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arXiv:cond-mat/9505049v1 11 May 1995

[M

0

PUPT 1536

Exact solution of A-D Temperley-Lieb Models

¨ R. KOBERLE1??

and

A. LIMA - SANTOS2?

1

Dept. of Physics, Princeton University, Princeton, NJ 08544

and

NEC Research Institute, 4 Independence Way, Princeton, NJ 08540

e-mail: roland@puhep1.princeton.edu

2

Departamento de F? ?sica, Universidade Federal de S?o Carlos a Caixa Postal 676, 13569-905 S?o Carlos, Brasil a PACS numbers: 75.10.Jm, 05.50.+q, 64.60.Cn

Abstract

We solve for the spectrum of quantum spin chains based on representations of the Temperley-Lieb algebra associated with the quantum groups Uq (Xn ) for Xn = A1 , Bn , Cn and Dn . We employ a generalization of the coordinate Bethe-Ansatz developed previously for the deformed biquadratic spin one chain. As expected, all these models have equivalent spectra, i.e. they di?er only in the degeneracy of their eigenvalues. This is true for ?nite length and open boundary conditions. For periodic boundary conditions the spectra of the lower dimensional representations are containded entirely in the higher dimensional ones. The Bethe states are highest weight states of the quantum group, except for some states with energy zero.

?

On leave of abscence from Instituto de F? ?sica de S?o Carlos, Universidade de S?o Paulo, a a

Caixa Postal 369, S?o Carlos 13560-970 , Brasil. a

?

Supported in part by CNPq-BRASIL.

1

Introduction

The recent interplay between the ?eld of solvable two-dimensional lattice ( or quantum spin chain ) models and quantum groups, has generated a lot of interesting results. One particular way of building models, which are quantum group invariant, uses the Temperley-Lieb (TL) algebra[1], satis?ed by the Hamitonian density Uk :

2 Uk =

QUk , Uk Uk±1 Uk = Uk , (1.1)

[Uk , Ul ] = 0, |k ? l| ≥ 2. The Hamiltonian is now given by the following sum over N sites:

N

H(q) =

k=1

Uk .

(1.2)

The paper is organized as follows. In section 2, we describe the representations of the TL algebra, constructed as projectors on total spin zero of two neighbouring spins. In section 3, we discuss a modi?ed coordinate BA and show the reasons, why the techniques develloped for the spin 1/2 XXZ model don’t work here. Section 3 contains the algebraic details of the computation and section 4 is reserved for the conclusions.

2

Representations of the Temperley-Lieb algebra as spin zero projectors.

Representations of the TL algebra, commuting with quantum groups, can be constructed in the following way[2]. Suppose Uq (Xn ) is the universal envelopping algebra of a ?nite dimensional Lie algebra Xn , equipped with the coproduct ? : Uq → Uq ?Uq [3]. If now π : Uq → EndVΛ is a ?nite dimensional irreducible representation with highest weight Λ and we assume that the decomposition VΛ ? VΛ is multiplicity free and includes one trivial representation on V0 , then the projector P0 from 1

VΛ ? VΛ onto V0 is a representation of the TL algebra. The deformation parameter q, which plays the role of a coupling constant in the Hamiltonian, is related to Q as: Q = T rV (q ?2ρ ), where ρ is half the sum of the positive roots. By construction P0 commutes with the quantum group Uq (Xn ). Since we are not going to use any group-theoretical machinery, we will just lift the relevant formulas o? Batchelor and Kuniba[4] in order to display explicitly the Hamiltonians to be diagonalized. We will consider the following speci?c cases, (VΛ , Uq (Xn )) = (V2sΛ1 , Uq (A1 )) for spin s, (VΛ1 , Uq (Bn ),(VΛ1 , Uq (Cn ) and (VΛ1 , Uq (Dn ). I.e, we treat the q-deformations of the spin-s representation of sl(2) and the vector representations of so(2n + 1), sp(2n) and so(2n). VΛ denotes the Uq (Xn ) module with highest weight Λ. Λ1 is a highest weight of Xn . Introduce the following notation. Let ei , i = 1, . . . , n be orthonomal vectors and express the fundamental weight, Π = Λ1 + . . . + Λn , the set A of weights and the √ coupling constant Q ≡ ?2? as: (2.1)

A1 : A =

{s(e1 ? e2 ), (s ? 1)(e1 ? e2 ), . . . , ?s(e1 ? e2 )}, Λ1 = (e1 ? e2 )/2, ρ = (e1 ? e2 )/2, J = {s, s ? 1, . . . , ?s},

? ?(?) = (?1)? ,

Q = [2s + 1]; Bn (n ≥ 2) : A = {0, ±e1 , . . . , ±en }, Λi = e1 + . . . + ei , (1 ≤ i < n),

2

=

(e1 + . . . + en )/2,

(i = n),

ρ = (n ? 1/2)e1 + . . . + en /2, J = {0, ±1, . . . , ±n},

? ?(?) = (?1)? ,

Q = [2n ? 1][n + 1/2]/[n ? 1/2]; Cn : A = {±e1 , . . . , ±en }, Λi = e1 + . . . + ei , ρ = ne1 + . . . + en , J = {±1, . . . , ±n}, ?(?) = sign(?), Q = [n][2n + 2]/[n + 1]; Dn : A = = = {±e1 , . . . , ±en }, Λi = e1 + . . . + ei , (1 ≤ i < n ? 1), (i = n ? 1), (i = n), (e1 + . . . + en?1 ? en )/2, (e1 + . . . + en?1 + en )/2, ρ = (n ? 1)e1 + . . . + en?1 , J = {0, ±1, . . . , ±n}, Q = [2n ? 2][n]/[n ? 1]; ?(?) = 1. For ? ∈ J the symbol ? is de?ned as ? = ? + (1 ± 1)/4 for A1 with s ∈ Z + (1 ± 1)/4 ? ? (1 ≤ i ≤ n), (2.2)

and ? = 0 with the exception of ? = 1 for Bn . The q-number notation is [x] ≡ ? 0 by setting e?? = ?e? (hence e0 = 0). Using the index set J, above. we can write A = {?(e1 ? e2 )} for A1 and A = {e? |? ∈ J)} for Bn , Cn , Dn .

(q x ?q ?x )/(q?q ?1 ) . For Xn = Bn , Cn , Dn , we extend the su?x of e? to ?n ≤ ? ≤ n

Denoting by E?ν ∈ End VΛ the matrix unit, having all elements zero, except at 3

row ? and column ν, the projector can be written as P0 = Q?1/2 ?(?)?(ν)q ?<e? +eν ,ρ> E?ν ? E???ν . (2.3)

?,ν∈J

In the following we will refer to all models generically as higher spin models for simplicity, even when not talking about A1 Consider a one-dimensional chain of lenght N with a ”spin” at each site. The spin variables range over the set of weight vectors v? |? ∈ J and our Hilbert space is an N-fold tensor product VΛ ? . . . ? VΛ . For A1 , these are the q-analogs of the usual spin states. The Hamiltonian densities acting on two neighboring sites are then given by: k, l|U|i, j = ?(i)?(k)q ?<ei +ek ,ρ> δi+j,0 δk+l,0 . (2.4)

3

The coordinate Bethe - Ansatz

All the above Hamiltonians are U(1) invariant and we can classify their spectra according to sectors. For A1 (s = 1) the commuting operator is the total spin Sz =

N k=1

it equals r = N ? ω ? S z for A1 and Bn and r = N ? (ω ? 1/2) ? S z for Cn and Dn . We set ω = maxJ. Therefore, there exists a reference state |? , satisfying H |? = E0 |? , with E0 = 0. We take |? to be |? =

N k

z Sk and we set the conserved quantum number r = N ? S z . In general

|ω, k .

In every sector r there are eigenstates degenerate with |? . They contain a set of impurities. We call impurity any state obtained by lowering some of the |ω, k ’s, such that the sum of any two neighboring spins is non-zero. Since H(q) is a projector on spin zero, all these states are annihilated by H(q). In particular, they do not move under the action of H(q), which is the reason for their name. We will now start to diagonalize H(q) in every sector. Nothing interesting happens in sector r < 2ω. Sector r = 2ω is more interesting, although still trivial, 4

since it contains one free pseudoparticle. The main result of this paper is to show that H(q) can be diagonalized in a convenient basis, constructed from products of single pseudoparticle wavefunctions. The energy eigenvalues will be parametrized as a sum of single pseudoparticle contributions.

3.1

The sector r = 2ω, containing one pseudoparticle

Starting with r = 2ω, we encounter the situation, where the states |j, k and |?j, k ± 1 , j = ω occur in neighboring pairs. They do move under the action of H(q) and mix with states containing one |?ω, k . Eigenstates are a superposition of |x[?ω] = ( . . . ωωωω ?ω ωω . . .) and |x[j,?j] = (. . . ωωω +j ? jωω . . .), x x i.e. |2ω; . . . = where

′ j x ′

{aω (x) |x[?ω] +

j

bj (x) |x[j,?j] },

(3.1)

means j ∈ J ? = J ? {±ω} and the ellipses stand for parameters the

eigenvector is going to depend on. When H(q) now acts on |2ω; . . . it sees the reference con?guration, except in the vicinity of x and we obtain the eigenvalue equations

(E ? q 2<?ω ,ρ> ? q ?2<?ω ,ρ> ) aω (x) = aω (x + 1) + aω (x ? 1) +

′ ′ l

?(ω)?(l)q ?<eω +el ,ρ> bl (x ? 1) +

?(?ω)?(l)q ?<e?ω +el ,ρ> bl (x)

l

Ebj (x) = ?(ω)?(l)q ?<ej +eω ,ρ> aω (x + 1) + ?(?ω)?(l)q ?<ej ?e?ω ,ρ> aω (x) +

′

?(j)?(l)q ?<ej +el ,ρ> bl (x),

l

j ∈ J ?.

(3.2)

Eliminating the bj ’s, we get an equation very similar to the XXZ model: (E ? q ?2<el ,ρ> )aω (x) = aω (x + 1) + aω (x ? 1). (3.3)

j∈J

5

We will treat periodic boundary conditions maintaining translational invariance in the following sections. They demand aω (x + N) = aω (x) and bj (x + N) = bj (x). We parametrize as: aω (x) = aω ξ x and bl (x) = bl ξ x , l ∈ J ? . Substituting this into equ.(3.2) we get two eigenstates and their energies aw = ?(?ω)q <eω ,ρ> + ?(ω)q ?<eω ,ρ> ξ ?1 ≡ Γ(ξ ?1) bl = ?(l)q ?<el ,,ρ> , l ∈ J ?

′

(3.4) (3.5)

E1 =

l

q ?<el,ρ> + Γ(ξ)Γ(ξ ?1) =

l∈J

q ?2<el ,ρ> + ξ + ξ ?1

and a highly degenerate solution with E2 = 0, with the following constraint on the parameters:

′

?(l)?(ω)q ?<el+ω ,ρ> bl + Γ(ξ)?(ω)q ?<eω ,ρ> aω = 0.

l

(3.6)

Here ξ = eiθ , θ being the momentum determined from the periodic condition to be: θ = 2πl/N, with l integer. We describe this situation by saying that we have two types of pseudoparticles with energies E1 and E2 . Whereas the pseudoparticle |2ω; θ with energy E1 = ?2? + 2 cos θ, 2? ≡ ? q ?2<el ,ρ> .

l∈J 2

is degenerate with

|? , i.e. propagates with energy E2 = 0, the pseudoparticle |2ω; θ 1, propagates (3.7)

As mentioned before, the energy eigenvalues are going to be parametrized as a sum of single pseudoparticle energies. Thus we write:

p ′

E=

n=1

?n (

l

?1 q ?<el ,ρ> + Γ(ξn )Γ(ξn )),

(3.8)

where ?n depends on which pseudoparticle we use: ?n = 1 for E = E1 and E = E2 = 0.

3.2

Two pseudoparticles and the XXZ Bethe - Ansatz

The next higher sector would be r = 2ω + 1, but let us treat r = 4ω ?rst, since then we can compare it with the ?rst nontrivial sector in the XXZ model. 6

This sector contains states, which consist of two interacting pseudoparticles. We seek these eigenstates in the form:

′

|4ω; . . .

′ j

?1 ?2

=

x1 <x2

{aωω (x1 , x2 ) |x1 [?ω], x2 [?ω] +

′ ′ j

i

bωi (x1 , x2 ) |x1 [?ω], x2 [i,?i] + (3.9)

bjω (x1 , x2 ) |x1 [j,?j], x2 [?ω] +

i

bij (x1 , x2 ) |x1 [i,?i], x2 [j,?j] }.

Translational invariance now speci?es aωω (x1 , x2 ) = ξ x1 aωω (n) and similarly for the other wave functions, where n = x2 ? x1 . Periodic boundary conditions require that aωω (n) = ξ n aωω (N ? n), biω (n) = ξ n bωi (N ? n), bij (n) = ξ n bji (N ? n), (3.10)

where ξ = ξ1 ξ2 (ξi = eiθi , i = 1, 2) and the total momentum is θ1 + θ2 = 2πl/N, with l integer. According to equ.(3.8), we will parametrize the energy as

2 ′

E=

n=1

?n [

l

?1 q ?<el,ρ> + Γ(ξn )Γ(ξn )].

(3.11)

Let us take the block ?1 = ?2 = 1 ?rst. We try to build 2-pseudoparticle eigenstates out of translationally invariant products of 1-pseudoparticle excitations at x1 and x2 with weight functions Di (x1 , x2 ), i = 1, 2:

′

|4ω; θ1 , θ2

11

=

x1 <x2 ′ j

?1 { D1 (x1 , x2 )[Γ(ξ1 )|x1 [?ω] +

i

?(i)q ?<ei ,ρ> |x1 [i,?i] ]

?1 [ Γ(ξ2 )|x2 [?ω] +

?(j)q ?<ej ,ρ> |x2 [j,?j] ] +

′

?1 D2 (x1 , x2 )[ Γ(ξ2 )|x1 [?ω] + j ′ ?1 [ Γ(ξ1 )|x2 [?ω] + i

?(j)q ?<ej ,ρ> |x1 [j,?j] ]

?(i)q ?<ei ,ρ> |x2 [i,?i] ] }.

7

Comparing this with equ.(3.9) and using translational invariance, implying D2 (n) = ξ n D1 (N ? n), we get

?1 ?1 aωω (n) = Γ(ξ1 ) Γ(ξ2 ) D(n), ?1 ?1 bωi (n) = ?(i) q ?<ei ,ρ> [ Γ(ξ1 ) D1 (n) + Γ(ξ2 ) D2 (n)], ?1 ?1 biω (n) = ?(i)q ?<ei ,ρ> [ Γ(ξ2 )D1 (n) + Γ(ξ1 )D2 (n) ],

bij (n) = ?(i)?(j) q ?<ei +ej ,ρ> D(n), 3 ≤ n ≤ N ? 3, where D(n) = D1 (n) + D2 (n).

(3.12)

Applying H(q) to the state of (3.9), we obtain a set of coupled equations for aωω (n), bij (n). Following [5], we split the equations into far equations, when excitations do not meet and near equations, containing terms when they are neighbors. The far equations are: (E ? 2q ?2<eω ,ρ> ? 2q 2<eω ,ρ> )aωω (n) = (1 + ξ)?1)aωω (n + 1) + (1 + ξ)aωω (n ? 1) +

′ l ′ l

?(l)?(ω)q ?<el +eω ,ρ> [ ξ ?1 blω (n + 1) + bωl (n ? 1) ] + ?(l)?(?ω)q ?<el ?eω ,ρ> [ blω (n) + bωl (n) ], 2 ≤ n ≤ N ? 2, (3.13)

(E ? q ?2<eω ,ρ> ? q <2eω ,ρ> )bωj (n) = ξ ?1bωj (n + 1) + ξbωj (n ? 1) + ?(j)?(?ω)q ?<ej ?eω ,ρ> aωω (n) + ?(j)?(ω)q ?<ej +eω ,ρ> aωω (n + 1) +

′

?(l)q ?<el ,ρ> ?(ω)q ?<eω ,ρ> ξ ?1blj (n + 1) + ?(?ω)q <eω ,ρ> blj (n)+

l

?(j)q ?<ej ,ρ> ξ ?1 bωl (n) ,

2 ≤ n ≤ N ? 2,

(3.14)

(E ? q ?2<eω ,ρ> ? q 2<eω ,ρ> )bjω (n) = bjω (n ? 1) + bωj (n + 1) + ?(j)?(?ω)q ?<ej ?eω ,ρ> aωω (n) + ?(j)?(ω)q ?<ej +eω ,ρ> ξ aωω (n ? 1) +

′ l

?(l)q ?<el ,ρ> ?(ω)q ?<eω ,ρ> bjl (n ? 1) + ?(?ω)q <eω ,ρ> bjl (n) + ?(j)q ?<ej ,ρ> blω (n) , 3 ≤ n ≤ N ? 3, (3.15)

Ebij (n) = ?(ω)q ?<eω ,ρ> [?(i)q ?<ei ,ρ> ξbωj (n ? 1) + ?(j)q ?<ej ,ρ> bωi (n + 1)] + ?(?ω)q <eω ,ρ> [?(i)q ?<ei ,ρ> bωj (n) + ?(j)q ?<ej ,ρ> bωi (n)] + 8

′

?(l)q ?<el ,ρ> [?(i)q ?<ei ,ρ> blj (n) + ?(j)q ?<ej ,ρ> bil (n)],

l

3 ≤ n ≤ N ? 3.

(3.16)

We already know them to be satis?ed, if we parametrize D1 (n) and D2 (n) by plane waves:

n N n D1 (n) = ξ2 , D2 (n) = ξ2 ξ1 .

(3.17)

The real problem arises of course, when pseudoparticles are neighbors, so that they interact and we have no guarantee that the total energy is a sum of single pseudoparticle energies. Let us now have a lightning review of the XXZ coordinate BA in order to be able to comment on the features, which are not going to survive generalizations to the present models. The equations in the sector rXXZ = 2 are (E ? 2q 2 ? 2q ? 2 ) a(x1 , x2 ) = a(x1 + 1, x2 ) + a(x1 ? 1, x2 ) + a(x1 , x2 + 1) + a(x1 , x2 ? 1), if x1 and x2 are not neighbors. In case they are we get (E ? q 2 ? q ? 2 ) a(x1 , x1 + 1) = a(x1 ? 1, x1 + 1) + a(x1 , x1 + 2).

1 1 1 1

(3.18)

(3.19)

(3.20)

One now supposes, that the parametrization equ.(3.9) for s = 1/2 solves both the above equations. In this case we are allowed to set x2 = x1 + 1 in equ.(3.18) and subtract it from equ.(3.20), yielding the following consistency condition : ? (q 2 + q ? 2 )a(x1 , x1 + 1) = a(x1 , x1 ) + a(x1 + 1, x1 + 1).

1 1

(3.21)

This gives the BA equation for the XXZ model, determining the 2-body phase shift:

N ξ2

=?

1 + ξ + ξ2 (q 2 + q ? 2 ) 1 + ξ + ξ1 (q 2 + q ? 2 ) 9

1 1

1

1

.

(3.22)

This type of procedure only works for the spin 1/2 XXZ model, due to the following fact. When the two pseudoparticles come together in a con?guration like (. . . + + + ? ? + + + . . .) and when H is applied to the two down spins, it gives zero, since their total Sz equals ?1. But whenever two excitations approach each other becoming neighbors and the Hamiltonian applied to them yields a nonvanishing resul t, then the representation like equ.(3.12) cannot solve both the far and near equations. Yet this is exactly the situation arising for higher spins. As we shall see, in this case, the representation equ.(3.12) has to modi?ed[11], the two-body wavefunction developing a ”discontinuity” at minimum separation. We call this the spin zero rule. Now back to our problem to solve the near equations. They are: (E ? q ?2<eω ,ρ> ? q 2<eω ,ρ> )aωω (1) = (1 + ξ)?1aωω (2) +

′

?(l)q ?<el ,ρ> [?(ω)q ?<eω ,ρ> ξ ?1blω (2) + ?(?ω)q <eω ,ρ> bωl (1) ];

l

(E ? q ?2<eω ,ρ> )bωj (1) = ξ ?1bωj (2) + ?(j)?(?ω)q ?<ej ?eω ,ρ> aωω (1) + ?(j)?(ω)q ?<ej +eω ,ρ> aωω (2) +

′

?(l)q ?<el ,ρ> ?(ω)q ?<eω ,ρ> ξ ?1 blj (2) + ?(j)q ?<ej ,ρ> bωl (1) ;

l

(E ? q 2<eω ,ρ> )bjω (2) = bωj (3) + ?(j)?(?ω)q ?<ej ?eω ,ρ> aωω (2) + ?(j)?(ω)q ?<ej +eω ,ρ> ξ aωω (1) +

′

?(l)q ?<el ,ρ> ?(?ω)q <eω ,ρ> bjl (2) + ?(j)q ?<ej ,ρ> blω (2) ;

l

Ebij (2) = ?(ω)q ?<eω ,ρ> [?(i)q ?<ei ,ρ> ξbωj (1) + ?(j)q ?<ej ,ρ> biω (3)] + ?(?ω)q <eω ,ρ> [?(i)q ?<ei ,ρ> bωj (2) + ?(j)q ?<ej ,ρ> biω (2)] +

′

?(l)q ?<el ,ρ> [?(i)q ?<ei ,ρ> blj (2) + ?(j)q ?<ej ,ρ> bil (2)],

l

i = j; 10

Ebii (2) = ?(i)?(ω)q ?<ei +eω ,ρ> [ξbωi (1) + biω (3)] +

′

?(i)?(?ω)q

′ l

?<ei ?eω ,ρ>

[bωi (2) + biω (2)] +

l (l)

?(l)q ?<ei +el,ρ> [bli (2) + bil (2)] + (3.23)

?(?i)?(l)q ?<el ?ei ,ρ> Bi .

(l)

Here some new states are showing up. Bi are the wavefunctions of the states of the type (. . . ωωi l ? l ωω . . .), l = i. Applying H(q) to them we obtain the x system: (E?q 2<el ,ρ> )Bi =

j∈J ? ,j=?i (l)

?(l)?(j)q < el +ej ,ρ> Bi +?(l)?(?i)q ?<el ?ei,ρ> bii (2), (3.24) ?(l)?(?i)q <el ?ei ,ρ> bii (2). (E ? j∈J,j=?i q ?2<ej ,ρ> )

(l)

yielding Bi =

(l) (l)

(3.25)

Eliminating Bi from equation (3.23), we get E(E + 2?) bii (2) = ?(i)?(ω)q ?<ei+eω ,ρ> [ξbωi (1) + biω (3)] + E + 2? + q <ei ,ρ> ?(i)?(?ω)q ?<ei ?eω ,ρ> [bωi (2) + biω (2)] +

′

?(l)?(i)q ?<ei +el,ρ> [bli (2) + bil (2)].

l

(3.26)

In order to solve these equations, we follow [11] and now leave the value of the wavefunctions for nearest separation as arbitrary parameters: aωω (1) = Γ(ξ)Γ(ξ ?1)D(1) + Faωω (1), bωi (1) = ?(i)q ?<ei ,ρ> [Γ(ξ ?1 )D1 (1) + Γ(ξ ?1 )D2 (1)] + Fbωi (1), bωi (2) = ?(i)q ?<ei ,ρ> [Γ(ξ ?1 )D1 (2) + Γ(ξ ?1 )D2 (2)] + Fbiω (2), bij (2) = ?(i)?(j)q ?<ei +ej ρ> D(2) + Fbij (2). (3.27)

In order for this modi?cation to leave the far equations still satis?ed, the following conditions have to hold:

′

(1 + ξ ?1 )Faωω (1) +

l

?(l)?(ω)q ?<el +eω ,ρ> Fbωl (1) + 11

′ l

?(l)?(?ω)q ?<el ?eω ,ρ> Fblω (2) = 0,

′

ξFbωj (1) +

l ′

?(l)?(?ω)q ?<el ?eω ,ρ> Fblj (2) = 0, ?(l)?(ω)q ?<el +eω ,ρ> , Fbjl (2) = 0. (3.28)

Fbjω (2) +

l

Now using equs.(3.12) and equ.(3.27) in equ.(3.26) , we get the following equation D(2) , i ∈ J ?. E + 2? Doing the same with equ.(3.23), we get: Fbii (2) = Fbij (2) = 0, i = j. for Fbii (2):

(3.29)

(3.30)

These results for Fbii (2), Fbij (2) are reasonable. In the ?rst case the colliding excitations satisfy the zero spin rule and we get a non-zero result, wheras in the second case the rule is not satis?ed and we get zero. Using this in equ.(3.28), we obtain for the remaining constants: Faωω (1) = ?(q ?2<eω ,ρ> + q 2<eω ,ρ> + 2?) Fbii (2), Fbωj (1) = ??(j)?(?ω)q ?<ej ?eω ,ρ> Fbii (2), Fbjω (2) = ??(j)?(ω)q ?<ej +eω ,ρ> Fbii (2) , j ∈ J ? . (3.31)

Substituting ?nally the complete parametrization into the remaining near equatins, get the following Bethe-Ansatz equation: D(2) ξ = D(1), E + 2? 1+ξ which can also be rewritten as

N ξ2 = ?

(3.32)

ξ2 [(1 + ξ ?1 )ξ2 ? 2? ? E] . ξ1 [(1 + ξ ?1 )ξ1 ? 2? ? E]

(3.33)

Using the explicit form of the energy, the set of equations determining the spectrum are:

N ξ2 = ?

1 + ξξ2 ? 2?ξ2 , 1 + ξξ1 ? 2?ξ1 12

ξ N = 1.

(3.34)

Notice that this equation is independent of n or any other representation speci?c quantities. All the models considered show therefore an equivalent spectrum, when parametrized in terms of ?. In particular, this is the same consistency condition one ?nds for the XXZ model, showing that for ?1 = ?2 = 1, even for periodic boundary conditions, the spectra of all our models are equivalent to the spectrum of the XXZ model, if expressed in terms of ? . We will refrain from discussing the other two blocks: ?1 = 1, ?2 = 0 and ?1 = ?2 = 0, since the calculations are analogous to the ones presented above. For details the reader might consult reference[11]. Anyhow, the ?rst of the two cases doesn’t show up for the more interesting situation of free boundary conditions. It is too asymetric to satisfy free boundary conditions. This is the reason, why for free boundary conditions, the spectrum of all of our models is equivalent to the XXZ spectrum. The block ?1 = ?2 = 0 has E = 0 and the BA equation reduces to ξ N = 1, being highly degenerate. The eigenvalue E = 0 also occurs in the XXZ spectrum, albeit with di?erent degeneracy.

3.3

One pseudoparticle and impurities

Since the setup with pseudoparticles and impurities is a little di?erent from the case of two pseudoparticles, we will dedicate some space to it. The eigenstates sought for, will be like:

′

|2ω + 1; . . . =

x1 <x2

{aωk (x1 , x2 ) |x1 [?ω], x2 [k] +

′

i

bjk (x1 , x2 ) |x1 [j,?j], x2 [k] + (3.35)

akω (x1 , x2 ) |x1 [?k], x2 [?ω] +

j

bkj (x1 , x2 ) |x1 [k], x2 [j,?j] .

Translational invariance and periodic boundary conditions impose: aωk (x1 , x2 ) = ξ x1 aωk (n), bjk (x1 , x2 ) = ξ x1 bjk (n), 13 (3.36)

aωk (n) = ξ n akω (N ? n), θ1 + θ2 = 2πl/N, with l integer.

bjk (n) = ξ n bkj (N ? n),

(3.37)

where n = x2 ? x1 , ξ = ξ1 ξ2 (ξi = eiθi , i = 1, 2) and the total momentum is Let us take the block ?1 = 1 , building eigenstates out of translationally invariant products of 1-pseudoparticle excitations at x1 and an impurity at x2 with weight functions Di (x1 , x2 ), i = 1, 2 as in the previous section. This yields the parametrizations:

?1 aωk (n) = Γ(ξ1 ) D1 (n), ?1 akω (n) = Γ(ξ1 ) D2 (n),

bjk (n) = ?(j) q ?<ej ,ρ> D1 (n), bkj (n) = ?(j) q ?<ej ,ρ> D2 (n). The far equations for the impurity at the right are now: (E ? 2q ?2<eω ,ρ> ? 2q 2<eω ,ρ> )aωk (x1 , x2 ) = aωk (x1 ? 1, x2 )) + aωk (x1 + 1, x2 ) +

′ ′

(3.38)

?(l)?(ω)q

l

?<el +eω ,ρ>

blk (x1 ? 1, x2 ) +

?(l)?(?ω)q ?<el ?eω ,ρ> blk (x1 , x2 ),

l

x1 + 2 ≤ x2 ≤ N ? x1 ? 2, Ebjk (x1 , x2 ) = ?(j)?(ω)q ?<ej +eω ,ρ> aωk (x1 + 1, x2 ) + ?(j)?(?ω)q ?<ej ?ω ,ρ> aωk (x1 , x2 ) +

′

(3.39)

?(j)?(l)q ?<ej +el ,ρ> [?(i)q ?<ei ,ρ> blk (x1 , x2 ),

l

x1 + 3 ≤ x2 ≤ N ? x1 ? 3,

(3.40)

and analogous equations for the impurity at the left. Eliminating the b-functions, we get: (E ? (E ? q ?2<el ,ρ> )aωk (x1 , x2 ) = aωk (x1 ? 1, x2 ) + aωk (x1 + 1, x2 ), q ?2<el ,ρ> )akω (x1 , x2 ) = akω (x1 , x2 ? 1) + akω (x1 , x2 + 1), (3.41) 14

l∈J

l∈J

x1 + 3 ≤ x2 ≤ N ? x1 ? 3.

We know them to be satis?ed, if the energy is given by equ.(3.11). The near equations require of course the by now costumary treatment of modifying the ansatz of the wavefunctions at nearest separations. The near equations for the impurity at the right are: (E ? 2q ?2<eω ,ρ> )aωk (x, x + 1) =

′

aωk (x ? 1, x + 1)) +

l

?(l)?(ω)q ?<el+eω ,ρ> blk (x ? 1, x + 1),

(3.42)

Ebjk (x, x + 1) = ?(j)?(ω)q ?<ej +eω ,ρ> aωk (x + 1, x + 2) +

′

?(j)?(?ω)q ?<ej ?ω ,ρ> aωk (x, x + 2) +

l

?(j)?(l)q ?<ej +el ,ρ> blk (x, x + 2), (3.43)

Ebjj (x, x + 2) = ?(j)?(ω)q ?<ej +eω ,ρ> aωj (x + 1, x + 2) + ?(j)?(?ω)q ?<ej ?ω ,ρ> aωj (x, x + 2) + ?(?j)?(?ω)q <ej +eω ,ρ> ajω (x, x + 1) + ?(?j)?(ω)q <ej ?ω ,ρ> ajω (x, x + 2) +

′ ′

?(j)?(l)q ?<ej +el ,ρ> blj (x, x + 2) +

l l

?(?j)?(l)q <ej ?el,ρ> bjl (x, x + 1). (3.44)

They can be solved modifying the parametrization for nearest neighbors in the usual way. The result is:

2 aωk (1) = Γ(ξ ?1)ξ2 + Fakω , bjk (2) = ?(j)q ?<ej ,ρ> ξ2 + Fbjk , N N akω (1) = Γ(ξ ?1)ξ2 ξ1 + Fakω , bkj (1) = ?(j)q ?<ej ,ρ> ξ2 ξ1 + Fbkj ,

(3.45) (3.46)

where

N N Fakω = ??(ω)q ?<eω ,ρ> ξ2 , Fbkj = ?(k)q ?<ek ,ρ> ξ2 δk+j,0,

Faωk = ??(?ω)q <eω ,ρ> ξ2 , Fbjk = ?(?k)q <ek ,ρ> ξξ2δk?j,0,

N with ξ N = 1, ξ1 ?2ξ 2 = 1 resulting from periodic boundary conditions.

(3.47)

15

4

Free boundary conditions

It is for free boundary conditions, that the Hamiltonian H(Q) commutes with the quantum group Uq (Xn ). As expected, the Bethe states are highest weight states of Uq (Xn ), except some E = 0 states. Since the extension of the BA-procedure from the periodic boundary conditions to the free case, follows exactly the lines of ref.[11], we will only state the results for the sector r = 4. Take the block ?1 = ?2 = 1. The nearest approach constants to be added to the now standing waves are the same as in the periodic case, namely equs.(3.30),(3.31), only Fbii (2) is di?erent[11]). The BA equations are now:

r 2N ξa = b=1,b=a ?1 b(ξa , ξb ) , a = 1, 2, b(ξa , ξb )

(4.1)

where b(ξa , ξb ) = and Eab = 2

l

ξb ?1 ?1 ?1 [ξb + ξa ? 2? ? Eab ][ξb + ξa ? 2? ? Eab ]. ξa

′ ?1 ?1 q ?<el,ρ> + Γ(ξa )Γ(ξa ) + Γ(ξb )Γ(ξb ).

(4.2)

(4.3)

The only other block is ?1 = ?2 = 0, E = 0. It is again highly degenerate with Fbii (2), ξ1, ξ2 as free parameters? . Thus all models have spectra equivalent to the one of the XXZ model.

5

Conclusion

We obtained the spectra of quantum spin chain models, arising as representations of the Temperley-Lieb algebra associated with quantum groups. The tool is a modi?ed version of the coordinate Bethe Ansatz, since the simpler algebraic Bethe Ansatz is not immediately available for these models. We ?nd that all models have equivalent

?

Actually there is one more free parameter, called α5 in ref.[11].

16

spectra, i.e. they di?er at most in their degeneracies. The energy eigenvalues are given by E= where ?2? =

′ l p n=1 ?1 (?n (?2 ? ? + Γ(ξn )Γ(ξn )),

(5.1)

q ?<el,ρ> and the rapidities ξn are solutions of the BA equations.

In the sector r we may have p pseudoparticles Nω? ?1 , Nω? ?1 , · · · , N?ω? +1 impurities of the type (ω ? ? 1), (ω ? ? 2), · · · , (?ω ? + 1), respectively, such that Nω? ?1 + 2Nω? ?1 + · · · + (2ω ? ? 1)N?ω? +1 = r ? 2ω ? p. Here ω ? = ω for A1 and Bn and ω ? = ω ? 1/2 for Cn and Dn . For example, for periodic boundary conditions? , the total rapidity ξ = ξ1 ξ2 . . . ξp ξimp , ξimp = ξp+1 ξp+2 . . . ξr?σ obeys ξ N = 1 and the BA equations for E = 0 are:

r?σ N 2 ξa ξimp = b=1,b=a

(5.2)

?

ξa (1 + (ξa ξb )?1 ξa ? ?ab , ξb (1 + (ξa ξb )?1 ξb ? ?ab

(5.3)

where ?ab = Eab + 2? and σ can be ?xed for each allowed case from the equations (5.2).The Bethe eigenstates are highest weight states of the quantum group, except for the states with energy E = 0, for which this is not always true. in the sector r with p impurities and E = 0, the total rapidity ξ = ξ1 ξ2 . . . ξp ξimp , ξimp = ξp+1 ξp+2 . . . ξr?p obeys ξ N = 1 and the BA equations are:

N 2 ξa ξimp = b=1,b=a

?

ξa (1 + (ξa ξb )?1 ξa ? ? , ξb (1 + (ξa ξb )?1 ξb ? ?

(5.4)

where ? = E + 2?. The Bethe eigenstates are highest weight states of the quantum group, except for the states with energy E = 0, for which this is not always true. These results are expected, but as far as we know, unprooven using only TemperleyLieb algebraic statements as input. Acknowledgments It is R.K.’s pleasure to thank C.Callan and W. Bialek for their support.

?

We don’t list the equations for free boundary conditions, since they are identical to those of

ref.[11].

17

References

[1] Temperley H N V and Lieb E 1971 Proc. R. Soc. A 322 251. [2] Reshetikhin N Y 1988 Quantized universal enveloping algebras, the YangBaxter equation and invariants of links I,LOMI Preprint E 4. Proc. R. Soc. A 322 251. [3] Drinfeld V G 1986 ICM Proceedings Berkeley 798; Jimbo M 1985 Lett. Math. Phys. 10 63. [4] Batchelor M T and Kuniba A 1991 J. Phys. A24 2599. [5] Parkinson J.B. 1988 J. Phys. C: Solid State Phys. 21 3793. [6] K¨berle R. and Lima - Santos A. 1994 J. Phys.A: Math. Gen. 27 5409. o [7] Yang C N 1967 Phys. Rev. Lett. 19 1312; Baxter R J 1982 Exactly Solved Models in Statistical Mechanics ( Academic Press, London ). [8] Takhtajan L A and Faddeev L D 1979 Russ. Math. Surveys 34 11. [9] Bethe H. 1931 Z. Physik 71 205. [10] Alcaraz F.C., K¨berle R. and Lima - Santos A. 1992 Int. J. Mod. Phys. A7 o 7615. [11] K¨berle R. and Lima - Santos A. 1994 J. Phys.A: Math. Gen. 27 5409. o [12] Barber M. and Batchelor M.T. 1989 Phys. Rev. B40 4621. [13] Kl¨ mper A. 1990 Int. J. Mod. Phys. B4 171. u [14] Batchelor M.T.,Mezincescu L.,Nepomechie R.I., Rittenberg V. 1990 J. Phys. A: Math. Gen. 23 L141.

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[15] Kulish P. P., Reshetikhin N. Yu. and Sklyanin E.L. 1981 Lett. Math Phys. 5 393. [16] Alcaraz F.C., Barber M.N., Batchelor M.T., Baxter R.J. and Quispel G.R.W. 1987 J. Phys. A:Math. Gen. 20 6397. [17] Smirnov F. A. 1991 Int. J. Mod. Phys. A6 40. [18] Alcaraz F. C. and Malvezzi A. 1992 J. Phys. A: Math. Gen. 25 4535.

19

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