hiral P otts mo del in [1, 8 , 31]. In this work, w e s t ud y the CPM mainly f rom the asp ect of the herm i tian quantum c hain case as in [31]. The physic s of the quantum spin chain Hamiltonian is differen t from the statistical mo del with the p ositiv e Boltzmann w eigh t case, w hic h w as the main concern in [4 , 9, 14], (see, e.g. [3]). Th ese t w o cases o verlap only in the critical limiting case of [25]. In this pap er the issu e s ab out the p eculiaritie s of those v arious physical regimes we re not discussed . Here, ju st to k eep th i ngs simple, we restrict our atten tion only to the mathematical asp ect of eigen v alue sp ectrum of the fin it e-size transfer matrix. W e lea v e the physica l discussion of ours r e sults, an d p ossible generalizations to future work. Ac kno wledgemen ts The author is pleased to ac kno wledge the hospitalit y of Lab oratoire de Mat hematiques et Physique Theorique, CNRS/UMR, Univ ersit y of T ours , F r a nce (2007, fall), and U C Berk eley , U.S.A. (2008, spring), w h ere parts of this work were carried out. He wishes to thank Professor P . Baseilhac, and Professor S. Kobay ashi for their in vitatio n. This work is supp orted in part by Na tional Science Council of T aiw an under Gran t No NSC 96-2115- M-001-004. References [1] G. Alb ertini, B. M. McCo y , and J. H. H. Perk, Eigen v alue sp ectrum of the sup erin tegrable c hiral P otts mo del, In Inte gr able system in quantum field the ory and statistic al me chanics, Adv. Stud. Pure Math., 19, Kinokun iy a Academic, Academic Pr e ss, Boston, MA (1989) 1–55. 22 [2] H. Au-Y ang, B. M. McCo y , J . H. H. P erk and S. T ang, Solv able mo d el s in statistical mec hanics and Riemann surfaces of genus greater than one, Algebr aic Analysis , V ol. 1 , eds. M. Kashiw ara and T. Ka wa i, Academic Press, San Diego (1988), 29–40. [3] H. Au-Y ang and J. H. H. Pe rk, Onsager’s star-triangle equation: Master k ey to inte grabilit y , In Inte gr able system in quantum field the ory and statistic al me chanics, Adv. Stud. Pure Math., 19, Kinokun i ya Academic, Academic Press, Boston, MA (1989) 57–94. [4] H. Au-Y ang, B. Q. Jin and J. H. H. P erk, Baxter’s solution for th e free energy of the c hiral P otts mo del, J. Stat. Phys. 102 (2001) 471–499. [5] R. J. Baxter, Partitio n function of the eight vertex mo del, Ann. Phys. 70 (1972) 193–228. [6] R. J. Baxter, Eight-v ertex mo del in lattice statistic and on e-dimen sio nal anisotropic Heisenberg c hain I: S o me fundamental eigen v alues, I I. Equiv alence to a generalize d Ice-t yp e lattice mo del, I I I. Eigen v alues of the trans fer matrix and Hamiltonian, Ann. Phys. 76 (1973 ) 1–71. [7] R. J. Baxter, E x actly solv ed mo dels in s tatistical mec hanics, Academic Press (1982). [8] R. J . Baxte r, Sup erin tegrable c hiral Pot ts mo del: T hermod ynamic pr op erties, an ”In v erse” mo del, and a simp l e asso ciated Hamiltonian, J. Stat. Phys. 57 (198 9) 1–39 . [9] R. J. Baxter, C alculation of the eigen v alues of the transfer matrix of the c hiral Po tts mo del, Pr o c . F ourth Asia-Pacific Physics Confer enc e (Seoul, Korea, 1990) V ol 1, W orld - Scient ific, Singap ore (1991) 42–58. [10] R. J. Baxter, Chiral Potts mo del: eigen v alues of the transfer matrix, Ph ys. Lett. A 146 (1990) 110–1 14. [11] R. J. Baxter, Chiral Potts mo del w i th skew ed b oundary conditions, J. Stat. Ph ys. 73 (1993) 461–4 95. [12] R. J. Baxter, I n terfacial tension of the chiral P otts mo del, J. Phys. A: Math. Gen. 27 (1994) 1837– 1849. [13] R. J. Baxter, Th e ”inv ersion r e lation” metho d for obtaining the free energy of the chiral P otts mo del, Physica A 322 (2003) 407–431; cond-mat/02 12107 5. [14] R. J. Baxter, The Riemann su r fac e of the c hiral P otts mo del free energy function, J. S ta t. Ph ys. 112 (2003) 1–26; cond-mat/021 2076 . [15] R. J. Baxter, T r a nsfer matrix functional r e lation for the generalized τ 2 ( t q ) mo del, J. Stat. Ph ys. 117 (2004) 1–25; cond-mat/040 9493 . [16] R. J . Baxter, Deriv ation of the ord e r parameter of the c hiral P otts mo del, Ph ys.Rev.Lett. 94 (2005 ) 130602; cond -mat/0501227. 23 [17] R. J. Baxter, V.V. Bazhano v and J.H.H. Perk, F un ct ional relations for transfer matrices of the c hiral Pott s mo del, Int. J. Mo d. Phys. B 4 (1990) 803–870 . [18] R. J. Ba xter, J. H. H. Pe rk and H. Au-Y ang, New solutions of the star-triangle relati ons for the c hiral Pott s mo del, Phys. Lett. A 128 (1988) 138–142. [19] V.V. Bazhano v and Y u.G. Str o gano v, Ch i ral P otts mod e l as a descendan t of the six-v ertex mo del, J. S tat. Phys. 59 (1990) 799–817 . [20] T. Deguc hi, K. F abricius and B. M. McCo y , Th e sl 2 lo o p algebra symm e try for the six-ve rtex mo del at r oots of unity , J . Stat. Phys. 102 (2001) 701–736 ; cond - mat/99121 41 . [21] K. F abricius and B. M. McCoy , Ev aluation parameters and Bethe ro ots for the six v ertex mo del at ro ots of unity , Pr o gr ess in Mathematic al Physics V ol 23, eds. M. Kashiwara and T. Miw a, Birkh¨ auser Boston (2002) , 119–144 ; cond -mat/0108057 . [22] K. F abricius and B. M. McCo y , New devel opments in th e eigh t vertex mo del, J . Stat. Phys. 111 (2003) 323–337 ; cond -mat/0207177. [23] K. F abricius and B. M. McCoy , F unctional equations and fusion m a trices f o r the eigh t v ertex mo del, Pu bl. RIMS, 40 (2004) 905–932 ; cond - mat/03111 22 . [24] K. F abricius, A new Q -op erator in the eight -v ertex mo del, J. Phys. A: Math. Theor. 40 (2007) 4075– 4086; cond-mat/06104 81 v3. [25] V. A. F ateev and A. B. Zamolo dc hiko v, Self-dual solutions of the star-triangle r elations in Z N -mo dels, Phys. Lett. A 92 (1982) 37–39. [26] G. v on Gehlen, N. Iorgo v, S. P akuliak and V. Shadur a : Baxter-Baz hanov- Strogano v mo del: Separation of v ariables and Baxter equation, J. Phys. A: Math. Gen. 39 (2006 ) 7257–7282 ; nlin.SI/0603028. [27] G. vo n Gehlen, N. Iorgo v, S. Pakuliak, V. Shadur a and Y u T ykhyy: F orm - factors in the Baxter-Baz hanov- Strogano v mo del I: Norms and matrix elemen ts, J. Phys. A: Math. T heor. 40 (2007) 14117–14 138; arXiv:0708.362 5 . [28] G. vo n Gehlen, N. Iorgo v, S. Pakuliak, V. Shadur a and Y u T ykhyy: F orm - factors in the Baxter-Baz hanov- Strogano v mo del I I: Ising mo del on the finite lattice, J. Phys. A: Math. Theor. 41 (2008) 095003; arXiv:0711.045 7 . [29] V. B. Matv eev and A. O. Simn o v, Some comments on the s olv able chiral Potts mo del, Lett. Math. Phys. 19 (1990) 179–185. [30] B. M. McCo y , J. H. H. Pe rk, S. T ang and C. H. Sah, Comm uting transfer matrices for the four-state self-du a l c h iral P otts mo del with a genus-three uniformizing F ermat curve , Ph ys. Lett. A 125 (1987) 9–14. 24 [31] B. M. McCoy and S. S . Roa n, Excitation sp ectrum and phase structure of the c hiral Pot ts mo del. Phys. Lett. A 150 (1990) 347–3 54. [32] L. Onsager, C r ystal statistics.I. A tw o-dimens i onal m o del with an order-disorder transition. Ph ys. Rev. 65 (1944) 117– 149. [33] S. S. Roan, Ch iral P otts rapidity curve descended from six-v ertex mo del and symmetry group of rapidities, J. Phys. A: Math. Gen. 38 (2005) 7483 –7499; cond-mat/04100 11 . [34] S. S . R oan, The On sa ger algebra symmetry of τ ( j ) -matrices in the su perintegrable c hiral Po tts mo del, J. S tat. Mec h. (2005) P09007; cond-mat/050569 8 . [35] S. S. Roan, Bethe ansatz and symmetry in sup erin tegrable chiral Po tts mo del and ro ot -of- unit y s i x-ve rtex mo del, in Na nk ai T racts in Mathematics V ol. 10, Diffe r ential Ge ometry and Physics , eds. Mo-Lin Go an d W eiping Zhang, W orld Scien tific, Singap ore (2006), 399-40 9; cond-mat/051 1543 . [36] S. S . Roan, The Q-op erator for ro ot-o f-unity symmetry in six v ertex mo del, J. Phys. A: Math. Gen. 39 (2006) 12303-1 2325; co nd-mat/0602375. [37] S. S . Roan, F usion op erators in the generalized τ (2) -mo del and r o ot-of-unit y symmetry of the XXZ spin chai n of higher s p in, J. Phys. A: Math. Th eo r. 40 (2007) 1481-15 11; cond-mat/060 7258 . [38] S. S . Roan, Th e Q -op erator an d functional relations of the eigh t-v ertex mo del at ro ot-of-unit y η = 2 mK/ N f o r o dd N , J. Phys. A: Math. T heo r. 40 (2007) 11019-11 044; cond-mat/061131 6 . [39] S. S. Roan, O n Q-op erators of XXZ spin c hain of h igher sp in, cond-mat/ 070227 1 . [40] S. S . R oan, T h e transfer matrix of su per integrable c hiral Po tts mo del as the Q-op erator of ro ot- of-unity XXZ c hain with cyclic represen tation of U q ( sl 2 ), J. Stat. Mec h. (2007) P090 21; arXiv: 0705.2856. [41] S. S. Roan, On the equiv alen t theory of the generalized τ (2) -mo del and the c hiral Po tts m o del with tw o alternating v ertical r ap id it ies, arXiv: 0710.2764. [42] V. O. T araso v, Cyclic m o no drom y m atrices for the R-matrix of the six-ve rtex mo del and th e c hiral Pot ts mo del w it h fix spin b oundary cond i tions, In tern. J. Mo d. Phys. A7 Su ppl. 1B (1992 ) 963–975 . 25