cond-mat.stat-mech 2011-02-11 0

Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method

For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by…

Authors: Tetsuo Deguchi, Chihiro Matsui

Correlation F unctions of the integrable higher-spin XXX and XXZ spin c hains through the fusion metho d T etsuo Deguc hi 1 ∗ and Chihiro Matsui 2 , 3 † 1 Departmen t of Ph ysics, Graduate Sc ho ol of Humanities and Sciences, Ochanomiz u Univ ersit y 2-1-1 Oh tsuk a, Bunky o-ku, T okyo 112-861 0 , Japan 2 Departmen t of Ph ysics, Graduate Sc ho ol of Science, the Univ ersit y of T okyo 7-3-1 Hongo, Bunky o-ku, T okyo 113-003 3 , Japan 3 CREST, JST, 4-1-8 Honc ho Kaw aguc hi, Saitama, 332-001 2, Japan Abstract F or the in tegrable higher-sp in XXX and XXZ spin c hains w e pr esen t m ultiple-in tegral represent ations for the correlation function of an arbitrary p rod uct of Hermitian ele- men tary matrices in the massless ground state. W e giv e a formula expressing it b y a single term of m ultiple in tegrals. In particular, we exp lici tly derive the emptiness forma- tion probabilit y (EFP). W e assume 2 s -strings for the ground-state solution of the Bethe ansatz equations for the spin- s XXZ c hain, and s olve the in tegral equations for the spin - s Gaudin matrix. In terms of the XXZ coupling ∆ we define ζ b y ∆ = cos ζ , and pu t it in a region 0 ≤ ζ < π / 2 s of the gapless regime: − 1 < ∆ ≤ 1 (0 ≤ ζ < π ), where ∆ = 1 ( ζ = 0) corresp onds to the an tiferromagnetic p oin t. W e calculate the zero-temp erature correlation functions b y the alg ebraic Bethe ansatz, introd ucing the Herm itia n elementa ry matrices in the massless r egi me, and taking adv an tage of the fu sion construction of the R -matrix of the higher-spin rep resen tations of the affine quantum group . ∗ e-mail deguchi@ph ys.o cha.ac.jp † e-mail matsui@spin.phys.s.u-tokyo.ac.jp 1 1 In tro duc t ion The correlation functions of the spin-1/2 XXZ spin c hain ha v e b een studied extensiv ely through the algebraic Bethe-ansatz during the last decade [1, 2, 3 , 4, 5 , 6]. The m ultiple-in tegral repre- sen tations of the correlation functions for the infinite la ttice at zero temp erature first deriv ed through the affine quantum-group symme try [7, 8] and also b y solving the q - KZ equations [9, 10] ha v e b een rederiv ed and then g eneralized in to those for the finite-size latt ice under non-zero magnetic field. They are also extended into those at finite temp eratures [11]. F urt hermore, the asymptotic expansion of a correlat io n f unction has b een systematically discussed [12]. Th us, the exact study of the correlation functions of the XXZ spin c hain should b e not only very fruitful but also quite fundamen tal in t he mathematical phys ics of in tegrable mo dels. Recen tly , the correlation functions and form fa cto r s of the integrable higher-spin XXX spin c hains and the form factors of the in tegrable higher-spin XXZ spin c hains ha v e b een deriv ed b y the algebraic Bethe-ansatz metho d [1 3, 14, 15]. In t he spin-1/ 2 XXZ c hain the Hamiltonian under the p erio dic b oundary conditions is giv en by H XXZ = 1 2 L X j =1  σ X j σ X j +1 + σ Y j σ Y j +1 + ∆ σ Z j σ Z j +1  . (1.1) Here σ a j ( a = X , Y , Z ) are the Pauli matrices defined on the j th site and ∆ denotes the XXZ coupling. W e define parameter q b y ∆ = ( q + q − 1 ) / 2 . (1.2) W e define η and ζ b y q = exp η and η = iζ , respectiv ely . W e th us hav e ∆ = cos ζ . In the massless regime: − 1 < ∆ ≤ 1, we hav e 0 ≤ ζ < π for the spin-1/2 XXZ spin c hain (1.1). A t ∆ = 1 (i.e. q = 1), the Hamilto nian (1.1) corresponds to the antiferromagnetic Heisen b erg (XXX) chain. The solv able higher-spin g eneralizations of the XXX and XXZ spin c hains ha v e b een studied by the fusion metho d in sev eral references [16, 17, 18 , 19, 20, 21, 22, 23]. The spin- s XXZ Hamiltonian is deriv ed fr om the spin- s fusion transfer matrix (see also section 2.6). F o r instance, the Hamiltonian of the integrable spin-1 XXX spin c hain is given by H (2) XXX = 1 2 N s X j =1  ~ S j · ~ S j +1 − ( ~ S j · ~ S j +1 ) 2  . (1.3) Here ~ S j denotes the spin-1 spin-angular momen tum op erator acting on the j th site among the N s lattice sites of t he spin- s c hain. F or the general spin- s case, the integrable spin- s XXX and XXZ Hamiltonians denoted H (2 s ) XXX and H (2 s ) XXZ , resp ectiv ely , can also b e derived systematically . The correlation functions of integrable higher-spin XXX and XXZ spin chains are asso ci- ated with v a rious topics of mathematical ph ysic s. F or the in tegrable spin-1 XXZ spin c hain correlation functions ha v e b een deriv ed by the metho d of q -v ertex op erators thro ugh some no v el results of the represen tation theory of the quan tum algebras [24, 25, 2 6, 2 7 , 28]. They 2 should b e closely related to the higher-spin solutions of the quantum Knizhnik-Zamolo dc hik ov equations [1 0 ]. F or the fusion eigh t-v ertex mo dels, correlation functions ha v e b een discussed b y an algebraic metho d [29]. Moreov er, the partition function of the six-v ertex mo del under domain w all b oundary conditions hav e b een extended in to the hig her-spin case [30]. In a massless regio n 0 ≤ ζ < π / 2 s , the lo w-lying excitation sp ectrum a t zero temp erature of the integrable spin- s XXZ c hain should corresp ond to the lev el- k S U (2) WZWN mo del with k = 2 s . By a ss uming the string h ypo t hes is it is conjectured that the ground state of the in tegrable spin- s XXX Hamiltonian is giv en b y N s / 2 sets of 2 s - strings [3 1 ]. It has also b een extended in to the XXZ case [32 ]. The ground-state solution of 2 s -strings is deriv ed for the spin- s XXX c hain throug h the zero-temp erature limit of the thermal Bethe ansatz [18]. The lo w-lying excitation sp ectrum is discussed in terms of spinons for t he spin- s XXX and XXZ spin c hains [31, 32]. Numerically It w as sho wn that the finite-size corrections to the ground-state energy of the in tegrable spin- s XXX c hain ar e consisten t with the conf o rmal field theory (CFT) with c = 3 s/ ( s + 1) [33, 34, 35, 36]. Here c denotes the cen tral c harge of the CFT. It is also the case with the in tegrable spin- s XXZ c hain in t he region 0 ≤ ζ < π / 2 s [37, 38, 39]. The r esults are consisten t with the conjecture that the ground state of the integrable spin- s XXZ c hain with 0 ≤ ζ < π / 2 s is giv en by N s / 2 sets of 2 s -strings [22, 32, 3 7 , 38, 39, 40, 4 1, 42]. F urthermore, it was sho wn ana lytically that the lo w-lying excitation sp ectrum of the in tegrable spin- s XXZ c hain in the region 0 ≤ ζ < π / 2 s is consisten t with the CFT of c = 3 s/ ( s + 1) [4 1 , 42]. In fact, the lo w-lying excitation sp ectrum o f spinons for the spin- s XXX c hain is described in terms of the lev el- k S U (2) WZWN mo del with k = 2 s [43]. In the pap er w e calculate zero-temp erature correlation functions for the in tegrable higher- spin XXZ spin chains by the algebraic Bethe-ansatz metho d. F or a given pro duct of elemen tary matrices w e presen t the m ultiple-in tegral represen tations of the correlation function in the re- gion 0 ≤ ζ < π / 2 s of the massless regime near the an tiferromagnetic p oin t ( ζ = 0). F o r an illustration, w e deriv e the m ultiple-in tegral represen tations of the emptiness formation proba- bilit y (EFP) o f the spin- s XXZ spin c hain, explicitly . Here the spin s is giv en b y an arbitra ry p ositiv e integer or half-integer. Assuming the conjecture that the ground-state solution of the Bethe ansatz equations is giv en b y 2 s -strings for the regime of ζ , w e deriv e the spin- s EFP for a finite c hain and then take the thermo dynamic limit. W e solv e the integral equations asso- ciated with the spin- s Gaudin matrix for 0 ≤ ζ < π / 2 s , and express the diag onal elemen ts in terms of the densit y of strings. Here w e remark that the integral equations asso ciated with the spin- s Gaudin matrix hav e not b een explicitly solv ed, y et, ev en for the case of the integrable higher-spin XXX spin c hains [13]. W e also calculate the spin- s EFP for the homogeneous chain where all inhomogeneous para meters ξ p are giv en b y zero. Here w e shall intro duce inhomoge- neous parameters ξ p for p = 1 , 2 , . . . , N s , in § 2.4. F urthermore, w e take adv an tage of the fusion construction of the spin- s R -matrix in t he algebraic Bethe-ansatz deriv ation of the correlation functions [15]. Giv en t he spin- s XXZ spin c hain on the N s lattice sites, w e define L b y L = 2 sN s and 3 consider the spin-1/2 XXZ spin c hain on the L sites with inhomogeneous parameters w j for j = 1 , 2 , . . . , L . In the f usion metho d we express any giv en spin- s lo cal op erator as a sum of pro ducts of op erator-v alued elemen ts of the spin-1/2 mono dromy matrix in the limit of sending inhomogeneous parameters w j to sets of complete 2 s -strings as sho wn in Ref. [15]. Here, w e apply the spin-1/2 form ula o f the quan tum in v erse scattering problem [4], whic h is v alid at least for generic inhomogeneous para meters. Therefore, sending inhomogeneous parameters w j in to complete 2 s -strings, w e can ev aluate the v a cuum exp ectation v alues or the f orm fa cto r s of spin- s lo cal op erators which are expresse d in terms of the spin-1/2 mono drom y matrix elemen ts with generic inhomogeneous parameters w j . Here, the rapidities of the gro und state satisfy the Bethe ansatz equations with inhomogeneous parameters w j . W e assume in t he pap er that the Bethe ro ots are contin uous with resp ect to inhomogeneous parameters w j , in particular, in the limit of sending w j to complete 2 s -strings. W e can construct higher-spin transfer matrices b y the fusion metho d [22, 23]. Here w e recall that the spin-1 / 2 XXZ Hamiltonian (1.1) is derived fro m the logarithmic deriv ative of the ro w- to-row transfer mat r ix of the six-v ertex mo del. W e call it the spin-1/2 transfer matrix and denote it by t (1 , 1) ( λ ). Let us express by V ( ℓ ) an ( ℓ + 1)-dimensional v ector space. W e denote b y T ( ℓ, 2 s ) ( λ ) the spin- ℓ/ 2 mono drom y matrix acting on the tensor pro duct of the auxiliary space V ( ℓ ) and the N s th tensor product of the quan tum space s, ( V (2 s ) ) ⊗ N s . W e call it of t yp e ( ℓ, (2 s ) ⊗ N s ), whic h w e ex press ( ℓ, 2 s ) in the sup erscript. T aking the trace o f the spin- ℓ/ 2 mono drom y matrix T ( ℓ, 2 s ) ( λ ) o v er the auxiliary space V ( ℓ ) , w e define the spin- ℓ/ 2 transfer matrix, t ( ℓ, 2 s ) ( λ ). F or ℓ = 2 s , w e hav e the spin- s tra nsfer matr ix t (2 s, 2 s ) ( λ ), and w e deriv e the spin- s Hamiltonian from its logarithmic deriv ative. W e construct the ground state | ψ (2 s ) g i of the spin- s XXZ Hamiltonian b y the B opera t o rs of the 2-by-2 mono drom y matr ix T (1 , 2 s ) ( λ ). As shown by Babujan, t he spin- s t ransfer matrix t (2 s, 2 s ) ( λ ) comm utes with the spin-1/2 transfer matrix t (1 , 2 s ) ( λ ) due to the Y ang-Baxter relations, and hence they ha v e eigen v ectors in common [18 ]. The ground state | ψ (2 s ) g i of the spin- s XXZ spin c hain is originally an eigen ve ctor of the spin- s transfer matrix t (2 s, 2 s ) ( λ ), and consequen tly it is also an eigen v ector of the spin-1 / 2 transfer matrix t (1 , 2 s ) ( λ ). Therefore, the ground state | ψ (2 s ) g i of the spin- s XXZ spin c hain can be constructe d b y applying the B op erators o f the 2-b y-2 mo no dro my matrix T (1 , 2 s ) ( λ ) to the v acuum. W e can sho w that the fusion R -matrix corresp onds to the R -matrix o f the affine quantum group U q ( c sl 2 ). W e recall that b y the fusion metho d, we can construct the R - matrix acting on the tensor pro duct V ( ℓ ) ⊗ V (2 s ) [16, 17, 18, 19, 20, 21, 22, 23]. W e denote it b y R ( ℓ, 2 s ) . In the affine quan tum group, the R -mat r ix is defined as the in tert winer of the tensor pro duct of tw o represen tations V and W [44, 45, 46]. Due to the conditions of the in tert winer the R -matrix of the affine quan tum group is determined uniquely up to a scalar factor [47 ], whic h w e denote b y R V ,W . Therefore, show ing that the fusion R -matr ix satisfies all the conditions of the in tert winer, w e pro v e that the fusion R -ma t rix coincides with the R -matrix of the quantum group, R V ,W . Consequen t ly , for ℓ = 2 s the fusion R -matrix, R (2 s, 2 s ) ( λ ), b ecomes the p erm utation op erator 4 when sp ectral parameter λ is giv en by zero. This prop ert y o f the R -matr ix plays a cen tral role in the deriv ation of the in tegrable spin- s Hamiltonian. It is also fundamen tal in the inv erse scattering problem in the spin- s case [48]. There a re sev eral relev ant and in teresting studies of the integrable spin- s XXZ spin c hains. The expression o f eigenv alues of the spin- s XXX t r ansfer matrix t (2 s, 2 s ) ( λ ) w as derive d b y Babu- jan [17, 18, 21] through the algebraic Bethe-ansatz metho d. It w as a lso deriv ed b y solving the series of functiona l relations a mong the spin- s transfer matrices [2 2]. The functional relations are system atically generalize d to the T systems [49]. Recen tly , the a lgebraic Bethe ansatz for the spin- s XXZ transfer matrix has b een thoroughly review ed a nd reconstructed from the viewpoint of the alg ebraic Bethe a nsatz o f the U (1)-inv arian t in tegrable mo del [50, 51]. Quite in terestingly , it has also b een applied to construct t he inv ariant subspaces asso ciated with the Ising-lik e sp ectra of the superintegrable c hiral P otts mo del [52]. The conten t of the pap er consists of the follo wing. In section 2, w e in tro duce the R - matrix for the spin-1/ 2 XXZ spin c hain. W e then in tro duce conjugate basis vectors in order to form ulate Hermitian elemen tary matrices e E m, n in the massless regime where | q | = 1. W e define the massless higher-spin mono drom y matrices e T ( ℓ, 2 s ) ( λ ) in terms of the conjugate v ectors, a fter reviewing the fusion construction of the massiv e higher-spin mono drom y matrices T ( ℓ, 2 s ) ( λ ) and higher-spin XXZ tra nsfer matrices, t ( ℓ, 2 s ) ( λ ), for ℓ = 1 , 2 , . . . , as f ollo ws. W e express the matrix elemen ts o f T ( ℓ, 2 s ) 0 , 12 ··· N s ( λ ) in terms of those o f the spin-1/2 mo no dromy matrix T (1 , 1) 0 , 12 ··· L ( λ ). Here T (1 , 1) 0 , 12 ··· L ( λ ) is defined on the tensor pro duct o f the tw o- dimens ional auxiliary space V (1) 0 and the L th tensor pro duct of the 2 - dimens ional quan tum space, ( V (1) ) ⊗ L . Here w e r ecall L = 2 s × N s . In the f usion construction [15], mono drom y matrix T (1 , 2 s ) 0 , 12 ··· N s ( λ ) acting on the N s lattice sites is derive d fro m mono drom y matrix T (1 , 1) 0 , 12 ··· L ( λ ) acting on the 2 sN s lattice sites by setting inhomo g eneous parameters w j to N s sets of complete 2 s -strings and by m ultiplying it by the N s th tensor pro duct of pro jection op erators whic h pro ject ( V (1) ) ⊗ 2 s to V (2 s ) . In section 3, w e explain the metho d for calculating the exp ectation v alues of giv en pro ducts of spin- s lo cal op erators. W e exp ress the lo cal op erators in terms of global op erators with inhomogeneous parameters w (2 s ; ǫ ) j , whic h are defined to b e close to complete 2 s -strings with small deviations of O ( ǫ ), and ev aluate the scalar pro ducts and the exp ectation v alues for the Bethe state with the same inhomogeneous parameters w (2 s ; ǫ ) j . Then, w e obtain the expectation v alues, sending ǫ to 0. Here w e note that the pro jection op erators intro duced in the fusion construction comm ute with the matrix elemen ts of mono drom y matrix T (1 , 1) of inhomogeneous parameters w (2 s ; ǫ ) j with O ( ǫ ) corrections. In section 4 w e calculate the emptiness formation probabilit y (EFP) for the spin- s XXZ spin chain for a lar ge but finite ch ain, and then ev aluate the matrix S whic h is introduced for expressin g the EFP of an infinite spin- s XXZ c hain in the massless regime with ζ < π / 2 s . Here w e solv e explicitly the inte gral equations for the spin- s Gaudin matrix, expressing the 2 s -strings of the ground- stat e solution systematically in terms of the string cen ters. In section 5, we presen t explicitly the m ultiple-in tegral represen tation of the spin- s EFP . W e also derive it for the inhomogeneous chain where all the inhomogeneous parameters ξ p are giv en by 0. 5 F o r an illustration, w e calculate the m ultiple-inte gral represen tation of spin-1 EFP for m = 1, h e E 2 , 2 i , explicitly . In the XXX limit, the v alue of h e E 2 , 2 i a pproac hes 1/3, which is consisten t with the XXX result of Ref. [13]. In section 6, w e presen t t he multiple-in tegral represen tations of the inte grable spin- s XXZ cor r elation functions. W e express the correlation function of an arbitrary pro duct of elemen tary matrices b y a single term of m ultiple in tegrals. F or instance, w e calculate the m ultiple-in tegral represen tation of the spin-1 ground-state exp ectation v a lue, h e E 1 , 1 i , explicitly , and show that it is consisten t with the v alue of spin-1 EFP in section 5 , i.e. w e show h e E 1 , 1 i + 2 h e E 2 , 2 i = 1. Finally in section 7, w e give concluding remarks. 2 F usion transfer matrices 2.1 R -matrix and the mono drom y matrix of t yp e (1 , 1 ⊗ L ) Let us in tro duce the R -matrix o f the XXZ spin ch ain [1, 3 , 4 , 5]. W e denote b y e a, b a unit matrix that has only one nonzero elemen t equal to 1 at en try ( a, b ) where a, b = 0 , 1. L et V 1 and V 2 b e t w o-dimensional vec tor spaces. The R - mat rix acting on V 1 ⊗ V 2 is giv en by R + ( λ 1 − λ 2 ) = X a,b,c,d =0 , 1 R + ( u ) a b c d e a, c ⊗ e b, d =      1 0 0 0 0 b ( u ) c − ( u ) 0 0 c + ( u ) b ( u ) 0 0 0 0 1      , (2.1) where u = λ 1 − λ 2 , b ( u ) = sinh u/ sinh( u + η ) and c ± ( u ) = exp( ± u ) sinh η / sinh( u + η ). In t he massless regime, w e set η = iζ by a real n um b er ζ , and w e ha v e ∆ = cos ζ . In the pap er w e mainly consider the region 0 ≤ ζ < π / 2 s . In the massiv e regime, w e assign η a real nonzero n um b er a nd we hav e ∆ = cosh η > 1. Here w e remark tha t the R + ( λ 1 − λ 2 ) is compatible with the homogeneous grading of U q ( b sl 2 ), whic h is explained in App endix A [15]. W e denote b y R ( p ) ( u ) or simply by R ( u ) the symmetric R -ma t rix where c ± ( u ) of (2.1) are replaced by c ( u ) = sinh η / sinh( u + η ) [15]. The symmetric R -matrix is compatible with the affine quan tum gr o up U q ( b sl 2 ) of the principal grading [15]. Let s b e an in teger or a half-in teger. W e shall ma inly consider the tensor pro duct V (2 s ) 1 ⊗ · · · ⊗ V (2 s ) N s of (2 s + 1)-dimensional v ector spaces V (2 s ) j with L = 2 sN s . In general, w e consider the tensor pro duct V (2 s 0 ) 0 ⊗ V (2 s 1 ) 1 ⊗ · · · ⊗ V (2 s r ) r with 2 s 1 + · · · + 2 s r = L , where V (2 s j ) j ha v e sp ectral parameters λ j for j = 1 , 2 , . . . , r . F or a given set of matrix elemen ts A a, α b, β for a, b = 0 , 1 , . . . , 2 s j and α, β = 0 , 1 , . . . , 2 s k , w e define op erator A j,k b y A j,k = ℓ X a,b =1 X α,β A a, α b, β I (2 s 0 ) 0 ⊗ I (2 s 1 ) 1 ⊗ · · · ⊗ I (2 s j − 1 ) j − 1 ⊗ e a,b j ⊗ I (2 s j +1 ) j +1 ⊗ · · · ⊗ I (2 s k − 1 ) k − 1 ⊗ e α,β k ⊗ I (2 s k +1 ) k +1 ⊗ · · · ⊗ I (2 s r ) r . (2.2) W e no w consider the ( L + 1)th tensor pro duct of spin-1/ 2 represen tations, whic h consists of t he tensor pr o duct of auxiliary space V (1) 0 and the L th tensor pro duct of quantum spaces 6 V (1) j for j = 1 , 2 , . . . , L , i.e. V (1) 0 ⊗  V (1) 1 ⊗ · · · ⊗ V (1) L  . W e call it the tensor pro duct of t yp e (1 , 1 ⊗ L ) and denote it b y the fo llowing sym bo l: (1 , 1 ⊗ L ) = (1 , L z }| { 1 , 1 , . . . , 1) . (2.3) Applying definition ( 2 .2) for matrix eleme nts R ( u ) ab cd of a giv en R - ma t rix, w e define R - matrices R j k ( λ j , λ k ) = R j k ( λ j − λ k ) for in tegers j and k with 0 ≤ j < k ≤ L . F or inte gers j, k and ℓ with 0 ≤ j < k < ℓ ≤ L , the R -matrices satisfy the Y ang-Baxter equations R j k ( λ j − λ k ) R j ℓ ( λ j − λ ℓ ) R k ℓ ( λ k − λ ℓ ) = R k ℓ ( λ k − λ ℓ ) R j ℓ ( λ j − λ ℓ ) R j k ( λ j − λ k ) . (2.4) W e define the mo no dromy matrix of t yp e (1 , 1 ⊗ L ) asso ciated with homogeneous grading by T (1 , 1 +) 0 , 12 ··· L ( λ 0 ; w 1 , w 2 , . . . , w L ) = R + 0 L ( λ 0 − w L ) · · · R + 02 ( λ 0 − w 2 ) R + 01 ( λ 0 − w 1 ) . (2.5) Here w e hav e set λ j = w j for j = 1 , 2 , . . . , L , where w j are arbitrary parameters. W e call them inhomogeneous parameters. W e ha v e expressed the sym bol of type (1 , 1 ⊗ L ) as (1 , 1) in sup ersc ript. The sym b ol (1 , 1 +) denotes that it is consisten t with homog eneous grading . W e express op erator-v alued matrix elemen ts of the mono dromy matr ix as follows . T (1 , 1 +) 0 , 12 ··· L ( λ ; { w j } L ) = A (1+) 12 ··· L ( λ ; { w j } L ) B (1+) 12 ··· L ( λ ; { w j } L ) C (1+) 12 ··· L ( λ ; { w j } L ) D (1+) 12 ··· L ( λ ; { w j } L ) ! . (2.6) Here { w j } L denotes the set of L parameters, w 1 , w 2 , . . . , w L . W e also denote the ma t r ix elemen ts of the mono dromy matrix by [ T (1 , 1+) 0 , 12 ··· L ( λ ; { w j } L )] a,b for a, b = 0 , 1. W e deriv e the mono drom y matrix consisten t with principal grading, T (1 , 1 p ) 0 , 12 ··· L ( λ ; { w j } L ), from that of homogeneous g rading via similarity transformation χ 01 ··· L as follow s [15]. T (1 , 1 +) 0 , 12 ··· L ( λ ; { w j } L ) = χ 012 ··· L T (1 , 1 p ) 0 , 12 ··· L ( λ ; { w j } L ) χ − 1 012 ··· L = χ 12 ··· L A (1 p ) 12 ··· L ( λ ; { w j } L ) χ − 1 12 ··· L e − λ 0 χ 12 ··· L B (1 p ) 12 ··· L ( λ ; { w j } L ) χ − 1 12 ··· L e λ 0 χ 12 ··· L C (1 p ) 12 ··· L ( λ ; { w j } L ) χ − 1 12 ··· L χ 12 ··· L D (1 p ) 12 ··· L ( λ ; { w j } L ) χ − 1 12 ··· L ! . (2.7) Here χ 01 ··· L = Φ 0 Φ 1 · · · Φ L and Φ j are giv en b y diagona l tw o- b y-t w o matrices Φ j = diag (1 , exp( w j )) acting on V (1) j for j = 0 , 1 , . . . , L , and we set w 0 = λ 0 . In Ref. [15] op erator A (1 +) ( λ ) has b een written as A + ( λ ). Hereafter w e shall often abbreviate the sym b ols p in superscripts whic h sho ws the principal grading, and denote (2 s p ) simply b y (2 s ). Let us in tro duce useful notatio n for expressing pro ducts of R - matrices as fo llows. R ( w ) 1 , 23 ··· n = R ( w ) 1 n · · · R ( w ) 13 R ( w ) 12 , R ( w ) 12 ··· n − 1 ,n = R ( w ) 1 n R ( w ) 2 n · · · R ( w ) n − 1 n . (2.8) Here R ( w ) ab denote the R -matrix R ( w ) ab = R ( w ) ab ( λ a − λ b ) for a, b = 1 , 2 , . . . , n , where w = + a nd w = p in sup erscripts sho w the ho mogeneous and the principal grading, respectiv ely . Then, 7 the mono drom y matrix of t yp e (1 , 1 ⊗ L w ) is express ed as follows. T (1 , 1 w ) 0 , 12 ··· L ( λ 0 ; { w j } L ) = R ( w ) 0 , 12 ··· L ( λ 0 ; { w j } L ) = R ( w ) 0 L R ( w ) 0 L − 1 · · · R ( w ) 01 . (2.9) F o r instance w e hav e B (1 w ) 12 ··· L ( λ 0 ; { w j } L ) = [ R (1 w ) 0 , 12 ··· L ( λ 0 ; { w j } L )] 0 , 1 . 2.2 Pro jection op erators and the massiv e fusion R -matrices Let V 1 and V 2 b e (2 s + 1)-dimensional v ector spaces. W e define p erm utation op erator Π 1 , 2 b y Π 1 , 2 v 1 ⊗ v 2 = v 2 ⊗ v 1 , v 1 ∈ V 1 , v 2 ∈ V 2 . (2.10) In the case of spin-1/2 represen tations, w e define op erator ˇ R + 12 ( λ 1 − λ 2 ) b y ˇ R + 12 ( λ 1 − λ 2 ) = Π 1 , 2 R + 12 ( λ 1 − λ 2 ) . (2.11) W e now introduce pro jection op erators P ( ℓ ) 12 ··· ℓ for ℓ ≥ 2. W e define P (2) 12 b y P (2) 12 = ˇ R + 1 , 2 ( η ). F o r ℓ > 2 w e define pro jection op erators inductiv ely with resp ect to ℓ as follows [46, 23]. P ( ℓ ) 12 ··· ℓ = P ( ℓ − 1) 12 ··· ℓ − 1 ˇ R + ℓ − 1 , ℓ (( ℓ − 1) η ) P ( ℓ − 1) 12 ··· ℓ − 1 . (2.12) The pro jection op erator P ( ℓ ) 12 ··· ℓ giv es a q -ana lo gue of the f ull symmetrizer of the Y oung op erators for the Hec k e algebra [46]. W e shall sho w the idemp otency:  P ( ℓ ) 12 ··· ℓ  2 = P ( ℓ ) 12 ··· ℓ in App endix B. Hereafter w e denote P ( ℓ ) 12 ··· ℓ also b y P ( ℓ ) 1 for short. Applying pro jection op erator P ( ℓ ) a 1 a 2 ··· a ℓ to ve ctors in the tensor pro duct V (1) a 1 ⊗ V (1) a 2 ⊗ · · · ⊗ V (1) a ℓ , w e can construct the ( ℓ + 1)-dimensional v ector space V ( ℓ ) a 1 a 2 ··· a ℓ asso ciated with the spin- ℓ/ 2 represen tation of U q ( sl 2 ). F or instance, we hav e P (2) a 1 a 2 | + −i a = ( q / [2 ] q ) || 2 , 1 i a , where w e hav e in tro duced | + −i a = | 0 i a 1 ⊗ | 1 i a 2 . The sym b ols suc h as q - integers are defined in App endix C. Moreo v er, the basis ve ctors || ℓ, n i ( n = 0 , 1 , . . . , ℓ ) a nd their dual vec tors h ℓ, n || are giv en for arbitrary nonzero integers ℓ in App endix C. W e denote V ( ℓ ) a 1 a 2 ··· a ℓ also b y V ( ℓ ) a or V ( ℓ ) 0 for short. Since P ( ℓ ) 12 ··· ℓ is consisten t with the spin- ℓ/ 2 represen tation of U q ( sl (2)) (see ( C.6)), we ha v e P ( ℓ ) 12 ··· ℓ = ℓ X n =0 || ℓ, n i h ℓ, n || . (2.13) Applying pro jection op erator P (2 s ) 2 s ( j − 1)+1 ··· 2 s ( j − 1)+2 s to tensor pro duct V (1) 2 s ( j − 1)+1 ⊗ · · · ⊗ V (1) 2 s ( j − 1)+2 s , w e construct the spin- s represen tation V (2 s ) 2 s ( j − 1)+1 ··· 2 s ( j − 1)+2 s . W e denote it also b y V (2 s ) j , briefly . In the tensor pro duct of quan tum spaces V (2 s ) 1 ⊗ · · · ⊗ V (2 s ) N s , w e define P (2 s ) 12 ··· L b y P (2 s ) 12 ··· L = N s Y i =1 P (2 s ) 2 s ( i − 1)+1 . (2.14) 8 Here w e recall L = 2 sN s . W e hav e put 2 s in place of ℓ . W e no w intro duce the massiv e fusion R -matrix R ( ℓ, 2 s +) 0 , j on the tensor pro duct V ( ℓ ) 0 ⊗ V (2 s ) j ( j = 1 , 2 , . . . , N s ). It is v alid in the massiv e regime with ∆ > 1. W e first set rapidities λ a j of auxiliary spaces V (1) a j b y λ a k = λ a 1 − ( k − 1) η for k = 1 , 2 , . . . , ℓ − 1, and then rapidities λ 2 s ( j − 1)+ k of quan tum spaces V (1) 2 s ( j − 1)+ k b y λ 2 s ( j − 1)+ k = λ 2 s ( j − 1)+1 − ( k − 1 ) η for k = 1 , 2 , . . . , 2 s and j = 1 , 2 , . . . , N s . W e define the massiv e fusion R - matrix R ( ℓ, 2 s +) 0 , j as follow s. R ( ℓ, 2 s +) 0 j ( λ a 1 − λ 2 s ( j − 1)+1 ) = P ( ℓ ) a 1 ··· a ℓ P (2 s ) 2 s ( j − 1)+1 R + a 1 ··· a ℓ , 2 s ( j − 1)+1 ··· 2 sj P ( ℓ ) a 1 ··· a ℓ P (2 s ) 2 s ( j − 1)+1 = P ( ℓ ) a 1 ··· a ℓ P (2 s ) 2 s ( j − 1)+1 R + a 1 ··· a ℓ , 2 sj · · · R + a 1 ··· a ℓ , 2 s ( j − 1)+2 R + a 1 ··· a ℓ , 2 s ( j − 1)+1 P ( ℓ ) a 1 ··· a ℓ P (2 s ) 2 s ( j − 1)+1 . (2.15) 2.3 Conjugate v ectors and the massless fus ion R -matrices In order to construct Hermitian elemen tary ma t rices in the massless regime where | q | = 1, w e no w in tro duce vec tors ^ || ℓ, n i whic h are Hermitian conjugate to h ℓ, n || when | q | = 1 fo r p ositiv e in tegers ℓ with n = 0 , 1 , . . . , ℓ . Setting the norm of ^ || ℓ, n i suc h that h ℓ, n || ^ || ℓ, n i = 1, w e hav e ^ || ℓ, n i = X 1 ≤ i 1 < ··· n or m < n are giv en in App endix E. When we ev alua te exp ectation v alues, w e w an t to remo v e the pro jection op erators intro- duced in order to express the spin- s lo cal o perato r in terms of spin-1/2 global op erators such as in (3.18). Then, we shall make use of the follow ing lemma. Lemma 3.1. Pr oje ction op er a tors P (2 s ) 12 ··· L and e P (2 s ) 12 ··· L c ommute with the matrix elements of the mono dr omy matrix T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ; ǫ ) j } L ) such as A (2 s +; ǫ ) ( λ ) in the limit of ǫ go i n g to 0. P (2 s ) 12 ··· L T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ; ǫ ) j } L ) P (2 s ) 12 ··· L = P (2 s ) 12 ··· L T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ; ǫ ) j } L ) + O ( ǫ ) , (3.19) P (2 s ) 12 ··· L T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ; ǫ ) j } L ) e P (2 s ) 12 ··· L = P (2 s ) 12 ··· L T (1 , 1) 0 , 12 ··· L ( λ ; { w (2 s ; ǫ ) j } L ) + O ( ǫ ) . (3.20) F or i n stanc e we have P (2 s ) 12 ··· L B (2 s +; ǫ ) ( λ ) P (2 s ) 12 ··· L = P (2 s ) 12 ··· L B (2 s +; ǫ ) ( λ ) + O ( ǫ ) . Pr o of. T aking deriv a tiv es with resp ect to inhomogeneous para meters w j , w e can sho w T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s +; ǫ ) j } L ) = T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ) j } L ) + O ( ǫ ) , (3.21) where T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ) j } L ) comm utes with the pro jection op erator P (2 s ) 12 ··· L as f o llo ws [15]. P (2 s ) 12 ··· L T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ) j } L ) = P (2 s ) 12 ··· L T (1 , 1 +) 0 , 12 ··· L ( λ ; { w (2 s ) j } L ) P (2 s ) 12 ··· L . (3.22) W e show (3 .20) making use of (2.22). 15 3.3 Exp ectation v alue of a lo c al op erator through the limit: ǫ → 0 In t he massless regime, w e define the expectatio n v alue o f pro duct of op erators Q m k =1 e E i k , j k (2 s +) k with resp ect to an eigenstate | ] { λ α } (2 s +) M i b y h m Y k =1 e E i k , j k (2 s +) k i  { λ α } (2 s +) M  = h ] { λ α } (2 s +) M | Q m k =1 e E i k , j k (2 s +) k | ] { λ α } (2 s +) M i h ] { λ α } (2 s +) M | ] { λ α } (2 s +) M i . (3.23) W e ev aluate the exp ectation v a lue of a giv en spin- s lo cal op erator fo r a Bethe-ansatz eigen- state |{ λ α } (2 s ) M i , a s follo ws. W e first assume that the Bethe ro ots { λ α ( ǫ ) } M are contin uous with resp ect to small parameter ǫ . W e express the spin- s lo cal op erator in terms of spin-1/2 global op erators suc h as form ula (3.18) with generic inhomo g eneous parameters w (2 s ; ǫ ) j . Applying (3.19) and (3.19) we r emov e the pro jection op erators out of the pro duct of global op erators. W e next calculate the scalar pro duct for the Bethe state | { λ k ( ǫ ) } (2 s ; ǫ ) M i whic h has the same inhomogeneous parameters w (2 s ; ǫ ) j , making use of the formulas of the spin-1/2 case. Then w e tak e the limit of sending ǫ to 0, and obtain the exp ectation v alue of the spin- s lo cal op erator. F o r an illustration, let us consider the exp ectation v alue o f e E n, n (2 s +) 1 . First, applying pro jection op erator P (2 s ) 12 ··· L to | ] { λ α } (2 s +) M i = Q M α =1 e B (2 s +) ( λ α ) | 0 i w e sho w P (2 s ) 1 ··· L | ] { λ α } (2 s +) M i = P (2 s ) 1 ··· L M Y α =1 B (2 s +; ǫ ) ( λ α ( ǫ )) | 0 i + O ( ǫ ) = e − P M α =1 λ α ( ǫ ) P (2 s ) 1 ··· L χ 12 ··· L M Y α =1 B (2 s ; ǫ ) ( λ α ( ǫ )) | 0 i + O ( ǫ ) . (3.24) Second, making use o f the relat io n h 0 | = h 0 | P (2 s ) 12 ··· L , w e sho w h ] { λ α } (2 s +) M | = h 0 | M Y α =1 C (2 s +; ǫ ) ( λ α ( ǫ )) P (2 s ) 1 ··· L + O ( ǫ ) = h 0 | M Y α =1 C (2 s ; ǫ ) ( λ α ( ǫ )) χ − 1 12 ··· L P (2 s ) 1 ··· L e P M α =1 λ α ( ǫ ) + O ( ǫ ) . (3.25) Making use of (3.18) we hav e h ] { λ α } (2 s +) M | e E n n (2 s +) 1 | ] { λ α } (2 s +) M i = 2 s n ! h 0 | M Y α =1 C (2 s +; ǫ ) ( λ α ( ǫ )) P (2 s ) 1 ··· L e P (2 s ) 12 ··· L n Y k =1 D (2 s +; ǫ ) ( w (2 s ; ǫ ) k ) 2 s Y k = n +1 A (2 s +; ǫ ) ( w (2 s ; ǫ ) k ) × 2 sN s Y α =2 s +1 ( A (2 s +; ǫ ) + D (2 s +; ǫ ) )( w (2 s ; ǫ ) α ) e P (2 s ) 1 ··· L · M Y α =1 e B (2 s +) ( λ α ) | 0 i + O ( ǫ ) . (3.26) Here w e ha v e Q 2 sN s j =1 ( A (2 s +; ǫ ) + D (2 s +; ǫ ) )( w (2 s ; ǫ ) j ) = I ⊗ L for generic ǫ . W e apply pro jection op erators P (2 s ) to e P (2 s ) from the left, whic h are underlined in (3.2 6), and mak e use of (2.22). 16 W e then mov e the pro jection op erators P (2 s ) in the leftw ard direction, making use of ( 3 .19). Th us, the righ t-hand side of (3.2 6) is now give n b y the following: = 2 s n ! h 0 | M Y α =1 C (2 s +; ǫ ) ( λ α ( ǫ )) n Y k =1 D (2 s +; ǫ ) ( w (2 s ; ǫ ) k ) 2 s Y k = n +1 A (2 s +; ǫ ) ( w (2 s ; ǫ ) k ) × 2 sN s Y j =2 s +1 ( A (2 s +; ǫ ) + D (2 s +; ǫ ) )( w (2 s ; ǫ ) j ) M Y β =1 B (2 s +; ǫ ) ( λ β ( ǫ )) | 0 i + O ( ǫ ) . (3.27) After applying the gauge t ransformation χ − 1 1 ··· L in v erse t o (2 .7) [15], w e obtain h ] { λ α } (2 s +) M | e E n n (2 s +) 1 | ] { λ α } (2 s +) M i = 2 s n ! lim ǫ → 0 h 0 | M Y α =1 C (2 s ; ǫ ) ( λ α ( ǫ )) n Y k =1 D (2 s ; ǫ ) ( w (2 s ; ǫ ) k ) 2 s Y k = n +1 A (2 s ; ǫ ) ( w (2 s ; ǫ ) k ) × 2 sN s Y j =2 s +1 ( A (2 s ; ǫ ) + D (2 s ; ǫ ) )( w (2 s ; ǫ ) j ) M Y β =1 B (2 s ; ǫ ) ( λ β ( ǫ )) | 0 i . (3.28) Here A (2 s ; ǫ ) and D (2 s ; ǫ ) denote matrix elemen ts A (2 s p ; ǫ ) and D (2 s p ; ǫ ) of the mono drom y matrix with principal grading, resp ectiv ely . In the last line of (3.27), we hav e ev aluated the eigen v a lue of transfer matrix A (2 s ; ǫ ) ( w (2 s ; ǫ ) j ) + D (2 s ; ǫ ) ( w (2 s ; ǫ ) j ) on t he eigenstate |{ λ β ( ǫ ) } (2 s ; ǫ ) M i as follow s. 2 sN s Y j =2 s +1  A (2 s ; ǫ ) ( w (2 s ; ǫ ) j ) + D (2 s ; ǫ ) ( w (2 s ; ǫ ) j )  |{ λ β ( ǫ ) } (2 s ; ǫ ) M i = 2 sN s Y j =2 s +1 M Y α =1 b − 1 ( λ α ( ǫ ) − w (2 s ; ǫ ) j ) ! |{ λ β ( ǫ ) } (2 s ; ǫ ) M i . (3.29) Before sending ǫ to 0, w e expand the pro ducts o f C op erators multiplied by op erators A and D by the commutation relations b et w een C and A a s w ell as C and D , resp ectiv ely . W e then ev a luate the scalar pro duct of B and C op erators with inhomo g eneous para meters w (2 s ; ǫ ) j . Finally , w e deriv e the exp ectation v a lue in the limit of sending ǫ to 0 . Sending ǫ to 0, w e calculate the exp ectation v alue o f A (2 s ) ( λ ) + D (2 s ) ( λ ) at λ = w (2 s ) 2 . F or instance, w e calculate A (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 ) + D (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 ) on t he v acuum | 0 i as follows. lim ǫ → 0 h 0 |  A (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 ) + D (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 )  | 0 i = lim ǫ → 0 h 0 |  A ( w (2 s ; ǫ ) 2 ; { w (2 s ; ǫ ) j } L ) + D (2 s ) ( w (2 s ; ǫ ) 2 ; { w (2 s ; ǫ ) j } L )  | 0 i = lim ǫ → 0 1 + ℓN s Y j =1 b ( w (2 s ; ǫ ) 2 − w (2 s ; ǫ ) j ) ! h 0 | 0 i = (1 + 0) h 0 | 0 i . (3.30) 17 If w e put λ = w (2 s ) 2 after sending ǫ to 0, the result is differen t from ( 3 .30) as follows . lim λ → w (2 s ) 2 h 0 |  A (2 s ) ( λ ; { w (2 s ) j } L ) + D (2 s ) ( λ ; { w (2 s ) j } L )  ) | 0 i = 1 + N s Y p =1 b ℓ ( w (2 s ) 2 − ξ p ) ! h 0 | 0 i . (3.31) 4 Deriv atio n of matrix S 4.1 The ground-state solution of 2 s -strings W e shall introduce ℓ -strings for an in teger ℓ . Let us shift rapidities λ j b y sη suc h as ˜ λ j = λ j + sη . Then, the Bethe ansatz equations (3.9) are giv en by N s Y p =1 sinh( ˜ λ j − ξ p + sη ) sinh( ˜ λ j − ξ p − sη ) = n Y β =1; β 6 = α sinh( ˜ λ j − ˜ λ β + η ) sinh( ˜ λ j − ˜ λ β − η ) , fo r j = 1 , 2 , . . . , n . (4.1) W e define a n ℓ -string by the fo llo wing set of rapidities. ˜ λ ( α ) a = µ a + ( ℓ + 1 − 2 α ) η 2 + ǫ ( α ) a for α = 1 , 2 , . . . , ℓ. (4.2) W e call µ a the cen ter of t he ℓ -string and ǫ ( α ) a string deviations. W e assume t hat ǫ ( α ) a are ve ry small for large N s : lim N s →∞ ǫ ( α ) a = 0 . (4.3) If they are zero, then we call t he set of rapidities of (4.2) a complete ℓ -string. The string cen ter µ a corresp onds to the cen tral p osition a mong the ℓ complex n um b ers: ˜ λ (1) a , ˜ λ (2) a , . . . , ˜ λ ( ℓ ) a . F urthermore w e assume that µ a are real. If inhomogeneous par ameters, ξ p , are small enough, then the Bethe ansatz equations should ha v e a n ℓ - strings as a solution. In t erms o f rapidities λ j whic h are not shifted, an ℓ -string is expressed in the follo wing form: λ ( α ) a = µ a − ( α − 1 / 2) η + ǫ ( α ) a for α = 1 , 2 , . . . , ℓ . (4.4) W e denote λ ( α ) a also b y λ ( a,α ) . Let us no w intro duce the conjecture that t he gr o und state of the spin- s case | ψ (2 s ) g i is give n b y N s / 2 sets of 2 s -strings: λ ( α ) a = µ a − ( α − 1 / 2) η + ǫ ( α ) a , for a = 1 , 2 , . . . , N s / 2 and α = 1 , 2 , . . . , 2 s. (4.5) In terms of λ ( α ) a s in the massless regime, for w = + and p , w e hav e | ψ (2 s w ) g i = N s / 2 Y a =1 2 s Y α =1 e B (2 s w ) ( λ ( α ) a ; { ξ p } ) | 0 i . (4.6) Hereafter w e set M = 2 sN s / 2 = sN s . 18 According to analytic and nume rical studie s [40, 41, 4 2], w e ma y assume the follow ing prop erties of string deviations ǫ ( α ) a s. When N s is v ery la rge, the deviations a re given by ǫ ( α ) a = i δ ( α ) a , (4.7) where i denotes √ − 1, and δ ( α ) a are real. Moreo v er, δ ( α ) a − δ ( α +1) a > 0 for α = 1 , 2 , . . . , 2 s − 1, and | δ ( α ) a | > | δ ( α +1) a | for α < s , while | δ ( α ) a | < | δ ( α +1) a | for α ≥ s . In the thermo dynamic limit: N s → ∞ , the Bethe ansatz equations fo r the ground state of t he higher-spin XXZ chain b ecome the in tegral equation for the string cente rs, as sho wn in App endix F [53]. The densit y o f string cen ters, ρ tot ( µ ), is giv en by ρ tot ( µ ) = 1 N s N s X p =1 1 2 ζ cosh( π ( µ − ξ p ) /ζ ) (4.8) Th us, the sum o v er all the Bethe ro ots of the ground state is ev aluated b y in tegrals in the thermo dynamic limit, N s → ∞ , as follows. 1 N s M X A =1 f ( λ A ) = 1 N s 2 s X α =1 N s / 2 X a =1 f ( λ ( a,α ) ) = 2 s X α =1 Z ∞ −∞ f ( µ a − ( α − 1 / 2) η + ǫ ( α ) a ) ρ tot ( µ a ) dµ a + O (1 / N s ) . (4.9) F o r the homogeneous c hain where ξ p = 0 for p = 1 , 2 , . . . , N s , w e denote the density of string cen ters by ρ ( λ ). ρ ( λ ) = 1 2 ζ cosh( π λ/ζ ) . (4.10) Let us in tro duce useful notat io n of the suffix of rapidities. F or rapidities λ ( α ) a = λ ( a,α ) w e define in tegers A b y A = 2 s ( a − 1) + α for a = 1 , 2 , . . . , N s / 2 and for α = 1 , 2 , . . . , 2 s . W e thus denote λ ( a,α ) also b y λ A for A = 1 , 2 , . . . , sN s , and put λ ( a,α ) in increasing order with resp ect to A = 2 s ( a − 1) + α suc h a s λ (1 , 1) = λ 1 , λ (1 , 2) = λ 2 , . . . , λ ( N s / 2 , 2 s ) = λ sN s . In the ground state rapidities λ A for A = 1 , 2 , . . . , M , are no w expresse d by λ 2 s ( a − 1)+ α = µ a − ( α − 1 / 2) η + ǫ ( α ) a for a = 1 , 2 , . . . , N s / 2 and α = 1 , 2 , . . . , 2 s. (4.11) F o r a giv en real num b er x , let us denote b y [ x ] the greatest integer less t han or equal to x . When A = 2 s ( a − 1) + α with 1 ≤ α ≤ 2 s , in teger a is given b y a = [( A − 1) / 2 s ] + 1, a nd in teger α is giv en by α = A − 2 s [( A − 1) / 2 s ]. 4.2 Deriv ation of the spin- s EF P for a finite c hain W e define t he emptiness formation probability (EFP) for the spin- s case b y τ (2 s +) ( m ) = h ψ (2 s +) g | e E 2 s, 2 s (2 s +) 1 · · · e E 2 s, 2 s (2 s +) m | ψ (2 s +) g i h ψ (2 s +) g | ψ (2 s +) g i . (4.12) 19 W e shall denote τ (2 s +) ( m ) b y τ (2 s ) ( m ). Let us assume that Bethe ro ots { λ α ( ǫ ) } M with inhomogeneous parameters w (2 s ; ǫ ) j ( j = 1 , 2 , . . . , L ; L = 2 sN s ) b ecome the ground-state solution o f the spin- s XXZ spin chain, { λ α } M , in the limit of sending ǫ to 0 . W e denote the Bethe v ector with Bethe ro ots { λ α ( ǫ ) } M b y | ψ (2 s +; ǫ ) g i = M Y α =1 B (2 s ; ǫ ) ( λ α ( ǫ )) | 0 i = e − P M α =1 λ α ( ǫ ) χ 12 ··· L · M Y α =1 B (2 s p ; ǫ ) ( λ α ( ǫ )) | 0 i = e − P M α =1 λ α ( ǫ ) χ 12 ··· L | ψ (2 s ; ǫ ) g i . (4.13) Here we recall t he transformation in v erse to ( 2 .7). W e now calculate the norm of the spin- s ground state from that of the spin-1/2 case through the limit of sending ǫ to 0 as follow s. h ψ (2 s +) g | ψ (2 s +) g i = lim ǫ → 0 h ψ (2 s ; ǫ ) g | ψ (2 s ; ǫ ) g i = lim ǫ → 0 h 0 | M Y k =1 C (2 s ; ǫ ) ( λ k ) M Y j =1 B (2 s ; ǫ ) ( λ j ) | 0 i = lim ǫ → 0 sinh M η M Y j,k =1; j 6 = k b − 1 ( λ j ( ǫ ) , λ k ( ǫ )) · detΦ (1) ′  { λ k ( ǫ ) } M ; { w (2 s ; ǫ ) j } L  = sinh M η M Y j,k =1; j 6 = k b − 1 ( λ j , λ k ) · detΦ (2 s ) ′ ( { λ k } M ; { ξ p } N s ) (4.14) where matrix elemen ts of the spin- s Gaudin matr ix for j, k = 1 , 2 , . . . , M , are giv en b y Φ (2 s ) ′ j,k ( { λ l } M ; { ξ p } ) = − ∂ ∂ λ k log a (2 s ) ( λ j ) d (2 s ) ( λ j ) Y t 6 = j sinh( λ t − λ j + η ) sinh( λ t − λ j − η ) ! = δ j,k N s X p =1 sinh(2 sη ) sinh( λ j − ξ p ) sinh( λ j − ξ p + 2 sη ) − M X C =1 sinh 2 η sinh( λ j − λ C + η ) sinh ( λ j − λ C − η ) ! + sinh 2 η sinh( λ j − λ k + η ) sinh( λ j − λ k − η ) . (4.15) By applying form ula ( 3 .18) with n = 2 s , the n umerator o f (4.12) is given b y h ψ (2 s +; ǫ ) g | e E 2 s, 2 s (2 s +) 1 · · · e E 2 s, 2 s (2 s +) m | ψ (2 s +; ǫ ) g i = lim ǫ → 0 h ψ (2 s ; ǫ ) g | m Y k =1 E 2 s, 2 s (2 s ) k | ψ (2 s ; ǫ ) g i = lim ǫ → 0 h ψ (2 s ; ǫ ) g | P (2 s ) 12 ··· L m Y i =1 2 s ( i − 1) Y α =1  A (2 s ; ǫ ) + D (2 s ; ǫ )  ( w (2 s ; ǫ ) α ) · 2 s Y k =1 D (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 s ( i − 1)+ k ) · 2 sN s Y α =1  A (2 s ; ǫ ) + D (2 s ; ǫ )  ( w (2 s ; ǫ ) α ) ! P (2 s ) 12 ··· L | ψ (2 s ) g i = m Y j =1 M Y α =1 b 2 s ( λ α , ξ j ) lim ǫ → 0 h ψ (2 s ; ǫ ) g | D (2 s ; ǫ ) ( w (2 s ; ǫ ) 1 ) · · · D (2 s ; ǫ ) ( w (2 s ; ǫ ) 2 sm ) | ψ (2 s ; ǫ ) g i . (4.16) 20 Let us set λ M + j ( ǫ ) = w (2 s ; ǫ ) j for j = 1 , 2 , . . . , 2 sm . Applying formu la (G.1) to (4.16) w e hav e h 0 | M Y α =1 C (2 s ; ǫ ) ( λ α ( ǫ )) 2 sm Y j =1 D (2 s ; ǫ ) ( λ M + j ( ǫ )) M Y β =1 B (2 s ; ǫ ) ( λ β ( ǫ )) | 0 i = M X c 1 =1 M X c 2 =1; c 2 6 = c 1 · · · M X c 2 sm =1; c 2 sm 6 = c 1 ,...,c 2 sm − 1 G c 1 ··· c 2 sm ( λ 1 ( ǫ ) , · · · , λ M +2 sm ( ǫ ); { w (2 s ; ǫ ) j } L ) ×h 0 | M +2 sm Y k =1; k 6 = c 1 ,...,c 2 sm C (2 s ; ǫ ) ( λ k ( ǫ )) M Y α =1 B (2 s ; ǫ ) ( w (2 s ; ǫ ) j ) | 0 i , (4.17) where G c 1 ··· c 2 sm ( λ 1 , · · · , λ M +2 sm ; { w j } L ) = 2 sm Y j =1 d ( λ c j ; { w j } L ) Q M + j − 1 t =1; t 6 = c 1 ,...,c j − 1 sinh( λ c j − λ t + η ) Q M + j t =1; t 6 = c 1 ,...,c j sinh( λ c j − λ t ) ! . (4.18) W e remark that from (4.18) the set of integers c 1 , . . . , c 2 sm of t he most dominan t terms in (4.17) are giv en by m sets of 2 s -strings. If they are not, the num erator of ( 4 .18) and hence the righ t-hand-side of (4 .1 7) b ecomes smaller at least by the order of 1 / N s in the large N s limit. Ho w ev er, eac h of the most dominan t terms diverges with resp ect to N s in the larg e- N s limit, and they should cancel each other so that the final result b ecomes finite. W e therefore calculate all p ossible con tributions with resp ect to the set of in tegers, c 1 , c 2 , . . . , c 2 sm . Let us tak e a sequence of distinct in tegers c j satisfying 1 ≤ c j ≤ M f or j = 1 , 2 , . . . , 2 sm . W e denote it b y ( c j ) 2 sm , i.e. ( c j ) 2 sm = ( c 1 , c 2 , . . . , c 2 sm ). Let us denote b y Σ M the set o f in tegers, 1 , 2 , . . . , M : Σ M = { 1 , 2 , . . . , M } . W e t hen consider the complemen tary set of integers Σ M \ { c 1 , . . . , c 2 sm } , and put the elemen ts in increasing order suc h as z 1 < z 2 < · · · < z M − 2 sm . W e then extend the sequence z n of M − 2 sm in tegers in to that of M in tegers b y s etting z j + M − 2 sm = c j for j = 1 , 2 , . . . , 2 sm . W e shall denote z n also b y z ( n ) for n = 1 , 2 , . . . , M . In t erms o f sequence ( z n ) M w e express the scalar pro duct in the last line of (4.1 7) as follo ws. h 0 | M +2 sm Y k =1; k 6 = c 1 ,...,c 2 sm C (2 s ; ǫ ) ( λ k ( ǫ )) M Y α =1 B (2 s ; ǫ ) ( λ α ( ǫ )) | 0 i = h 0 | M − 2 sm Y k =1 C (2 s ; ǫ ) ( λ z ( k ) ( ǫ )) 2 sm Y j =1 C (2 s ; ǫ ) ( w (2 s ; ǫ ) j ) M − 2 sm Y i =1 B (2 s ; ǫ ) ( λ z ( i ) ( ǫ )) 2 sm Y j =1 B (2 s ; ǫ ) ( λ c j ( ǫ )) | 0 i . (4.19) W e ev aluate scalar pro duct (4.1 9), sending ν j to λ z ( j ) ( ǫ ) for j ≤ M − 2 sm and to w (2 s ; ǫ ) j − M +2 sm for j > M − 2 sm in the followin g matrix: H (1) (( λ z ( k ) ( ǫ )) M , ( ν z (1) , . . . , ν z ( M − 2 sm ) , ν M − 2 sm +1 , . . . , ν M ); ( w (2 s ; ǫ ) j ) L ) . (4.20) 21 Here w e define the matr ix elemen ts H (2 s ) ab ( { λ α } n , { µ j } n ; { ξ k } N s ) for a, b = 1 , 2 , . . . , n , b y H (2 s ) ab ( { λ α } n , { µ j } n ; { ξ k } N s ) = sinh η sinh( λ a − µ b ) a ( µ b ) d (2 s ) ( µ b ; { ξ k } ) n Y k =1; k 6 = a sinh( λ k − µ b + η ) − n Y K = 1; k 6 = a sinh( λ k − µ b − η ) ! . (4.21) Let us denote M − 2 sm b y M ′ . W e write the compo site of t w o sequences ( a ( i )) M and ( b ( j ) ) N as ( a ( i )) M #( b ( j )) N . Explicitly w e hav e ( a ( i )) M #( b ( j )) N = ( a (1) , . . . , a ( M ) , b (1) , . . . , b ( N )) . (4.22) F o r j > M ′ = M − 2 sm , w e ha v e lim ν j → w (2 s ; ǫ ) j − M ′ d ( ν j ; { w (2 s ; ǫ ) j } L ) H (1) i, j (( λ z ( k ) ( ǫ )) M , ( ν k ) M ′ #( ν k + M ′ ) 2 sm ; ( w (2 s ; ǫ ) j ) L ) = M Y α =1 sinh( λ α ( ǫ ) − w (2 s ; ǫ ) j − M ′ + η ) sinh η sinh( λ z ( i ) ( ǫ ) − w (2 s ; ǫ ) j − M ′ ) sinh( λ z ( i ) ( ǫ ) − w (2 s ; ǫ ) j − M ′ + η ) − d ( w (2 s ; ǫ ) j − M ′ ; { w (2 s ; ǫ ) j } L ) M Y t =1 sinh( λ t ( ǫ ) − w (2 s ; ǫ ) j − M ′ − η ) sinh( λ t ( ǫ ) − w (2 s ; ǫ ) j − M ′ + η ) × sinh η sinh( λ z ( i ) ( ǫ ) − w (2 s ; ǫ ) j − M ′ − 1 ) sinh( λ z ( i ) ( ǫ ) − w (2 s ; ǫ ) j − M ′ − 1 + η ) ! . (4.23) The second term of (4.23) for matrix elemen t ( i, j ) v anishes since w e hav e d ( w (2 s ; ǫ ) j − M ′ ; { w (2 s ; ǫ ) k } L ) = 0. Here w e r emark that if we directly ev aluate matrix H (2 s ) at ǫ = 0 , the second term of (4.23) for matrix eleme nt ( i, j ) fo r j 6 = 2 s ( n − 1) + 1 + M ′ with n = 1 , 2 , . . . , m , do es not v anish, although it is deleted b y subtracting column j b y column j − 1 , as discussed for the XXX case in Ref. [13]. W e th us ha v e lim ǫ → 0 det H (1) (( λ z ( k ) ( ǫ )) M , ( λ z (1) ( ǫ ) , . . . , λ z ( M − 2 sm ) ( ǫ ) , w (2 s ; ǫ ) 1 , . . . , w (2 s ; ǫ ) 2 sm ); ( w (2 s ; ǫ ) j ) 2 sN s ) = ( − 1) M − 2 sm M − 2 sm Y b =1 M Y k =1 sinh( λ k − λ z b − η ) 2 sm Y j =1 M Y k =1 sinh( λ k − w (2 s ) j + η ) × detΨ (2 s ) ′ (( λ z ( i )) M , ( λ z ( i )) M ′ #( w (2 s ) j ) 2 sm ; { ξ p } N s ) (4.24) where ( i, j ) eleme nt of Ψ (2 s ) ′ (( λ z ( k ) ) M , ( λ z ( k ) ) M ′ #( w (2 s ) k ) 2 sm ; { ξ p } N s ) for i = 1 , 2 , . . . , M , are giv en by Ψ (2 s ) ′ i, j (( λ z (1) , . . . , λ z ( M − 2 sm ) , λ c 1 , . . . , λ c 2 sm ) , ( λ z (1) , . . . , λ z ( M − 2 sm ) , w (2 s ) 1 , . . . , w (2 s ) 2 sm ); ( ξ p ) N s ) =          Φ (2 s ) ′ z ( i ) , z ( j ) (( λ k ) M ; { ξ p } ) for j ≤ M − 2 sm, sinh η sinh( λ z ( i ) − w (2 s ) j − M ′ ) sinh( λ z ( i ) − w (2 s ) j − M ′ + η ) for j > M − 2 sm . (4.25) 22 Therefore, for i, j = 1 , 2 , . . . , M − 2 sm , w e ha ve   Φ (2 s ) ′ (( λ z ( k ) ) M ; { ξ p } )  − 1 Ψ (2 s ) ′ (( λ z ( k ) ) M , ( λ z ( k ) ) M ′ #( w (2 s ) j ) 2 sm ; { ξ p } )  i, j = δ i, j . (4.26) In terms of sequence ( c j ) 2 sm , w e express the dep endence of matrix (Φ (2 s ) ′ ) − 1 Ψ (2 s ) ′ on the sequence of Bethe ro ots ( λ z ( i ) ) M etc., briefly , as follow s. (Φ (2 s ) ′ ) − 1 Ψ (2 s ) ′ (( c j ) 2 sm , { ξ p } ) = (Φ (2 s ) ′ ) − 1 Ψ (2 s ) ′  ( λ z ( j ) ) M , ( λ z ( j ) ) M ′ #( w (2 s ) j ) 2 sm ; { ξ p }  . (4.27) Recall M ′ = M − 2 sm . Similarly , we define Φ (2 s ) ′ (( c j ) 2 sm , { ξ p } ) and Ψ (2 s ) ′ (( c j ) 2 sm , { ξ p } ) b y Φ (2 s ) ′ i, j (( c l ) 2 sm , { ξ p } ) = Φ (2 s ) ′ i, j ( { λ z ( k ) } M ; ( ξ p ) N s ) = Φ (2 s ) ′ z ( i ) , z ( j ) ( { λ k } M ; ( ξ p ) N s ) , Ψ (2 s ) ′ i, j (( c k ) 2 sm , { ξ p } ) = Ψ (2 s ) ′ i, j (( λ z ( k )) M , ( λ z ( k )) M ′ #( w (2 s ) k ) 2 sm ; ( ξ p ) N s ) , (4.28) for i, j = 1 , 2 , . . . , M . Here w e r emark a g ain that sequence ( z ( i )) M is determined b y sequence ( c 1 , . . . , c 2 sm ) b y the definition that { z (1 ) , z (2) , . . . , z ( M ′ ) } = { 1 , 2 , . . . , M } \ { c 1 , . . . , c 2 sm } , and z (1) < · · · < z ( M ′ ) while z ( j + M ′ ) = c j for j = 1 , 2 , . . . , 2 sm . F ro m prop ert y (4.26) we define a 2 sm -b y-2 sm matrix φ (2 s ; m ) (( c j ) 2 sm ; { ξ p } ) b y φ (2 s ; m ) (( c j ) 2 sm ; { ξ p } ) j, k =   Φ (2 s ) ′  − 1 Ψ (2 s ) ′ (( c j ) 2 sm ; { ξ p } )  j + M ′ , k + M ′ for j, k = 1 , 2 , . . . , 2 sm. (4.29) Making use of (G.2), we obta in the spin- s EFP fo r the finite-size c hain a s follows. τ (2 s ) N s ( m ) = 1 Q 1 ≤ j M ′ w e hav e set z ( j ) = c j − M ′ . Lemma 4.4. I n the r e gion 0 < ζ < π / 2 s , a solution to the i n te gr al e quations (4.53) for inte ge rs A = 2 s ( a − 1) + α (i.e. ( a, α ) ) and B = 2 s ( b − 1) + β + M ′ with 1 ≤ α , β ≤ 2 s and 1 ≤ b ≤ m is given by ϕ A, B = ϕ ( β ) α ( µ a , ξ b ) = ρ ( µ a − ξ b ) δ α, β . (4.54) Pr o of. (i) In ( α, α ) case, i.e. when in tegers A = 2 s ( a − 1) + α and B = 2 s ( b − 1 ) + α corresp ond to indices ( a, α ) and ( b, α ), respective ly , assuming that ϕ ( α ) γ ( µ c , ξ b ) = 0 for γ 6 = α , w e reduce integral equations (4.53) to the Lieb equation (4 .3 5). The refore, w e hav e ϕ ( α ) α ( µ a , ξ b ) = ρ ( µ a − ξ b ). (ii) In ( α, β ) case, i.e. when A = ( a, α ) and B = ( b, β ) with β 6 = α , assuming ϕ ( β ) γ ( µ c , ξ b ) = 0 for γ 6 = β , w e ha v e from (4.53) Z ∞ −∞ K 2 ( µ a − µ c + ( β − α ) η + ǫ AB ) ϕ ( β ) β ( µ c , ξ b ) dµ c = 1 2 π i Ψ (2 s ) A, B . (4.55) F o r α < β , w e ha v e ǫ AB = iǫ . Shifting µ a analytically suc h as µ a → µ a − ( β − α − 1) η , w e hav e Z ∞ −∞ K 2 ( µ a − µ c + η + iǫ ) ϕ ( β ) β ( µ c , ξ b ) dµ c = 1 2 π i sinh η sinh( µ a + η − ξ b − η / 2) sinh( µ a + η − ξ b + η / 2) . (4.56) Making use of (4.37 ) w e reduce it essen tially to the Lieb equation. W e thus obta in ϕ ( β ) β ( µ a , ξ b ) = ρ ( µ a − ξ b ). F or α > β , w e hav e ǫ AB = − iǫ , and show it similarly , shifting µ a analytically as µ a → µ a − ( α − β + 1) η . Prop osition 4.5. L et us take a set of inte gers, c 1 , . . . , c 2 sm , satisfying 0 < c j ≤ M for j = 1 , 2 , . . . , 2 sm . Supp o s e that the numb er of c j which satisfy c j − 2 s [( c j − 1) / 2 s ] = α is giv e n by m for e a c h inte ger α satisfying 1 ≤ α ≤ 2 s . T hen, wh en 0 < ζ < π / 2 s , the solution to inte gr al e quations (4.53) for A = c j with j = 1 , 2 , . . . , 2 sm and for B = B ′ + M ′ wher e B ′ = 1 , 2 , . . . , 2 sm , is give n by ϕ ( β ) α ( µ a j , ξ b ) = ρ ( µ a j − ξ b ) δ α, β , (4.57) wher e a j = [( c j − 1 ) / 2 s ] + 1 , α = c j − 2 s [( c j − 1) / 2 s ] and B ′ = 2 s ( b − 1 ) + β w i th 1 ≤ β ≤ 2 s . 27 Pr o of. It f o llo ws from lemma 4.4 that ϕ c j , B of (4.57) giv es a solutio n to the in tegral equations. T aking the F ourier tr a nsform of (4.5 3), w e show in § 5.5 that the set of in tegral equations (4.53) for A = c j for j = 1 , 2 , . . . , 2 sm and B ′ = 1 , 2 , . . . , 2 sm has a unique solution. Th us w e obtain the unique solution (4.57). Let us recall the assumption that function ϕ ( β ) α ( µ a , ξ b ) is con tin uous with resp ect to µ a and ξ b . Then, fo r an y g iv en set of in tegers, c 1 , . . . , c 2 sm satisfying 0 < c j ≤ M f o r j = 1 , 2 , . . . , 2 sm , w e ma y approximate the matrix elemen ts of (Φ (2 s ) ′ ) − 1 Ψ (2 s ) ′ as follo ws. F or integers j and k with 1 ≤ j, k ≤ 2 sm , we define a j , α j , b k and β k as f o llo ws. a j = [( c j − 1) / 2 s ] + 1 , α j = c j − 2 s [( c j − 1) / 2 s ] , b k = [( k − 1) / 2 s ] + 1 , β k = k − 2 s [( k − 1) / 2 s ] . (4 .5 8) Then, w e hav e  (Φ (2 s ) ′ ) − 1 Ψ (2 s ) ′ (( c j ) 2 sm )  j + M ′ , k + M ′ = 1 N s ρ ( µ a j − ξ b k ) ρ tot ( µ a j ) δ α j , β k + O (1 /N 2 s ) . (4.59) Here w e recall M ′ = M − 2 s m . F o r a give n 2 s -string, λ ( a,α ) , with α = 1 , 2 , . . . , 2 s , we define λ ′ ( a,α ) b y the ‘regular part’ of λ ( a,α ) : λ ′ ( a,α ) = µ a − ( α − 1 / 2) η . (4.60) Let us in tro duce a 2 sm -by-2 sm ma t rix S b y S j, k ( c 1 , . . . , c 2 sm ; ( ξ p ) N s ) = ρ ( λ ′ c j − w (2 s ) k + η / 2) δ α j , β k for j, k = 1 , 2 , . . . , 2 sm. (4.61) Here a j , α j , b k and β k are giv en b y (4.58). Then, w e obta in φ (2 s ; m ) j, k (( c k ) 2 sm ; { ξ p } ) = 1 N s 1 ρ tot ( µ a ) S j, k (( c k ) 2 sm ; ( ξ p ) N s ) + O (1 /N 2 s ) . (4.62) and w e hav e det  φ (2 s ; m ) (( c k ) 2 sm ; { ξ p } )  = 2 sm Y j =1  1 N s 1 ρ tot ( µ a j )  ·  det S (( c j ) 2 sm ; ( ξ p ) N s ) + O (1 / N s )  . (4.63) 28 4.5 F ourier transform in the cases of spin- 1 and general spin- s The in tegral equations (4.53) for the spin-1 case a re giv en b y                                                        ϕ (1) 1 ( µ, ξ p ) + Z ∞ −∞ K 2 ( µ − λ ) ϕ (1) 1 ( λ, ξ p ) dλ + Z ∞ −∞ K 2 ( µ − λ + η + ǫ (1 , 2) ) ϕ (1) 2 ( λ, ξ p ) dλ = 1 2 π i sinh( η ) sinh( µ − ξ p + η 2 ) sinh( µ − ξ p − η 2 ) ϕ (2) 1 ( µ, ξ p ) + Z ∞ −∞ K 2 ( µ − λ ) ϕ (2) 1 ( λ, ξ p ) dλ + Z ∞ −∞ K 2 ( µ − λ + η + ǫ (1 , 2) ) ϕ (2) 2 ( λ, ξ p ) dλ = 1 2 π i sinh( η ) sinh( µ − ξ p + η 2 ) sinh( µ − ξ p + 3 η 2 ) ϕ (1) 2 ( µ, ξ p ) + Z ∞ −∞ K 2 ( µ − λ − η + ǫ (2 , 1) ) ϕ (1) 1 ( λ, ξ p ) dλ + Z ∞ −∞ K 2 ( µ − λ ) ϕ (1) 2 ( λ, ξ p ) dλ = 1 2 π i sinh( η ) sinh( µ − ξ p − 3 η 2 ) sinh( µ − ξ p − η 2 ) ϕ (2) 2 ( µ, ξ p ) + Z ∞ −∞ K 2 ( µ − λ − η + ǫ (2 , 1) ) ϕ (2) 1 ( λ, ξ p ) dλ + Z ∞ −∞ K 2 ( µ − λ ) ϕ (2) 2 ( λ, ξ p ) dλ = 1 2 π i sinh( η ) sinh( µ − ξ p − η 2 ) sinh( µ − ξ p + η 2 ) . (4.64) W e solve in tegral equations (4.53) via the F ourier transform. Let us expres s the F ourier transform of function ϕ ( β ) α ( µ, ξ ) by b ϕ ( β ) α ( ω , ξ ) = Z ∞ −∞ e iµω ϕ ( β ) α ( µ, ξ ) dµ, for α , β = 1 , 2 , . . . , 2 s . (4.65) W e denote b y b K n ( ω ) the F ourier transform of k ernel K n ( λ ). W e define mat r ix M (2 s ) ˆ ϕ b y  M (2 s ) ˆ ϕ  αβ = b ϕ ( β ) α ( ω , ξ ) for α, β = 1 , 2 , . . . , 2 s . (4.66) W e in tro duce a 2 s -b y-2 s matrix M (2 s ) K 2 . W e define matr ix elemen t ( j, k ) fo r j, k = 1 , 2 , . . . , 2 sm , b y  M (2 s ) K 2  j,k =                        1 + Z ∞ −∞ e iµω K 2 ( µ ) dµ for j = k , Z ∞ −∞ e iµω K 2 ( µ + ( k − j ) η + i 0) dµ f or j < k , Z ∞ −∞ e iµω K 2 ( µ − ( j − k ) η − i 0) d µ for j > k . (4.67) When 0 < ζ < π / 2 s , w e calculate the matrix elemen ts of M (2 s ) K 2 as follow s.  M (2 s ) K 2  j,k = δ j,k (1 + b K 2 ( ω )) + (1 − δ j,k ) e ( k − j ) ζ ω  b K 2 ( ω ) − e sgn( j − k ) ζ ω  for j, k = 1 , 2 , . . . , 2 s. (4.68) 29 Here we define sgn( j − k ) by the following: sgn( j − k ) = − 1 for j − k < 0, and sgn( j − k ) = +1 for j − k > 0. Here, b K 2 ( ω ), is giv en b y b K 2 ( ω ) = Z ∞ −∞ e iω µ K 2 ( µ ) dµ = sinh( π 2 − ζ ) ω sinh( π ω 2 ) . (4.69) Similarly , w e define a 2 s -b y-2 s matrix M (2 s ) K 1 b y  M (2 s ) K 1  j,k = Z ∞ −∞ e iω µ K 1 ( µ − ξ b + ( k − j ) η ) dµ for j, k = 1 , 2 , . . . , 2 s. (4.7 0) When 0 < ζ < π / 2 s , w e can show  M (2 s ) K 1  j,k = e iξ b ω n δ j,k b K 1 ( ω ) + (1 − δ j,k ) e ( k − j ) ζ ω  b K 1 ( ω ) − e sgn( j − k ) ζ ω / 2 o (4.71) for j, k = 1 , 2 , . . . , 2 s . Here b K 1 ( ω ) is giv en b y b K 1 ( ω ) = Z ∞ −∞ e iω µ K 1 ( µ ) dµ = sinh  π 2 − ζ 2  ω sinh( π ω 2 ) . (4.72) T aking the F ourier transform of integral equations (4.53) we ha v e the fo llo wing matrix equation. M (2 s ) K 2 M (2 s ) ˆ ϕ = M (2 s ) K 1 . (4.73) F o r the spin-1 case, from (4.64) we hav e 1 + b K 2 ( ω ) e ζ ω b K 2 ( ω ) − 1 e − ζ ω b K 2 ( ω ) − 1 1 + b K 2 ( ω ) ! b ϕ (1) 1 ( ω ) b ϕ (2) 1 ( ω ) b ϕ (1) 2 ( ω ) b ϕ (2) 2 ( ω ) ! = e iξ b ω b K 1 ( ω ) e ζ ω b K 1 ( ω ) − e ζ ω / 2 e − ζ ω b K 1 ( ω ) − e − ζ ω / 2 b K 1 ( ω ) ! . (4.74) It is easy to sho w that matrix M (2) ( b ϕ ) is giv en by the f ollo wing: b ϕ (1) 1 ( ω ) b ϕ (2) 1 ( ω ) b ϕ (1) 2 ( ω ) b ϕ (2) 2 ( ω ) ! =     e iξ b ω 2 cosh( ζ ω / 2) 0 0 e iξ b ω 2 cosh( ζ ω / 2)     . (4.75) W e calculate the determinant of M (2) K 2 (the spin-1 case) as follows. det 1 + b K 2 ( ω ) e ζ ω b K 2 ( ω ) − 1 e − ζ ω b K 2 ( ω ) − 1 1 + b K 2 ( ω ) ! = det 1 + b K 2 ( ω ) e ζ ω b K 2 ( ω ) − 1 − (1 + e − ζ ω ) 1 + e − ζ ω ! = det b K 2 ( ω )(1 + e ζ ω ) e ζ ω b K 2 ( ω ) − 1 0 1 + e − ζ ω ! = b K 2 ( ω )(1 + e − ζ ω )(1 + e ζ ω ) . (4.76) 30 Here, w e first subtract the 2nd ro w b y the 1 st row multiplie d by e − ζ ω . W e next add the 2 nd column to the 1st column. Fina lly , the determinan t is factorized and w e ha v e the result. By the same metho d w e can calculate the determinan t of M (2 s ) K 2 for the spin- s case. W e ha v e det M (2 s ) K 2 = (2 cosh( ζ ω / 2)) 2 s b K 2 s ( ω ) . (4.77) The determ inant is nonzero generically , and hence the solution to matrix equation (4.73) is unique. Therefore, w e obtain the solution of in tegral equation (4.5 3). 5 The EFP of the s pin- s XXZ sp i n chain near AF p oin t 5.1 Multiple-in tegral represen tations of the spin- s EFP Let us deriv e m ultiple-in tegral represen tations for the emptiness formation probability of the spin- s XXZ spin c hain. W e shall tak e the larg e N s limit o f the EFP (4.30) for a finite-size system, and w e replace rapidity λ c j with complex v ariable λ j for j = 1 , 2 , . . . , 2 sm , as follows . F o r a give n rapidit y of 2 s -string, λ A = µ a − ( α − 1 / 2) η + ǫ A , w e define its regular part λ ′ A b y λ ′ A = µ a − ( α − 1 / 2) η . In the large- N s limit, we first replace λ c k = λ ′ c k + ǫ c k b y λ ′ k + ǫ c k where λ ′ k are complex in tegral v a riables corresp onding to complete strings suc h as λ ′ k = µ k − ( β − 1 / 2) η for some in teger β with 1 ≤ β ≤ 2 s where η = iζ with 0 < ζ < π and µ k is real. W e express λ ′ k and ǫ c k simply b y λ k and ǫ k , resp ectiv ely , and then w e obtain m ultiple-in tegral r epresen tations. Applying (4.63) w e deriv e the emptiness formation probability for arbitrar y spin- s in the thermo dynamic limit N s → ∞ , as follow s. τ (2 s ) ∞ ( m ; { ξ p } ) = 1 Q 1 ≤ j 0 , − iǫ for I m ( λ k − λ l ) < 0 . (5.4) In the homogeneous case w e ha v e ǫ p = 0 for p = 1 , 2 , . . . , N s . W e ha v e th us defined inhomogeneous parameters ξ p . W e recall that in the homogeneous case, the spin- s Hamiltonian is deriv ed f r o m the logarit hmic deriv ativ e of the spin- s t r a nsfer matrix. Here w e should remark tha t an expression of the matr ix elemen ts of S similar to (5.3) has b een giv en in eq. (6.14) of Ref. [13] for the correlation functions of the in tegrable spin- s XXX spin c hain. 5.2 Symmetric expression of the spin- s EFP W e shall express the spin- s EFP (5.1) in a simpler w ay , making use of p ermutations of 2 s m in tegers, 1 , 2 , · · · , 2 sm , and the formula o f the Cauc hy determinant. Let us take a set o f intege rs a ( j,k ) = a 2 s ( j − 1)+ k satisfying 1 ≤ a ( j,k ) ≤ N s / 2 fo r j = 1 , 2 , . . . , m and k = 1 , 2 , . . . , 2 s . Here w e remark that indices a ( j,k ) corresp ond to the string cen ters µ ( j,k ) = µ 2 s ( j − 1)+ k . In order t o refo r mulate t he sum ov er in tegers, c 1 , . . . , c 2 sm , in eq. (4.30) in terms of indices a ( j,k ) , let us in tro duce ˆ c 1 , . . . , ˆ c 2 sm b y ˆ c 2 s ( j − 1)+ k = 2 s ( a ( j,k ) − 1) + k , for j = 1 , 2 , . . . , m and k = 1 , 2 , . . . , 2 s . (5.5) W e also define β ( z ) by β ( z ) = z − 2 s [( z − 1) / 2 s ]. Then, ˆ c j are express ed a s follows. ˆ c j = 2 s ( a j − 1) + β ( j ) , fo r j = 1 , 2 , . . . , 2 sm. (5.6 ) W e decomp ose the sum o v er c j in to 2 s sums ov er a j as follow s. M X c j =1 g ( c j ) = 2 s X k =1 N s / 2 X a j =1 g (2 s ( a j − 1) + k ) . (5.7) Let us consider suc h a function f ( c 1 , c 2 , . . . , c 2 sm ) of sequence of in tegers ( c j ) 2 sm that v anishes unless c j ’s are distinct. W e also assume that f ( c 1 , c 2 , . . . , c 2 sm ) v a nishe s unless the n um b er of c j ’s satisfying β ( c j ) = α is giv en b y m for eac h in teger α satisfying 1 ≤ α ≤ 2 s . Here w e recall that the tw o prop erties ar e in common with the summand of (4.30), in pa rticular, with det( φ (2 s ; m ) (( c j ) 2 sm ; { ξ p } ). Then, w e hav e M X c 1 =1 M X c 2 =1 · · · M X c 2 sm =1 f ( c 1 , · · · , c 2 sm ) = 1 ( m !) 2 s N s / 2 X a 1 =1 N s / 2 X a 2 =1 · · · N s / 2 X a 2 sm =1 X P ∈S 2 sm f ( ˆ c P 1 , · · · , ˆ c P (2 sm ) ) = N s / 2 X a 1 =1 N s / 2 X a 2 =1 · · · N s / 2 X a 2 sm =1 X π ∈S 2 sm / ( S m ) 2 s f ( ˆ c π 1 , · · · , ˆ c π (2 sm ) ) . (5.8) 32 Here an elemen t π of S 2 sm / ( S m ) 2 s giv es a permutation of integers 1 , 2 , . . . , 2 sm , where π j ’s suc h that π j ≡ k ( mod 2 s ) are put in increasing order in the seque nce ( π 1 , π 2 , . . . , π (2 sm )) for k = 0 , 1 , . . . , 2 s − 1. Reform ulating the sum ov er c j s in (4.30) in terms of a j ’s, in the large N s limit w e hav e τ (2 s ) ∞ ( m ) = 1 Q 1 ≤ α<β ≤ 2 s (sinh( β − α ) η ) m × 1 Q 1 ≤ k < l ≤ m Q 2 s α =1 Q 2 s β =1 sinh( ξ k − ξ l + ( α − β ) η ) × 2 s Y k =1 m Y j =1 Z ∞ −∞ dµ ( j,k ) X P ∈S 2 sm 1 ( m !) 2 s det S ( λ P 1 , . . . , λ P (2 sm ) ) H (2 s ) ( λ P 1 , . . . , λ P (2 sm ) ) = 1 Q 1 ≤ α<β ≤ 2 s (sinh( β − α ) η ) m × 1 Q 1 ≤ k < l ≤ m Q 2 s α =1 Q 2 s β =1 sinh( ξ k − ξ l + ( α − β ) η ) × 2 sm Y j =1 Z ∞ −∞ dµ j X π ∈S 2 sm / ( S m ) 2 s det S ( λ π 1 , . . . , λ π (2 sm ) ) H (2 s ) ( λ π 1 , . . . , λ π (2 sm ) ) , (5.9) where sym b ols λ j denote the follow ing λ j = µ j − ( β ( j ) − 1 2 ) η for j = 1 , 2 , . . . , 2 sm. (5.10) W e calculate det S applying the Cauc h y determinan t fo r mula det  1 sinh( λ a − ξ k )  = Q m k b sinh( λ a − λ b ) Q m a =1 Q m k =1 sinh( λ a − ξ k ) , (5.11) and w e obt a in the symmetric expression of the spin- s EFP as fo llows. τ (2 s ) ∞ ( m ; { ξ p } ) = 1 Q 1 ≤ α<β ≤ 2 s sinh m ( β − α ) η Y 1 ≤ k < l ≤ m sinh 2 s ( π ( ξ k − ξ l ) /ζ ) Q 2 s j =1 Q 2 s r =1 sinh( ξ k − ξ l + ( r − j ) η ) × i 2 sm 2 (2 iζ ) 2 sm 2 sm Y j =1 Z ∞ −∞ dµ j ! 2 s Y γ =1 Y 1 ≤ b m . Hence w e ma y consider that inhomogeneous parameters ξ p with p > m a r e all set to b e zero, af ter computing t he EFP: τ (2 s ) N s ( m ; { ξ p } m ). W e now show t hat the order of the homog eneous limit: ξ p → 0 and the thermodynamic limit N s → ∞ can b e rev ersed. W e can sho w the following relation: m Y p =1 lim ξ p → 0  lim N s →∞  τ (2 s ) N s ( m ; { ξ p } m )   = lim N s →∞ m Y p =1 lim ξ p → 0  τ (2 s ) N s ( m ; { ξ p } m )  ! . (5.14) In fact, when N s is v ery large, it follows from (4.63) that w e hav e τ (2 s ) ∞ ( m ; { ξ p } m ) = τ (2 s ) N s ( m ; { ξ p } m ) + O (1 / N s ) . (5.1 5 ) F urthermore, w e can explicitly sho w tha t τ (2 s ) N s ( m ; { ξ p } m ) is con tin uous with resp ect to ξ p at ξ p = 0 for p = 1 , 2 , . . . , m . W e first reform ulate the sum ov er c j in (4.3 0) in to the sum ov er a ( j,k ) b y relation (5.8 ) . M X c 1 =1 · · · M X c 2 sm =1 det S (( c j ) 2 sm ) H (2 s ) (( λ c j ) 2 sm ) = m Y j =1 2 s Y k =1   1 m ! N s / 2 X a ( j,k ) =1   X P ∈S 2 sm det S (( ˆ c P j ) 2 sm ) H (2 s ) (( λ ˆ c j ) 2 sm ) = m Y j =1 2 s Y k =1   1 m ! N s / 2 X a ( j,k ) =1   det S (( ˆ c j ) 2 sm ) X P ∈S 2 sm (sgn P ) H (2 s ) (( λ ˆ c j ) 2 sm ) . (5.16) 34 W e then apply the Cauc h y determinan t for mula to ev aluate det S ( ˆ c j ) as follo ws. det S (( ˆ c j ) 2 sm ) = det S (( λ b c j ) 2 sm ) = 2 s Y α =1 det S ( α ) (( λ 2 s ( a ( j,α ) − 1)+ α ) m ) =  Q j M δ β ( b ℓ ) , β ( k ) · ρ ( λ b ℓ − w (2 s ) k + η / 2) /N s ρ tot ( µ a ( b ℓ ) ) for b ℓ ≤ M (6.9) where µ j denote the cen ters o f λ j as follow s. λ j = µ j − ( β ( j ) − 1 2 ) η j = 1 , 2 , . . . , 2 sm. (6.10) W e recall that a ( j ) and β ( j ) hav e b een defined in terms of the Gauss’ sym b ol [ · ] b y a ( j ) = [( j − 1) / 2 s ] + 1 a nd β ( j ) = j − 2 s [( j − 1) / 2 s ], resp ectiv ely . W e remark that under the limit of sending ǫ to zero, the sum ov er v ariable c j is r es tricted up to M . 6.2 Multiple-in tegral represen tations of the spin- s XX Z correlatio n function for an arbitrary pro duct of elemen tary matrices Let us formu late mat r ix S for the correlation function of an arbitrary pro duct of elemen tary matrices. W e define the ( j, k ) elemen t of matrix S = S  ( λ j ) 2 sm ; ( w (2 s ) j ) 2 sm  b y S j,k = ρ ( λ j − w (2 s ) k + η / 2) δ ( α ( λ j ) , β ( k )) , fo r j, k = 1 , 2 , . . . , 2 sm . (6.11) Here δ ( α , β ) denotes the Kr onec k er delta. W e define α ( λ j ) b y α ( λ j ) = γ if λ j = µ j − ( γ − 1 / 2) η or λ j = w (2 s ) k where β ( k ) = γ (1 ≤ γ ≤ 2 s ). W e remark that µ j corresp ond to the cen ters o f complete 2 s - strings λ j . W e a lso remark that t he ab o v e definition of matrix S generalizes that of (5.3) since α ( λ j ) is no w also defined also for λ j = w (2 s ) k . Let Γ j b e a small contour rotating counterc lo c kwise around λ = w (2 s ) j . Sinc e the det S has simple p oles at λ = w (2 s ) j with residue 1 / 2 π i , w e therefor e ha v e Z ∞ + iǫ −∞ + iǫ det S (( λ k ) 2 sm ) dλ 1 = Z ∞− iǫ −∞− iǫ det S (( λ k ) 2 sm ) dλ 1 − I Γ 1 det S (( λ k ) 2 sm ) dλ 1 . (6.12) F o r sets α − and α + w e define ˜ λ j for j ∈ α − and ˜ λ ′ j for j ∈ α + , resp ectiv ely , b y the fo llowing sequence : ( ˜ λ ′ j ′ max , . . . , ˜ λ ′ j ′ min , ˜ λ j min , ˜ λ j max ) = ( λ 1 , . . . , λ 2 sm ) . (6.13) Th us, from the expression of the correlation function in terms of a finite sum (6 .8 ) we obtain the m ultiple-in tegral represen tation as follows. F (2 s +) ( { ǫ j , ǫ ′ j } ) = Z ∞ + iǫ −∞ + iǫ + · · · + Z ∞− i (2 s − 1) ζ + iǫ −∞− i (2 s − 1) ζ + iǫ ! dλ 1 · · · Z ∞ + iǫ −∞ + iǫ + · · · + Z ∞− i (2 s − 1) ζ + iǫ −∞− i (2 s − 1) ζ + iǫ ! dλ α + Z ∞− iǫ −∞− iǫ + · · · + Z ∞− i (2 s − 1) ζ − iǫ −∞− i (2 s − 1) ζ − iǫ ! dλ α + +1 · · · Z ∞− iǫ −∞− iǫ + · · · + Z ∞− i (2 s − 1) ζ − iǫ −∞− i (2 s − 1) ζ − iǫ ! dλ m × Q ( { ǫ j , ǫ ′ j } ; λ 1 , . . . , λ 2 sm ) det S ( λ 1 , . . . , λ 2 sm ) . (6.14) 38 Here w e hav e defined Q ( { ǫ j , ǫ ′ j } ; λ 1 , . . . , λ 2 sm ) in terms of small n umbers ǫ ℓ,k of (5.4) b y Q ( { ǫ j , ǫ ′ j } ; λ 1 , . . . , λ 2 sm )) = ( − 1) α + Q j ∈ α −  Q j − 1 k =1 ϕ ( ˜ λ j − w (2 s ) k + η ) Q 2 sm k = j +1 ϕ ( ˜ λ j − w (2 s ) k )  Q 1 ≤ k < ℓ ≤ 2 sm ϕ ( λ ℓ − λ k + η + ǫ ℓ,k ) × Q j ∈ α +  Q j − 1 k =1 ϕ ( ˜ λ ′ j − w (2 s ) k − η ) Q 2 sm k = j +1 ϕ ( ˜ λ ′ j − w (2 s ) k )  Q 1 ≤ k < ℓ ≤ 2 sm ϕ ( w (2 s ) k − w (2 s ) ℓ ) . (6.15 ) Th us, correlation functions (6.1 ) are expressed in the form of a single term o f m ultiple in tegrals. Similarly as the symmetric spin- s EFP , w e can sho w the symmetric expression of the m ultiple-in tegral represen tations of the spin- s correlation f unction as f o llo ws. F (2 s +) ( { ǫ j , ǫ ′ j } ) = 1 Q 1 ≤ α<β ≤ 2 s sinh m ( β − α ) η Y 1 ≤ k < l ≤ m sinh 2 s ( π ( ξ k − ξ l ) /ζ ) Q 2 s j =1 Q 2 s r =1 sinh( ξ k − ξ l + ( r − j ) η ) X σ ∈S 2 sm / ( S m ) 2 s α + Y j =1 Z ∞ + iǫ −∞ + iǫ + · · · + Z ∞− i (2 s − 1) ζ + iǫ −∞− i (2 s − 1) ζ + iǫ ! dµ σj 2 sm Y j = α + +1 Z ∞− iǫ −∞− iǫ + · · · + Z ∞− i (2 s − 1) ζ − iǫ −∞− i (2 s − 1) ζ − iǫ ! dµ σj × (sgn σ ) Q ( { ǫ j , ǫ ′ j } ; λ σ 1 , . . . , λ σ (2 sm ) )) 2 sm Y j =1 Q m b =1 Q 2 s − 1 β =1 sinh( λ j − ξ b + β η ) Q m b =1 cosh( π ( µ j − ξ b ) /ζ ) ! × i 2 sm 2 (2 iζ ) 2 sm 2 s Y γ =1 Y 1 ≤ b n w e ha v e e E m, n ( ℓ +) i = ℓ n ! " ℓ m # q " ℓ n # − 1 q e P ( ℓ ) 1 ··· L ( i − 1) ℓ Y α =1 ( A (1+) + D (1+) )( w α ) n Y k =1 D (1+) ( w ( i − 1) ℓ + k ) × m Y k = n +1 B (1+) ( w ( i − 1)2 s + k ) ℓ Y k = m +1 A (1+) ( w ( i − 1) ℓ + k ) ℓN s Y α = iℓ +1 ( A (1+) + D (1+) )( w α ) P ( ℓ ) 1 ··· L . (E.2) F o r m < n w e hav e e E m, n ( ℓ +) i = ℓ n ! " ℓ m # q " ℓ n # − 1 q e P ( ℓ ) 1 ··· L ( i − 1) ℓ Y α =1 ( A (1+) + D (1+) ( w α ) m Y k =1 D (1+) ( w ( i − 1) ℓ + k ) × n Y k = m +1 C (1+) ( w ( i − 1)2 s + k ) ℓ Y k = m +1 A (1+) ( w ( i − 1) ℓ + k ) ℓN s Y α = iℓ +1 ( A (1+) + D (1+) )( w α ) P ( ℓ ) 1 ··· L . (E.3) F Deriv ation of the d e nsit y of s tring ce n t ers In terms o f shifted rapidities ˜ λ A with A = 2 s ( a − 1) + α for a = 1 , 2 , . . . , N s / 2 a nd α = 1 , 2 , . . . , 2 s , the Bethe ansatz equations for the homog eneous chain ar e giv en b y sinh( ˜ λ A + sη ) sinh( ˜ λ A − sη ) ! N s = M Y B =1; B 6 = A sinh( ˜ λ A − ˜ λ B + η ) sinh( ˜ λ A − ˜ λ B − η ) , for A = 1 , 2 , . . . , M . (F.1) Putting λ A = µ a − (2 s + 1 − 2 α ) and taking the pro duct o v er α for α = 1 , 2 , . . . , 2 s , f o r the left-hand side of (F .1) a nd for the right-hand side of (F.1), w e hav e ( − 1) 2 s ( 2 s Y k =1  sinh(( k − 1 / 2) η − µ a ) sinh(( k − 1 / 2) η + µ a )  N s ) − 1 = ( − 1) 2 s + N s / 2 N s / 2 Y b =1 ( sinh(2 sη − ( µ a − µ b )) sinh(2 sη + ( µ a − µ b )) 2 s − 1 Y k =1  sinh( k η − ( µ a − µ b )) sinh( k η + ( µ a − µ b ))  2 ) − 1 . (F.2) 45 T aking the logarithm of (F.2) and ma king use of the follo wing relation K 2 k ( λ ) = d dλ 1 2 π i log  sinh( k η − λ ) sinh( k η + λ )  (F.3) w e hav e the inte gral equation for the densit y of string cen ters, ρ ( λ ), as follow s. ρ ( λ ) = ℓ X j =1 K 2 j − 1 ( λ ) − Z ∞ −∞ K 4 s ( λ − µ b ) + 2 s − 1 X k =1 2 K 2 k ( λ − µ b ) ! ρ ( λ ) dλ . (F.4 ) F o r 0 < ζ ≤ π /m we hav e the following F ourier transform: Z ∞ −∞ e iµω K m ( µ ) dµ = sinh(( π − mζ ) ω / 2) sinh( π ω / 2) . (F.5) T aking the F ourier transform of (F.4) w e hav e the F ourier transform b ρ ( ω ) of ρ ( λ ) as follows. b ρ ( ω ) = 2 s X k =1 b K 2 k − 1 ( ω ) ! / 1 + b K 4 s ( ω ) + 2 2 s − 1 X k =1 b K 2 k ( ω ) ! = 1 2 cosh( ζ ω / 2) . T aking the in v erse F ourier t r a nsform we obtain ρ ( λ ) = 1 / 2 ζ cosh( π λ/ζ ). G Some form ulas of the alg e braic B ethe ansatz Applying the comm utation relations b et w een C and D op erators w e hav e h 0 | M Y α =1 C ( λ α ) m Y j =1 D ( λ M + j ) = M +1 X a 1 =1 M +2 X a 2 =1; a 1 6 = a 1 · · · M + m X a m =1; a 1 6 = a 1 ,...,a m G a 1 ··· a m ( λ 1 , · · · , λ M + m ) where G a 1 ··· a m ( λ 1 , · · · , λ M + m ) = m Y j =1 d ( λ a j ; { w j } L ) Q M + j − 1 b =1; b 6 = a 1 ,...,a j − 1 sinh( λ a j − λ b + η ) Q M + j b =1; b 6 = a 1 ,...,a j sinh( λ a j − λ b ) ! . (G .1 ) Let { λ k } M b e a set of Bethe ro ots. W e hav e [4, 5] h 0 | M Y k =1; k 6 = a 1 ,...,a m C ( λ k ) m Y j =1 C ( w j ) M Y γ =1 B ( λ γ ) | 0 i / h 0 | n Y k =1 C ( λ k ) M Y γ =1 B ( λ γ ) | 0 i = det   Φ ′ ( { λ α } )  − 1 Ψ ′ ( { λ α } \ { λ a 1 , . . . , λ a m } ∪ { w 1 , . . . , w m } )  × m Y j =1 × M Y α =1; α 6 = a 1 ,...,a m sinh( λ α − w j + η ) sinh( λ α − λ a j + η ) m Y j =1 M Y α =1 sinh( λ α − λ a j ) sinh( λ α − w j ) Y 1 ≤ j

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment