The sl(2) loop algebra symmetry of the XXZ spin chain at roots of unity and applications to the superintegrable chiral Potts model 1

Transcription

1 The sl(2) loop algebra symmetry of the XXZ spin chain at roots of unity and applications to the superintegrable chiral Potts model 1 Tetsuo Deguchi Department of Physics, Ochanomizu Univ. In collaboration with Akinori Nishino. 1 Talk given at the Simon Center, Stony Brook, NY, Jan. 18, 2010

2 Contents (1) The sl(2) loop algebra symmetry of the XXZ spin chain at roots of unity (2) Proof that Bethe vectors are highest weight vectors of sl 2 loop algebra in the sector S Z (mod) 0. (3) Formulas useful for calculating norms for degenerate eigenstates (4) Inverse Scattering of the higher spin XXZ and application to the superintegrable chiral Potts (SCP) model

3 Main Reference of this talk [D1] T. D., Regular XXZ Bethe states at roots of unity as highest weight vectors of the sl 2 loop algebra, J. Phys. A: Math. Theor. Vol. 40 (2007) [D2] T.D., J. Stat. Mech. (2007) P05007 [ND2008] A. Nishino and T. D., An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model J. Stat. Phys. 133 (2008) pp [DM] T. D. and C. Matsui, Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method (arxiv: v3), to appear in Nucl. Phys. B [DFM] T. D., K. Fabricius and B. M. McCoy, The sl 2 loop algebra symmetry of the six-vertex model at roots of unity, J. Stat. Phys. 102 (2001)

4 Many interesting papers relevant to this talk [AY] H. Au-Yang and J. Perk, J. Phys. A 41 (2008) ; J. Phys. A 42 (2009) ; J. Phys. A 43 (2010) ; arxiv: [B] R. Baxter, arxiv: ; arxiv: ; arxiv: [vg] N. Iorgov, V. Shadura, Yu. Tykhyy, S. Pakuliak, and G. von Gehlen, arxiv: [FM2010] K. Fabricius and B.M. McCoy, arxiv:

5 1 The sl 2 loop algebra symmetry of the XXZ spin chain (P.B.C.) H XXZ = 1 2 L j=1 ( ) σ X j σj+1 X + σj Y σj+1 Y + σj Z σj+1 Z. When q is a root of unity: q 2N = 1 where = (q + q 1 )/2, the H XXZ (and τ 6V (z)) commutes with the generators of the sl 2 loop algebra, U(L(sl 2 )) in sector A: S Z 0 (mod N) (and in sector B: S Z N/2 (mod N)): S ±(N) = (S ± ) N /[N]!, T ±(N) = (T ± ) N /[N]! Here S ± and T ± are generators of U q (ŝl 2) in the Lth tensor product of the spin-1/2 representations S ± = T ± = L q σz /2 q σz /2 σ j ± q σz /2 q σz /2 j=1 L q σz /2 q σz /2 σ j ± qσz /2 q σz /2 j=1

6 and the q-integer [n] and the q-factorial [n]! are given by [n] = qn q n q q 1, [n]! = [1][2] [n] [DFM]: T. D, K. Fabricius and B. M. McCoy, J. Stat. Phys. 102 (2001)

7 1.1 S ±(N) and T ±(N) generate the sl 2 loop algebra U(L(sl 2 )) (the sl 2 affine Kac-Moody algebra with c = 0) Let us introduce the Chevalley generators as follows: E 0 + = T (N), E0 = T +(N), E 1 + = S +(N), E1 = S (N), H 0 = H 1 = 2 N SZ. E j ±, H j for j = 0, 1, satisfy the defining relations of the sl 2 loop algebra U(L(sl 2 )): H 0 + H 1 = 0, [H i, E j ± ] = ±a ije j ±, (i, j = 0, 1) [E i +, E j ] = δ ijh j, (i, j = 0, 1) [E i ±, [E± i, [E± i, E± j ]]] = 0, (i, j = 0, 1, i j) Here, the Cartan matrix (a ij ) of A (1) 1 is defined by ( ) ( ) a 00 a = a 10 a Ref: G. Lusztig, Contemp. Math. 82 (1989) pp

8 1.2 Anomalous spectral degeneracies Violation of the level non-crossing rule (in the sense of Heilmann and Lieb (1971)) (cf. von-neumann Wigner theorem) Novel level crossings with extremely large degeneracy (1) The vacuum state, 0, has the degeneracy of N2 L/N when L is a multiple of N. (the XXZ spin chain) (2) A regular Bethe vector R of R down spins with S Z 0 (mod N) has the degeneracy of 2 (L 2R)/N. In each sector (mod N) of the XXZ spin chain has the same degree of degeneracy. ([DFM], T.D., J. Phys. A 35 (2002) 879)

9 (3) The XYZ spin chain at roots of unity has the same spectral degeneracies such as the sl 2 loop algebra symmetry of the XXZ chain. ([DFM], T. D, J. Phys. A 35 (2002) 879; K. Fabricius and B.M. McCoy, J. Stat. Phys. 111 (2003) 323) H XY Z = 1 2 L j=1 ( ) JX σj X σj+1 X + J Y σj Y σj+1 Y + J Z σj Z σj+1 Z. A regular XYZ Bethe vector R with R down spins has the spectral degeneracy of 2 (L 2R)/N if L/N is an integer T.D., J. Phys. A 35 (2002) )

10 2 Representation theory 2.1 The sl 2 loop algebra and D-highest weight vectors Generators h k and x ± k (k Z) satisfy the defining relations of the sl 2 loop algebra (j, k Z) [h j, x ± k ] = ±2x± j+k, [x+ j, x k ] = h j+k [h j, h k ] = 0, [x ± j, x± k ] = 0 Correspondence between the Chevalley generators and h k, x ± k : E 1 ± x± 0, E+ 0 x 1, E 0 x+ 1, H 0 = H 1 h 0. We thus have x + 0 = S +(N), x 0 = S (N), x + 1 = T +(N), x 1 = T (N), h 0 = 2 N SZ.

11 Definition of D-highest weight vectors (Drinfeld-highest weight) We call a representation of U(L(sl 2 )) D-highest weight if it is generated by a vector Ω satisfying (i) Ω is annihilated by generators x + k : x + k Ω = 0 (k Z) (ii) Ω is a simultaneous eigenvector of generators, h k s: h k Ω = d k Ω, for k Z. Here d k denotes the eigenvalues of h k s. We call Ω a highest weight vector.

12 2.2 D-highest weight polynomials Definition 1 (D-highest weight polynomial). Let λ k be eigenvalues as follows: (x + 0 ) k (x 1 )k /(k!) 2 Ω = λ k Ω (k = 1, 2,..., r) For the sequence λ = (λ 1, λ 2,..., λ r ), we define polynomial P λ (u) by r P λ (u) = λ k ( u) k, (1) We call P λ (u) the D-highest weight polynomial for highest weight d k. k=0 When U(L(sl 2 ))Ω is irreducible, we call it the Drinfeld polynomial of the representation.

13 Definition 2 (D-highest weight parameters). We define parameters a k by s P λ (u) = (1 a k u) m k. (2) k=1 Here a 1, a 2,..., a s are distinct, and their multiplicities are given by m 1, m 2,..., m s, respectively, where we have r = m m s. We call parameters a j the D- highest weight parameters of Ω. If U(L(sl 2 ))Ω is irreducible, the dimensionality is given by s dimu(l(sl 2 ))Ω = (m j + 1). Example For r = 3, if we have P λ (u) = (1 a 1 u) 2 (1 a 2 u), we have m 1 = 2 and m 2 = 1. If the h.w. rep. is irreducible, then we have dimu(l(sl 2 ))Ω = (2 + 1)(1 + 1) = 6. j=1

14 2.3 R-matrix and monodromy matrix We define the R-matrix acting on the tensor product V V by R(u) = Rcd(u)e ab a,c e b,d. (3) We denote it by a,b,c,d=1, R 12 (λ 1, λ 2 ) = 0 b(u) c(u) 0 0 c(u) b(u) Here functions b(u) and c(u) are given by b(u) = sinh(u) sinh(u + η) [1,2], c(u) = sinh(η) sinh(u + η). The R-matrices satisfy the Yang-Baxter equations:. (4) R 12 (λ 1, λ 2 )R 13 (λ 1, λ 3 )R 23 (λ 2, λ 3 ) = R 23 (λ 2, λ 3 )R 13 (λ 1, λ 3 )R 12 (λ 1, λ 2 ) (5)

15 We define L-operators acting on the mth site for m = 1, 2,..., L, by L m (λ, ξ m ) = R 0m (λ, ξ m ). (6) We define the monodromy matrix acting on the L lattice-sites in one dimension by T 0,12 L (λ; ξ 1,..., ξ L ) = L L (λ, ξ L )L L 1 (λ, ξ L 1 ) L 2 (λ, ξ 2 )L 1 (λ, ξ 1 ). (7) We express the matrix elements of the monodromy matrix by ( ) A(u) B(u) T 0,12 L (u; ξ 1,..., ξ L ) = C(u) D(u) The transfer matrix, t(u), is given by the trace over the 0th space: [0] (8) t(u; ξ 1,..., ξ L ) = tr 0 (T 0,12 L (u; ξ 1,..., ξ L )) = A(u) + D(u). (9)

16 3 Rigorous methods for degenerate multiplicity 3.1 Conjecture on highest weight vectors proved in some sectors A conjecture by Fabricius and McCoy: Bethe eigenvectors are highest weight vectors of the sl 2 loop algebra, and they have the Drinfeld polynomials [FM]. [FM] K. Fabricius and B. M. McCoy, Progress in Math. Phys. Vol. 23 (MathPhys Odyssey 2001), (Birkhäuser, Boston, 2002) It was shown in sector A: S Z 0 (mod N) when q 2N = 1 Every regular Bethe ansatz eigenvector is highest weight (shown by the algebraic Bethe ansatz for the inhomogeneous 6V transfer matrix.) [D1] The degenerate eigenspace is mostly given by a Weyl module, which is either an irreducible or a reducible and indecomposable rep.

17 3.2 Quantum group generators through infinite rapidities Let us define function n ξ (z) by and function ĝ(z) by ĝ(z) = L n ξ (z) = sinh(z ξ j ), (10) j=1 { 2 exp( z) sinh 2η (Re z > 0), (11) 2 exp(z) sinh 2η (Re z < 0). We normalize operators A(z),..., D(z) as follows: Â(z) = A(z)/n ξ (z), ˆB(z) = B(z)/(ĝ(z)nξ (z)), ˆD(z) = D(z)/n ξ (z), Ĉ(z) = C(z)/(ĝ(z)n ξ (z)). (12) Taking the limit of infinite rapidities for the inhomogeneous case, we have Â(± ) = q ±SZ, ˆB( ) = T, ˆB( ) = S, ˆD(± ) = S Z, Ĉ( ) = S +, Ĉ( ) = T +, (13)

18 3.3 Complete N-strings Let N be a positive integer. Definition 3 (Complete N-string). We call a set of rapidities z j a complete N-string, if they satisfy z j = Λ + η(n + 1 2j) (j = 1, 2,..., N). (14) We call Λ the center of the N-string. f(z w) = sinh(z w η) sinh(z w)

19 3.4 Sufficient conditions of a highest weight vector Lemma 1. Suppose that x ± 0, x+ 1, x 1 and h 0 satisfy the defining relations of U(L(sl 2 )), and x ± k and h k (k Z) are generated from them. If a vector Φ satisfies the following: x + 0 Φ = x + 1 Φ = 0, (15) h 0 Φ = r Φ, (16) (x + 0 ) (n) (x 1 )(n) Φ = λ n Φ for n = 1, 2,..., r, (17) where r is a nonnegative integer and λ n are complex numbers. Then Φ is highest weight, i.e. we have x + k Φ = 0 (k Z), (18) h k Φ = d k Φ (k Z), (19) where d k are complex numbers.

20 3.5 Annihilation property We construct operators S +(N) ξ S +(N) ξ and T +(N) ξ 1 = lim q q0 [N] q! (Ĉ(, η))n, T +(N) ξ We evaluate the action of the operator S +(N) ξ procedure: S +(N) ξ { 1 R = lim q q0 [N] q! as follows: 1 = lim q q0 [N] q! (Ĉ(, η))n. (20) on the vector R by the limiting (Ĉ(, η) ) N B(t1, η) B(t R, η) 0 }. (21) R is the number of regular Bethe roots. We denote by Σ R = {1, 2,..., R} the set of indices of the regular Bethe roots, t 1, t 2,..., t R.

21 Lemma 2. Let t 1, t 2,..., t R be regular Bethe roots at generic q. For a given positive integer N c, we have 1 ) Nc (Ĉ(± ) B(t1 )B(t 2 ) B(t R ) 0 [N c ] q! = S N c =N c S N c Σ R l Σ R \S N c B(t l ) 0 exp(± ( 1) N c (q ±1 q 1 ) N c N c 1 j S N c t j ) j S N c a 6V ξ (t j ) k Σ R \S N c f(t j t k ) [ L 2 R + N c l] q. (22) l=0

22 3.6 Diagonal property In addition to regular Bethe roots at generic q, t 1, t 2,..., t R, we introduce kn rapidities, z 1, z 2,..., z kn, forming a complete kn-string: z j = Λ + (kn + 1 2j)η for j = 1, 2,..., kn. We calculate the action of (S +(N) ξ ) k (T (N) ξ ) k on the Bethe state at q 0, R = B( t 1 ) B( t R ) 0, as follows: ( S +(N) ξ ) k ( T (N) ξ ) k R = lim q q0 ( lim Λ 1 ([N] q!) k (Ĉ( ))kn 1 ([N] q!) k ˆB(z 1, η) ˆB(z kn, η n ) B(t 1, η) B(t R, η) 0 ). (23)

23 3.7 Highest weight polynomial of a regular Bethe state We denote inhomogeneous parameters by ξ l for l = 1, 2,..., L, where L is the lattice size. We define functions ϕ ± ξ (x) by L ϕ ± ξ (x) = (1 xe ±2ξ l ). (24) l=1 Recall that t 1, t 2,..., t R are regular Bethe roots at a given value of q in the inhomogeneous case. We define F ± (x) by R F ± (x) = (1 x exp(±2t j )). (25) j=1

24 Let R be a regular Bethe state at q 0 in sector A in the inhomogeneous case and t 1, t 2,..., t R the regular Bethe roots. At q = q 0, we express F ± (x) as F ± (x) = R j=1 (1 x exp(±2 t j )). We consider the following series with respect to x: ϕ ± ξ (x) F ± (xq 0 ) F ± (xq0 1 ) = m=0 χ ± ξ,m xm for x < min{ e ±2 t j }. (26) Here, we define coefficients χ ± ξ,m by power series (26). Explicitly, we have ( ) min(l,m) χ ± ξ,m = ρ ( 1) ρ exp ± 2ξ jk ρ=0 n 1 + +n R =m ρ 1 j 1 < <j ρ L e ± R j=1 2n j t j R j=1 k=1 [n j + 1] q0, (27) where symbol n 1 + +n R =m ρ denotes the sum over all nonnegative integers n 1, n 2,..., n R satisfying n n R = m ρ. When R = 0, we set ρ = m.

25 Proposition 1. Let q 0 be a root of unity with q 2N 0 = 1. For a regular Bethe state R at q 0 in sector A in the inhomogeneous case, weight d 0 is given by r = (L 2R)/N, and eigenvalues λ k are given by λ k = ( 1) kn χ + ξ,kn, (28) We thus obtain the D-highest weight polynomial P λ (u) of R r P λ (u) = χ + ξ,kn uk (29) k=0 The result was originally for the spin-1/2 chain [D1]. However, by the fusion construction, we can show it for spin-s case, setting the inhomogenous parameters into complete 2s-strings (also valid for mixed spin cases).

26 Review: A new criterion for irreducible representations Theorem 1 (D2). A highest weight rep. is irreducible if and only if the following holds: s ( 1) s j µ s j x j Ω = 0, j=0 where µ k (k = 1, 2,..., s) are given by µ k = a i1 a ik. 1 i 1 < <i k s Note: Making use of theorem 1 we can show that there exists a redicible but indecomposable highest weight representation in the degenerate spectrum of the inhmogeneous spin-1/2 XXZ chain [D1].

27 Example When r = 3 and P λ (u) = (1 a 1 u) 2 (1 a 2 u), we have m 1 = 2 and m 2 = 1. In any highest weight representation, we have the following: ( x 3 (a 1 + a 1 + a 2 )x 2 + (a a 1 a 2 )x 1 a2 1a 2 x 0 ) Ω = 0 However, if we have ( x 2 (a 1 + a 2 )x 1 + a 1a 2 x 0 ) Ω = 0, then the highest weight representation generated by Ω is irreducible.

28 3.8 Loop algebra generators with parameters Let α denote a finite sequence of complex parameters such as α = (α 1, α 2,..., α n ). n x ± m(α) = ( 1) k x ± m k α i1 α i2 α ik, h m (α) = k=0 n ( 1) k h m k k=0 {i 1,...,i k } {1,...,n}; i 1 < <i k n {i 1,...,i k } {1,...,n}; i 1 < <i k n α i1 α i2 α ik. (30) Let α and β be arbitrary sequences of n and p parameters, respectively. In terms of generators with parameters we express the defining relations of the sl 2 loop algebra as follows: [x + l (α), x m(β)] = h l+m (αβ), [h l (α), x ± m(β)] = ±2x ± l+m (αβ). (31) Here the symbol αβ denotes the composite sequence of α and β: αβ = (α 1, α 2,..., α n, β 1, β 2,..., β p ). (32) For a given sequence of m parameters, α = (α 1, α 2,, α m ), let us denote by

29 α j the sequence of parameters of α other than α j : α j = (α 1,..., α j 1, α j+1,..., α m ). (33) We now introduce the following symbol: ρ ± j (α) = x± m 1 (α j) for j = 1, 2,..., m. (34) The generators x j ρ j (α). for j = 0, 1,..., m 1, are expressed as linear combinations of

30 In the proof of the new criterion [D2], we use the following: Let a denote a set of distinct parameters. a = (a 1, a 2,..., a s ) Proposition 2. If x s (a)ω = 0, we have ( ρ j (a) ) m j +1 Ω = 0. (35) Lemma 3. Suppose that x s (a)ω = 0. For an integer n 0, we take such nonnegative integers l j, k j for j = 1, 2,..., s satisfying l 1 + +l s = k k s = n. Then, we have s ( ρ + j (a) ) s (l j ) ( ρ j (a) ) ( ) s s (k j ) m j Ω = δ lj,k j k j j=1 j=1 Here δ j,k denotes the Kronecker delta. j=1 t=1;t j It can be use to calculate the norm of an eigenvector. We note a j,t = a j a t a 2k j j,t Ω. (36)

31 4 Connection to SCP model 4.1 The 1D transverse Ising model H = A 0 + ha 1 where A 0 and A 1 are given by L L A 0 = σj z σj+1 z, A 1 = j=1 j=1 The A 0 and A 1 in (37) satisfy the Dolan-Grady conditions: [A i, [A i, [A i, A 1 i ]]] = 16[A i, A 1 i ], (i = 0, 1) and generate the Onsager algebra (OA). [L. Onsager (1944)] [A l, A m ] = 4G l m [G l, A m ] = 2A m+l 2A m l [G l, G m ] = 0 σ x j

32 L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65 (1944) The Z N -symmetric Hamiltonian by von Gehlen and Rittenberg: H ZN = A 0 + k A 1 = 4 N L i=1 N 1 m=1 1 1 ω mz2m i + k 4 N L i=1 N 1 m=1 1 1 ω mx m i Xi+1 m (37) Here, q N = 1 and ω = q 2, and the operators Z i, X i End((C N ) L ) are defined by Z i (v σ1 v σi v σl ) = q σ i v σ1 v σi v σl, X i (v σ1 v σi v σl ) = v σ1 v σi +1 v σl, for the standard basis {v σ σ = 0, 1,, N 1} of C N and under the P.B.C.s: Z L+1 = Z 1 and X L+1 = X 1. The Hamiltonian H ZN is derived from the expansion of the SCP transfer matrix with respect to the spectral parameter.

33 The A 0 and A 1 in (37) satisfy the Dolan-Grady conditions: [A i, [A i, [A i, A 1 i ]]] = 16[A i, A 1 i ], (i = 0, 1) and generate the Onsager algebra (OA). [A l, A m ] = 4G l m [G l, A m ] = 2A m+l 2A m l [G l, G m ] = 0 For H ZN, the Hilbert space (C N ) L is decomposed into the direct sum of finitedimensional irreducible representations of OA. The energy spectra of H ZN has the Ising-like form [Baxter(1988), Davies (1990), Roan (1991)]: n E =α+k β+2 m i 1+2k cos θ i +k 2, m i { l i, l i +2,, l i }. (38) i=1 Here, α, β R and θ i R/2πR are that are determined by the functional relations

34 of the SCP transfer matrix given by Baxter (1993). We introduce P SCP (z N ) = ω p b N 1 j=0 (1 z N ) L (zω j ) p a p b (1 zω j ) L F CP (zω j )F CP (zω j+1 ), (39) and call it the superintegrable chiral Potts polynomial. (SCP polynomial). Here F CP (z) = R (1 + zu i ω) i=1 and {u i } satisfy the Bethe ansatz equations. By factorizing the polynomial P SCP (ζ) as n P SCP (ζ) = (1 ζi 1 ζ) l i i=1 with zeros ζ 1,..., ζ n C, the parameters {θ i } in (38) are determined through ζ i = tan 2 (θ i /2). The dimensions of the corresponding representation space of OA, V OA are given

35 by [Davies (1990), Roan (1991)]. n dimv OA = (l i + 1) All the known results imply l i = 1 for all i. It is thus suggested that dimv OA = 2 degp CP(ζ). i=1 4.3 The superintegrable τ 2 model The transfer matix of the superintegrable τ 2 model (Bazhanov-Stroganov, Korepanov) commutes with that of the superintegrable chiral Potts (SCP) model. The L-opetrator of the superintegrable τ 2 model is equivalent to that of the integrable spin-(n 1)/2 XXZ spin chain with twisted boundary conditions.

36 4.4 Loop algebra symmetry of the superintegrable τ 2 model Proposition 3. If q N = 1 and L is a multiple of N, the transfer matrix τ τ2 (z) has the L(sl 2 ) symmetry in the sector with S Z 0 (mod N). Proposition 4. If L and R are multiples of N, the Bethe state of the τ 2 model, R; {z i }, is highest weight. The D-highest weight polynomial P D (z) of R; {z i } is given by P D (z N ) = L(N 1) 2R N k=0 χ kn z kn = N N 1 j=0 (1 z N ) L (1 zq 2j ) L F (zq 2j 1 )F (zq 2j+1 ). (40) Proposition 5. The superintegrable chiral Potts (SCP) polynomial P SCP (ζ) (39) (with p a + p b = 0) is equivalent to the D-highest weight polynomial (generalized Drinfeld polynomial) P D (ζ) in (40).

37 Observations: (1) SCP and the D-highest weight polynomials have the same Bethe roots. (2) The dimensions are the same for the OA representation and the highest weight representation of the loop algebra of the τ 2 model. Conjecture 1. The OA representation space V OA should correspond to the highest weight representation generated by the regular Bethe states R, {z i } of the τ 2 model. Corollary 1. The eigenvectors of the Z N symmetric Hamiltonian, V are given by V = (some element of L(sl 2 )) B(z 1 ) B(z R ) 0 Here B(z 1 ) B(z R ) 0 denotes a regular Bethe state of the τ 2 model and generates V loop. Proposition 6. Conjecture 1 is valid in the sector S Z 0 (mod N). [ND2008]

38 5 N = 2 case The Hamiltonians of the SCP model and the τ 2 -model are given in the forms L L L H SCP = σi z + λ σi x σi+1, x H τ2 = (σi x σ y i+1 σy i σx i+1), i=1 i=1 where σ x, σ y and σ z are Pauli s matrices. In terms of fermion operators: i 1 c i = σ i + σj z, c k = 1 L e ı(ki+π 4 ) c i, L j=1 i=1 i=1 the Hamiltonian H τ2 in the sector with S z 1 2 L i=1 σz i 0 mod 2 is written as H τ2 = k K sin(k) c k c k, where K = { π L, 3π L,..., (L 1)π L }. For even L, the generators of the sl 2 loop algebra symmetry of the τ 2 -model is given by h n = k K cot 2n ( k 2 ) (H) k, x + n = k K cot 2n+1 ( k 2 ) (E) k, x n = k K cot 2n 1 ( k 2 ) (F ) k,

39 where {(H) k, (E) k, (F ) k } is a two-dimensional representation of the sl 2 algebra given by (H) k = 1 c k c k c k c k, (E) k = c k c k, (F ) k = c k c k. The reference state 0, which is a highest weight vector, i.e. x + n 0 = 0 and h n 0 = k cot2n (k/2) 0, generates a 2 L/2 -dimensional irreducible representation corresponding to a degenerate eigenspace of the Hamiltonian H τ2. On the other hand, in the sector, the Hamiltonian H SCP is expressed as H SCP = 2 (H) k + 2λ ( cos(k)(h)k + sin(k) ( )) (E) k + (F ) k. k K k K It is clear that the Hamiltonian H SCP acts on the 2 L/2 -dimensional irreducible representation space. The 2 L/2 eigenvalues of H SCP and the corresponding eigenstates

40 are given by E(K + ; K ) = 2 K + ; K = k K + 1 2λ cos(k) + λ2 2 k K + (cos θ k + sin θ k (F ) k ) k K 1 2λ cos(k) + λ2, k K (sin θ k cos θ k (F ) k ) 0, where K + and K are such disjoint subsets of K that K = K + K and tan(2θ k ) = 2λ sin(k) 2(λ cos(k) 1). We remark: ρ + k = (E) k, ρ k = (F ) k,

41 6 Towards Form factors of SCP model 6.1 Norms Let us consider the monodromy matrix of the superintegrable tau 2 on L lattice sites. We embed the system in the spin-1/2 spin chain of (N 1)L lattice sites (with some twisted boundary conditions). Case of r distinct a j s (m 1 = m 2 = = m r = 1) In sector A we have Ψ = (α 1 + β 1 ρ 1 ) R j=1 B(λ j ) 0 (41) R Ψ = 0 C(λ j ) (α 1 + β 1 ρ + 1 ) (42) j=1

42 Ψ Ψ = = ( α1 2 + β1 2 ( α1 2 + β1 2 ) r R R (a 1 a t ) 2 0 C(λ j ) B(λ j ) 0 t=2 r t=2 a 2 1t ) j=1 j=1 det( Gaudin matrix of spin (N 1)/2 ) (43)

43 We thus have r R Ψ = (α j + β j ρ j ) B(λ k ) 0 (44) j=1 k=1 j=1 k=1 R r Ψ = 0 C(λ k ) (α j + β j ρ + j ) (45) Ψ Ψ = = r α j 2 + βj 2 j=1 r α j 2 + βj 2 j=1 r R R (a j a t ) 2 0 C(λ k ) B(λ j ) 0 t=1;t j r t=1;t j k=1 j=1 (a j a t ) 2 det( Gaudin matrix of spin (N 1)/2 ) (46)

44 6.2 QISP for the SCP model and application to Form factors We consider the spin-1/2 chain of (N 1)L lattice sites. We set l = N 1. For m = n (0 m N 1), upto gauge transformations, we have = n, n (N 1 +) Ẽi ( l n ) P (l) 1 (N 1)L l k=n+1 (i 1)l α=1 e φ A (l +) (w (i 1)l+k ) (e φ A (l +) + e φ D (l +) )(w ll α=il+1 (l +) α ) n k=1 (e φ A (l +) + e φ D (l +) )(w e φ D (l +) (l +) (w(i 1)l+k ) (l +) α ) P (l) 1 L(N 1). 6.3 Example We employ the same method for calculating the form factors of the higher-spin XXZ spin chain. Consider the case of N = 3, q 3 = 1 and ω = q 2

45 For i = 1,..., L, we have Z i = Ei 00 + ωei 11 + ω 2 Ei 22 2(i 1) = + ω P (2) 1 2L 2 k=1 P (2) 1 2L α=1 (e φ A (2 +) + e φ D (2 +) )(w e φ A (2 +) (2 +) (w 2(i 1)+k ) 2(i 1) α=1 e φ A (2 +) (2 +) (w 2(i 1)+2 ) + ω 2 P (2) 1 2L 2L α=2i+1 2(i 1) α=1 2L α=2i+1 (2 +) α ) (e φ A (2 +) + e φ D (2 +) (2 +) )(w (2 +) (e φ A (2 +) + e φ D (2 +) )(w α ) e φ D (2 +) (w 2L α=2i+1 (e φ A (2 +) + e φ D (2 +) )(w (e φ A (2 +) + e φ D (2 +) (2 +) )(w (e φ A (2 +) + e φ D (2 +) (2 +) )(w α ) (2 +) α ) 2 k=1 α ) (2 +) 2(i 1)+1 ) α ) P (2) 1 2L P (2) 1 2L e φ D (2 +) (2 +) (w 2(i 1)+k ) P (2) 1 2L (47)

46 7 Summary: Loop algebra approach to Correlation functions of SCP model (1) The infinite-dimenisonal symmetry U(L(sl 2 )) appears at roots of unity, q 2N = 1. In some sectors, we derive the spectral degeneracy of the loop algebra rigorously. (2) Highest-weight representation theory of the sl 2 loop algebra (3) A connection to the SCP model: The eigenvectors of the Z N symmetric Hamiltonian, V (more generally, those of the SCP model) are given by V = (some element of L(sl 2 )) B(z 1 ) B(z R ) 0 Here B(z 1 ) B(z R ) 0 denotes a regular Bethe state of the τ 2 model. (4) Quantum Inverse Scattering of local operators for the H ZN (and the SCP model) (5) In sector A, we can calculate the norm of a given eigenvectors. It is possible to calculate formfactors and correlation functions for finite chains.