The sl(2) loop algebra symmetry of the XXZ spin chainat roots of unity and applications to the

superintegrable chiral Potts model 1

Tetsuo Deguchi

Department of Physics, Ochanomizu Univ.

In collaboration with Akinori Nishino.

1Talk given at the Simon Center, Stony Brook, NY, Jan. 18, 2010

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 sl2 loop algebra in the

sector SZ ≡ (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

Main Reference of this talk

[D1] T. D., Regular XXZ Bethe states at roots of unity as highest weight vectors

of the sl2 loop algebra, J. Phys. A: Math. Theor. Vol. 40 (2007) 7473–7508

[D2] T.D., J. Stat. Mech. (2007) P05007

[ND2008] A. Nishino and T. D., An algebraic derivation of the eigenspaces as-

sociated with an Ising-like spectrum of the superintegrable chiral Potts model J.

Stat. Phys. 133 (2008) pp. 587–615

[DM] T. D. and C. Matsui, Correlation functions of the integrable higher-spin

XXX and XXZ spin chains through the fusion method (arXiv:0907.0582v3),

to appear in Nucl. Phys. B

[DFM] T. D., K. Fabricius and B. M. McCoy,

The sl2 loop algebra symmetry of the six-vertex model at roots of unity,

J. Stat. Phys. 102 (2001) 701-736.

Many interesting papers relevant to this talk

[AY] H. Au-Yang and J. Perk, J. Phys. A 41 (2008) 275201; J. Phys. A 42 (2009)

375208; J. Phys. A 43 (2010) 025203; arXiv:0907.0362

[B] R. Baxter, arXiv:0906.3551; arXiv:0912.4549; arXiv:1001.0281

[vG] N. Iorgov, V. Shadura, Yu. Tykhyy, S. Pakuliak, and G. von Gehlen, arXiv:0912.5027

[FM2010] K. Fabricius and B.M. McCoy, arXiv:1001.0614

1 The sl2 loop algebra symmetry of the XXZ spin chain (P.B.C.)

HXXZ =1

2

L∑j=1

(σXj σ

Xj+1 + σ

Yj σ

Yj+1 + ∆σ

Zj σ

Zj+1

).

When q is a root of unity: q2N = 1 where ∆ = (q + q−1)/2,

the HXXZ (and τ6V(z)) commutes with the generators of the sl2 loop algebra,U(L(sl2)) in sector A: S

Z ≡ 0 (mod N) (and in sector B: SZ ≡ N/2 (mod N)):

S±(N) = (S±)N/[N ]! , T±(N) = (T±)N/[N ]!

Here S± and T± are generators of Uq(ŝl2) in the Lth tensor product of the spin-1/2

representations

S± =L∑

j=1

qσZ/2 ⊗ · · · ⊗ qσZ/2 ⊗ σ±j ⊗ q−σ

Z/2 ⊗ · · · ⊗ q−σZ/2

T± =

L∑j=1

q−σZ/2 ⊗ · · · ⊗ q−σZ/2 ⊗ σ±j ⊗ qσ

Z/2 ⊗ · · · ⊗ qσZ/2

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) 701-736.

1.1 S±(N) and T±(N) generate the sl2 loop algebra U(L(sl2))

(the sl2 affine Kac-Moody algebra with c = 0)

Let us introduce the Chevalley generators as follows:

E+0 = T−(N), E−0 = T

+(N), E+1 = S+(N), E−1 = S

−(N),

−H0 = H1 =2

NSZ .

E±j , Hj for j = 0, 1, satisfy the defining relations of the sl2 loop algebra U(L(sl2)):

H0 + H1 = 0, [Hi, E±j ] = ±aijE±j , (i, j = 0, 1)

[E+i , E−j ] = δijHj , (i, j = 0, 1)

[E±i , [E±i , [E

±i , E

±j ]]] = 0, (i, j = 0, 1, i ̸= j)

Here, the Cartan matrix (aij) of A(1)1 is defined by(

a00 a01

a10 a11

)=

(2 −2−2 2

)Ref: G. Lusztig, Contemp. Math. 82 (1989) pp. 59–77.

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 N2L/N when L is a multipleof N . (the XXZ spin chain)

(2) A regular Bethe vector |R⟩ of R down spins with SZ ≡ 0 (mod N) has thedegeneracy 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)

(3) The XYZ spin chain at roots of unity has the same spectral degeneracies

such as the sl2 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)

HXY Z =1

2

L∑j=1

(JXσ

Xj σ

Xj+1 + JY σ

Yj σ

Yj+1 + JZσ

Zj σ

Zj+1

).

A regular XYZ Bethe vector |R⟩ with R ’down spins’ has the spectral degeneracyof 2(L−2R)/N if L/N is an integer T.D., J. Phys. A 35 (2002) 879-895.)

2 Representation theory

2.1 The sl2 loop algebra and D-highest weight vectors

Generators hk and x±k (k ∈ Z) satisfy the defining relations of the sl2 loop algebra

(j, k ∈ Z)[hj, x

±k ] = ±2x

±j+k , [x

+j , x

−k ] = hj+k

[hj, hk] = 0 , [x±j , x

±k ] = 0

Correspondence between the Chevalley generators and hk, x±k :

E±1 → x±0 , E+0 → x−1 , E−0 → x+1 , −H0 = H1 → h0 .

We thus have

x+0 = S+(N) , x−0 = S

−(N) , x+−1 = T+(N) , x−1 = T

−(N) , h0 =2

NSZ .

Definition of D-highest weight vectors (Drinfeld-highest weight)

We call a representation of U(L(sl2)) 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, hk’s:

hkΩ = dkΩ , for k ∈ Z.

Here dk denotes the eigenvalues of hk’s.

We call Ω a highest weight vector.

2.2 D-highest weight polynomials

Definition 1 (D-highest weight polynomial). Let λk be eigenvalues as follows:

(x+0 )k(x−1 )

k/(k!)2 Ω = λkΩ (k = 1, 2, . . . , r)

For the sequence λ = (λ1, λ2, . . . , λr), we define polynomial Pλ(u) by

Pλ(u) =r∑

k=0

λk (−u)k , (1)

We call Pλ(u) the D-highest weight polynomial for highest weight dk.

When U(L(sl2))Ω is irreducible, we call it the Drinfeld polynomial of the repre-

sentation.

Definition 2 (D-highest weight parameters). We define parameters ak by

Pλ(u) =s∏

k=1

(1 − aku)mk . (2)

Here a1, a2, . . . , as are distinct, and their multiplicities are given by m1,m2, . . . ,ms,

respectively, where we have r = m1 + · · · + ms. We call parameters aj the D-highest weight parameters of Ω.

If U(L(sl2))Ω is irreducible, the dimensionality is given by

dimU(L(sl2))Ω =

s∏j=1

(mj + 1).

Example

For r = 3, if we have

Pλ(u) = (1 − a1u)2(1 − a2u) ,

we have m1 = 2 and m2 = 1. If the h.w. rep. is irreducible, then we have

dimU(L(sl2))Ω = (2 + 1)(1 + 1) = 6.

2.3 R-matrix and monodromy matrix

We define the R-matrix acting on the tensor product V ⊗ V by

R(u) =∑

a,b,c,d=1,2

Rabcd(u)ea,c ⊗ eb,d . (3)

We denote it by

R12(λ1, λ2) =

1 0 0 0

0 b(u) c(u) 0

0 c(u) b(u) 0

0 0 0 1

[1,2]

. (4)

Here functions b(u) and c(u) are given by

b(u) =sinh(u)

sinh(u + η), c(u) =

sinh(η)

sinh(u + η).

The R-matrices satisfy the Yang-Baxter equations:

R12(λ1, λ2)R13(λ1, λ3)R23(λ2, λ3) = R23(λ2, λ3)R13(λ1, λ3)R12(λ1, λ2) (5)

We define L-operators acting on the mth site for m = 1, 2, . . . , L, by

Lm(λ, ξm) = R0m(λ, ξm) . (6)

We define the monodromy matrix acting on the L lattice-sites in one dimension by

T0,12···L(λ; ξ1, . . . , ξL) = LL(λ, ξL)LL−1(λ, ξL−1) · · ·L2(λ, ξ2)L1(λ, ξ1) . (7)

We express the matrix elements of the monodromy matrix by

T0,12···L(u; ξ1, . . . , ξL) =

(A(u) B(u)

C(u) D(u)

)[0]

(8)

The transfer matrix, t(u), is given by the trace over the 0th space:

t(u; ξ1, . . . , ξL) = tr0 (T0,12···L(u; ξ1, . . . , ξL))

= A(u) + D(u) . (9)

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 sl2 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) 119–144.

It was shown in sector A: SZ ≡ 0 (mod N) when q2N = 1

• Every regular Bethe ansatz eigenvector is highest weight (shown by the algebraicBethe ansatz for the inhomogeneous 6V transfer matrix.) [D1]

• The degenerate eigenspace is mostly given by a Weyl module, which is eitheran irreducible or a reducible and indecomposable rep.

3.2 Quantum group generators through infinite rapidities

Let us define function nξ(z) by

nξ(z) =

L∏j=1

sinh(z − ξj) , (10)

and function ĝ(z) by

ĝ(z) =

{2 exp(−z) sinh 2η (Re z > 0) ,−2 exp(z) sinh 2η (Re z < 0) .

(11)

We normalize operators A(z), . . . , D(z) as follows:

Â(z) = A(z)/nξ(z) , B̂(z) = B(z)/(ĝ(z)nξ(z)) ,

D̂(z) = D(z)/nξ(z) , Ĉ(z) = C(z)/(ĝ(z)nξ(z)) . (12)

Taking the limit of infinite rapidities for the inhomogeneous case, we have

Â(±∞) = q±SZ , B̂(∞) = T− , B̂(−∞) = S− ,D̂(±∞) = SZ , Ĉ(∞) = S+ , Ĉ(−∞) = T+ , (13)

3.3 Complete N-strings

Let N be a positive integer.

Definition 3 (Complete N -string). We call a set of rapidities zj a complete

N-string, if they satisfy

zj = 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)

3.4 Sufficient conditions of a highest weight vector

Lemma 1. Suppose that x±0 , x+−1, x

−1 and h0 satisfy the defining relations of

U(L(sl2)), and x±k and hk (k ∈ Z) are generated from them. If a vector |Φ⟩

satisfies the following:

x+0 |Φ⟩ = x+−1|Φ⟩ = 0 , (15)h0|Φ⟩ = r|Φ⟩ , (16)(x+0 )

(n)(x−1 )(n)|Φ⟩ = λn |Φ⟩ for n = 1, 2, . . . , r , (17)

where r is a nonnegative integer and λn are complex numbers. Then |Φ⟩ ishighest weight, i.e. we have

x+k |Φ⟩ = 0 (k ∈ Z) , (18)hk|Φ⟩ = dk|Φ⟩ (k ∈ Z) , (19)

where dk are complex numbers.

3.5 Annihilation property

We construct operators S+(N)ξ and T

+(N)ξ as follows:

S+(N)ξ = limq→q0

1

[N ]q!(Ĉ(∞, η))N , T+(N)ξ = limq→q0

1

[N ]q!(Ĉ(−∞, η))N . (20)

We evaluate the action of the operator S+(N)ξ on the vector |R⟩ by the limiting

procedure:

S+(N)ξ |R⟩ = limq→q0

{1

[N ]q!

(Ĉ(∞, η)

)NB(t1, η) · · ·B(tR, η) |0⟩

}. (21)

R is the number of regular Bethe roots. We denote by ΣR = {1, 2, . . . , R} theset of indices of the regular Bethe roots, t1, t2, . . . , tR.

Lemma 2. Let t1, t2, . . . , tR be regular Bethe roots at generic q. For a given

positive integer Nc, we have

1

[Nc]q!

(Ĉ(±∞)

)NcB(t1)B(t2) · · ·B(tR) |0⟩

=

|SNc|=Nc∑SNc⊆ΣR

∏ℓ∈ΣR\SNc

B(tℓ) |0⟩ exp(±∑

j∈SNc

tj)∏

j∈SNc

a6Vξ (tj) ∏k∈ΣR\SNc

f(tj − tk)

× (−1)Nc (q±1 − q∓1)Nc ×

Nc−1∏ℓ=0

[L

2− R + Nc − ℓ]q . (22)

3.6 Diagonal property

In addition to regular Bethe roots at generic q, t1, t2, . . . , tR, we introduce kN

rapidities, z1, z2, . . . , zkN , forming a complete kN -string: zj = Λ+(kN +1− 2j)ηfor j = 1, 2, . . . , kN .

We calculate the action of (S+(N)ξ )

k(T−(N)ξ )

k on the Bethe state at q0,

|R⟩ = B(t̃1) · · ·B(t̃R)|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

B̂(z1, η) · · · B̂(zkN , ηn) B(t1, η) · · ·B(tR, η) |0⟩)

. (23)

3.7 Highest weight polynomial of a regular Bethe state

We denote inhomogeneous parameters by ξℓ for ℓ = 1, 2, . . . , L, where L is the

lattice size. We define functions ϕ±ξ (x) by

ϕ±ξ (x) =

L∏ℓ=1

(1 − xe±2ξℓ) . (24)

Recall that t1, t2, . . . , tR are regular Bethe roots at a given value of q in the inho-

mogeneous case. We define F±(x) by

F±(x) =

R∏j=1

(1 − x exp(±2tj)) . (25)

Let |R⟩ be a regular Bethe state at q0 in sector A in the inhomogeneous case andt̃1, t̃2, . . . , t̃R the regular Bethe roots. At q = q0, we express F

±(x) as F̃±(x) =∏Rj=1(1 − x exp(±2t̃j)). We consider the following series with respect to x:

ϕ±ξ (x)

F̃±(xq0)F̃±(xq−10 )

=

∞∑m=0

χ̃±ξ,m xm for |x| < min{|e±2t̃j|}. (26)

Here, we define coefficients χ̃±ξ,m by power series (26). Explicitly, we have

χ̃±ξ,m =

min(L,m)∑ρ=0

(−1)ρ∑

1≤j1

Proposition 1. Let q0 be a root of unity with q2N0 = 1. For a regular Bethe

state |R⟩ at q0 in sector A in the inhomogeneous case, weight d0 is given byr = (L − 2R)/N , and eigenvalues λk are given by

λk = (−1)kN χ̃+ξ,kN , (28)

We thus obtain the D-highest weight polynomial Pλ(u) of |R⟩

Pλ(u) =r∑

k=0

χ̃+ξ,kN uk (29)

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).

Review: A new criterion for irreducible representations

Theorem 1 (D2). A highest weight rep. is irreducible if and only if the fol-

lowing holds:s∑

j=0

(−1)s−jµs−jx−j Ω = 0 ,

where µk (k = 1, 2, . . . , s) are given by

µk =∑

1≤i1

Example

When r = 3 and Pλ(u) = (1 − a1u)2(1 − a2u), we have m1 = 2 and m2 = 1.In any highest weight representation, we have the following:(

x−3 − (a1 + a1 + a2)x−2 + (a21 + 2a1a2)x−1 − a21a2x−0)

Ω = 0

However, if we have (x−2 − (a1 + a2)x−1 + a1a2x−0

)Ω = 0 ,

then the highest weight representation generated by Ω is irreducible.

3.8 Loop algebra generators with parameters

Let α denote a finite sequence of complex parameters such as α = (α1, α2, . . . , αn).

x±m(α) =

n∑k=0

(−1)kx±m−k∑

{i1,...,ik}⊂{1,...,n}; i1

αj the sequence of parameters of α other than αj:

αj = (α1, . . . , αj−1, αj+1, . . . , αm). (33)

We now introduce the following symbol:

ρ±j (α) = x±m−1(αj) for j = 1, 2, . . . ,m. (34)

The generators x−j for j = 0, 1, . . . ,m− 1, are expressed as linear combinations ofρj(α).

In the proof of the new criterion [D2], we use the following:

Let a denote a set of distinct parameters.

a = (a1, a2, . . . , as)

Proposition 2. If x−s (a)Ω = 0, we have(ρ−j (a)

)mj+1 Ω = 0 . (35)Lemma 3. Suppose that x−s (a)Ω = 0. For an integer n ≥ 0, we take suchnonnegative integers ℓj, kj for j = 1, 2, . . . , s satisfying ℓ1 + · · ·+ ℓs = k1 + · · ·+ks = n. Then, we have

s∏j=1

(ρ+j (a)

)(ℓj) × s∏j=1

(ρ−j (a)

)(kj) Ω = s∏j=1

δℓj,kj(

mj

kj

)s∏

t=1;t ̸=j

a2kjj,t

Ω . (36)Here δj,k denotes the Kronecker delta.

It can be use to calculate the norm of an eigenvector. We note

aj,t = aj − at

4 Connection to SCP model

4.1 The 1D transverse Ising model

H = A0 + hA1where A0 and A1 are given by

A0 =

L∑j=1

σzjσzj+1 , A1 =

L∑j=1

σxj

The A0 and A1 in (37) satisfy the Dolan-Grady conditions:

[Ai, [Ai, [Ai, A1−i]]] = 16[Ai, A1−i], (i = 0, 1)

and generate the Onsager algebra (OA). [L. Onsager (1944)]

[Aℓ, Am] = 4Gℓ−m

[Gℓ, Am] = 2Am+ℓ − 2Am−ℓ[Gℓ, Gm] = 0

L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder

Transition, Phys. Rev. 65 (1944) 117 – 149.

4.2 The ZN-symmetric Hamiltonian by von Gehlen and Rittenberg:

HZN = A0 + k′A1 =

4

N

L∑i=1

N−1∑m=1

1

1 − ω−mZ2mi + k

′ 4

N

L∑i=1

N−1∑m=1

1

1 − ω−mX−mi X

mi+1

(37)

Here, qN = 1 and ω = q2, and the operators Zi, Xi ∈ End((CN)⊗L) are defined by

Zi(vσ1 ⊗ · · · ⊗ vσi ⊗ · · · ⊗ vσL) = qσivσ1 ⊗ · · · ⊗ vσi ⊗ · · · ⊗ vσL,

Xi(vσ1 ⊗ · · · ⊗ vσi ⊗ · · · ⊗ vσL) = vσ1 ⊗ · · · ⊗ vσi+1 ⊗ · · · ⊗ vσL,

for the standard basis {vσ|σ = 0, 1, · · · , N − 1} of CN and under the P.B.C.s:ZL+1 = Z1 and XL+1 = X1.

The Hamiltonian HZN is derived from the expansion of the SCP transfer matrix

with respect to the spectral parameter.

The A0 and A1 in (37) satisfy the Dolan-Grady conditions:

[Ai, [Ai, [Ai, A1−i]]] = 16[Ai, A1−i], (i = 0, 1)

and generate the Onsager algebra (OA).

[Aℓ, Am] = 4Gℓ−m

[Gℓ, Am] = 2Am+ℓ − 2Am−ℓ[Gℓ, Gm] = 0

For HZN, the Hilbert space (CN)⊗L is decomposed into the direct sum of finite-dimensional irreducible representations of OA.

The energy spectra of HZN has the Ising-like form [Baxter(1988), Davies (1990),

Roan (1991)]:

E =α+k′β+2

n∑i=1

mi√

1+2k′ cos θi+k′2, mi ∈ {−li,−li+2, · · · , li}. (38)

Here, α, β ∈ R and θi ∈ R/2πR are that are determined by the functional relations

of the SCP transfer matrix given by Baxter (1993). We introduce

PSCP(zN) = ω−pb

N−1∑j=0

(1 − zN)L(zωj)−pa−pb(1 − zωj)LFCP(zωj)FCP(zωj+1)

, (39)

and call it the superintegrable chiral Potts polynomial. (SCP polynomial).

Here

FCP(z) =R∏

i=1

(1 + zuiω)

and {ui} satisfy the Bethe ansatz equations.By factorizing the polynomial PSCP(ζ) as

PSCP(ζ) =

n∏i=1

(1 − ζ−1i ζ)li

with zeros ζ1, . . . , ζn ∈ C, the parameters {θi} in (38) are determined throughζi = − tan2(θi/2).

The dimensions of the corresponding representation space of OA, VOA are given

by [Davies (1990), Roan (1991)].

dimVOA =

n∏i=1

(li + 1)

All the known results imply li = 1 for all i. It is thus suggested that

dimVOA = 2degPCP(ζ)

.

4.3 The superintegrable τ2 model

The transfer matix of the superintegrable τ2 model (Bazhanov-Stroganov, Ko-

repanov) commutes with that of the superintegrable chiral Potts (SCP) model.

The L-opetrator of the superintegrable τ2 model is equivalent to that of the

integrable spin-(N − 1)/2 XXZ spin chain with twisted boundary conditions.

4.4 Loop algebra symmetry of the superintegrable τ2 model

Proposition 3. If qN = 1 and L is a multiple of N , the transfer matrix ττ2(z)

has the L(sl2) symmetry in the sector with SZ ≡ 0 (mod N).

Proposition 4. If L and R are multiples of N , the Bethe state of the τ2

model, |R; {zi}⟩, is highest weight. The D-highest weight polynomial PD(z) of|R; {zi}⟩ is given by

PD(zN) =

L(N−1)−2RN∑

k=0

χkNzkN = N

N−1∑j=0

(1 − zN)L

(1 − zq2j)LF (zq2j−1)F (zq2j+1). (40)

Proposition 5. The superintegrable chiral Potts (SCP) polynomial PSCP(ζ)

(39) (with pa + pb = 0) is equivalent to the D-highest weight polynomial (gen-

eralized Drinfeld polynomial) PD(ζ) in (40).

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 τ2 model.

Conjecture 1. The OA representation space VOA should correspond to the

highest weight representation generated by the regular Bethe states |R, {zi}⟩ ofthe τ2 model.

Corollary 1. The eigenvectors of the ZN symmetric Hamiltonian, |V ⟩ aregiven by

|V ⟩ = (some element of L(sl2)) × B(z1) · · ·B(zR)|0⟩

Here B(z1) · · ·B(zR)|0⟩ denotes a regular Bethe state of the τ2 model and gen-erates Vloop.

Proposition 6. Conjecture 1 is valid in the sector SZ ≡ 0 (mod N). [ND2008]

5 N = 2 case

The Hamiltonians of the SCP model and the τ2-model are given in the forms

HSCP =

L∑i=1

σzi + λ

L∑i=1

σxi σxi+1, Hτ2 =

L∑i=1

(σxi σyi+1 − σ

yi σ

xi+1),

where σx, σy and σz are Pauli’s matrices. In terms of fermion operators:

ci = σ+i

i−1∏j=1

σzj , c̃k =1

L

L∑i=1

e−ı(ki+π4 )ci,

the Hamiltonian Hτ2 in the sector with Sz ≡ 12

∑Li=1 σ

zi ≡ 0 mod 2 is written as

Hτ2 =∑k∈K

sin(k)c̃†kc̃k,

where K = {πL,3πL , . . . ,

(L−1)πL }. For even L, the generators of the sl2 loop algebra

symmetry of the τ2-model is given by

hn =∑k∈K

cot2n(k

2

)(H)k, x

+n =

∑k∈K

cot2n+1(k

2

)(E)k, x

−n =

∑k∈K

cot2n−1(k

2

)(F )k,

where {(H)k, (E)k, (F )k} is a two-dimensional representation of the sl2 algebragiven by

(H)k = 1 − c̃†kc̃k − c̃†−kc̃−k, (E)k = c̃−kc̃k, (F )k = c̃

†kc̃

†−k.

The reference state |0⟩, which is a highest weight vector, i.e. x+n |0⟩ = 0 andhn|0⟩ =

∑k cot

2n(k/2)|0⟩, generates a 2L/2-dimensional irreducible representationcorresponding to a degenerate eigenspace of the Hamiltonian Hτ2. On the other

hand, in the sector, the Hamiltonian HSCP is expressed as

HSCP = 2∑k∈K

(H)k + 2λ∑k∈K

(cos(k)(H)k + sin(k)

((E)k + (F )k

)).

It is clear that the Hamiltonian HSCP acts on the 2L/2-dimensional irreducible rep-

resentation space. The 2L/2 eigenvalues of HSCP and the corresponding eigenstates

are given by

E(K+; K−) = 2∑k∈K+

√1 − 2λ cos(k) + λ2 − 2

∑k∈K−

√1 − 2λ cos(k) + λ2,

|K+; K−⟩ =∏

k∈K+

(cos θk + sin θk(F )k)∏

k∈K−

(sin θk − cos θk(F )k)|0⟩,

where K+ and K− are such disjoint subsets of K that K = K+∪K− and tan(2θk) =2λ sin(k)

2(λ cos(k)−1).

We remark:

ρ+k = (E)k, ρ−k = (F )k,

6 Towards Form factors of SCP model

6.1 Norms

Let us consider the monodromy matrix of the superintegrable tau2 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 aj s (m1 = m2 = · · · = mr = 1) In sector A we have

|Ψ⟩ = (α1 + β1ρ−1 ) ×R∏

j=1

B(λj)|0⟩ (41)

⟨Ψ| = ⟨0|R∏

j=1

C(λj) × (α1 + β1ρ+1 ) (42)

⟨Ψ|Ψ⟩ =

(α21 + β

21

r∏t=2

(a1 − at)2)⟨0|

R∏j=1

C(λj)

R∏j=1

B(λj)|0⟩

=

(α21 + β

21

r∏t=2

a21t

)det( Gaudin matrix of spin − (N − 1)/2 ) (43)

|Ψ⟩ =r∏

j=1

(αj + βjρ−j )

R∏k=1

B(λk)|0⟩ (44)

⟨Ψ| = ⟨0|R∏

k=1

C(λk)r∏

j=1

(αj + βjρ+j ) (45)

We thus have

⟨Ψ|Ψ⟩ =r∏

j=1

α2j + β2j r∏t=1;t ̸=j

(aj − at)2 ⟨0| R∏

k=1

C(λk)

R∏j=1

B(λj)|0⟩

=

r∏j=1

α2j + β2j r∏t=1;t ̸=j

(aj − at)2 det( Gaudin matrix of spin − (N − 1)/2 )

(46)

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

Ẽn, n (N−1 +)i

=

(ℓ

n

)P̃

(ℓ)1···(N−1)L

(i−1)ℓ∏α=1

(eφA(ℓ +) + e−φD(ℓ +))(w(ℓ +)α )

n∏k=1

e−φD(ℓ +)(w(ℓ +)(i−1)ℓ+k)

×ℓ∏

k=n+1

eφA(ℓ +)(w(i−1)ℓ+k)

ℓL∏α=iℓ+1

(eφA(ℓ +) + e−φD(ℓ +))(w(ℓ +)α ) P̃(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, q3 = 1 and ω = q2

For i = 1, . . . , L, we have

Zi = E00i + ωE

11i + ω

2E22i

= P̃(2)1···2L

2(i−1)∏α=1

(eφA(2 +) + e−φD(2 +))(w(2 +)α )

×2∏

k=1

eφA(2 +)(w(2 +)2(i−1)+k)

2L∏α=2i+1

(eφA(2 +) + e−φD(2 +))(w(2 +)α ) P̃(2)1···2L

+ ω P̃(2)1···2L

2(i−1)∏α=1

(eφA(2 +) + e−φD(2 +))(w(2 +)α ) e−φD(2 +)(w

(2 +)2(i−1)+1)

× eφA(2 +)(w(2 +)2(i−1)+2)2L∏

α=2i+1

(eφA(2 +) + e−φD(2 +))(w(2 +)α ) P̃(2)1···2L

+ ω2 P̃(2)1···2L

2(i−1)∏α=1

(eφA(2 +) + e−φD(2 +))(w(2 +)α )

2∏k=1

e−φD(2 +)(w(2 +)2(i−1)+k)

×2L∏

α=2i+1

(eφA(2 +) + e−φD(2 +))(w(2 +)α ) P̃(2)1···2L (47)

7 Summary: Loop algebra approach to Correlation functions of

SCP model

(1) The infinite-dimenisonal symmetry U(L(sl2)) appears at roots of unity, q2N = 1.

In some sectors, we derive the spectral degeneracy of the loop algebra rigorously.

(2) Highest-weight representation theory of the sl2 loop algebra

(3) A connection to the SCP model:

The eigenvectors of the ZN symmetric Hamiltonian, |V ⟩ (more generally, thoseof the SCP model) are given by

|V ⟩ = (some element of L(sl2)) × B(z1) · · ·B(zR)|0⟩

Here B(z1) · · ·B(zR)|0⟩ denotes a regular Bethe state of the τ2 model.

(4) Quantum Inverse Scattering of local operators for the HZN (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.