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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 910 (2016) 910–928

www.elsevier.com/locate/nuclphysb

ce:document-thread

Universal Bethe ansatz solution for the Temperley–Lieb

spin chain

Rafael I. Nepomechie

a,∗

, Rodrigo A. Pimenta

a,b

Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124, USA

Departamento de Física, Universidade Federal de São Carlos, Caixa Postal 676, CEP 13565-905, São Carlos, Brazil

Received 25

January 2016; received in revised form 27 April 2016; accepted 29 April 2016

Available

online 4 May 2016

Editor: Hubert

Saleur

Abstract

consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions as-

sociated

with the spin-s representation of quantum-deformed sl(2). We construct the transfer matrix, and

determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show

that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of

solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an

algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe

states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar

product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We ﬁnd

that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal

in the sense that they do not depend on the value of the spin. In an appendix, we brieﬂy consider the closed

TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be

improved so as to obtain the complete (albeit non-universal) spectrum.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

DOI of linked article: http://dx.doi.org/10.1016/j.nuclphysb.2016.04.044.

Corresponding author.

E-mail

addresses: nepomechie@physics.miami.edu (R.I. Nepomechie), pimenta@df.ufscar.br (R.A. Pimenta).

http://dx.doi.org/10.1016/j.nuclphysb.2016.04.045

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

R.I. Nepomechie, R.A. Pimenta / Nuclear Physics B 910 (2016) 910–928 911

1. Introduction

The generators {X

(1)

, ..., X

(N−1)

} of the unital Temperley–Lieb (TL) algebra TL

[1],

(i)

=cX

(i)

(i±1)

(i)

(j)

(i)

(j)

, |i − j |> 1 , (1.1)

can be used to deﬁne the Hamiltonian of an open quantum spin chain of length N with “free”

boundary conditions

H =

N−1



i=1

(i)

. (1.2)

This type of model has been the subject of many investigations. For simplicity, we focus here on

the models associated with U

) = U

sl(2). The generators X

(i)

have been constructed for

any value of spin s [2,3]. The s = 1/2 case is the well-known quantum-group-invariant spin-1/2

XXZ chain [4]. The s = 1 case is the quantum deformation [2] of the pure biquadratic spin-1

chain [5–7]. These models are integrable; and closed-chain versions with periodic boundary con-

ditions have been in

vestigated for s>1/2using inversion relations [8,9], numerically [10], by

coordinate Bethe ansatz [5,6,11], and by analytical Bethe ansatz [12]. Additional results can be

found in [13–17] for the open chain, and in [18–20] for the closed chain. TL models associated

with higher-rank algebras have also been investigated [21–23].

Despite these and further ef

forts, a number of fundamental problems related to these models,

such as the formulation of an algebraic Bethe ansatz solution, have remained unsolved. Moreover,

the analytical Bethe ansatz solution proposed in [12] does not give the complete spectrum.

The goal of this paper is to address some of these problems. We construct the transfer matrix

corresponding to the Hamiltonian (1.2), and we determine its eigen

values using analytical Bethe

ansatz. We prove that the transfer matrix has quantum group symmetry, which accounts for the

degeneracies of the spectrum. We propose an algebraic Bethe ansatz construction of the Bethe

states, which (when on-shell) we conjecture are highest-weight states of the quantum group. The

scalar product between an of

f-shell Bethe state and an on-shell Bethe state is also considered,

and we conjecture that it can be given in terms of a determinant formula; the square of the norm,

i.e., the scalar product between on-shell Bethe states, follo

ws as a limit.

We ﬁnd that all of these

results, except for the degeneracies and a constant factor in the scalar product, are universal in

the sense that they do not depend on the value of the spin.

Although most of this paper concerns the open TL chain, we brieﬂy consider the closed TL

chain with periodic boundary conditions in an appendix. There we re

visit the analytical Bethe

ansatz computation in [12], and show how the proposed solution can be improved so as to obtain

the complete spectrum. In contrast with the case of the open chain, the solutions of the closed-

chain Bethe equations are not uni

versal, as the Bethe roots depend on the value of the spin.

The outline of this paper is as follo

ws. In section 2 we describe the construction of the Hamil-

tonian (1.2) and the corresponding transfer matrix. In section 3 we use analytical Bethe ansatz

to determine the eigenvalues of the transfer matrix and the corresponding Bethe equations. In

section 4 we show that the transfer matrix has quantum group symmetry

, and we propose explicit

Such formulas are generally known as Slavnov [24] and Gaudin–Korepin [25–27] formulas, respectively.

912 R.I. Nepomechie, R.A. Pimenta / Nuclear Physics B 910 (2016) 910–928

formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-

matrix eigenvalues. In section 5 we present our proposals for the algebraic Bethe ansatz solution

and scalar products. We brieﬂy discuss these results and remaining problems in section 6. We

treat the closed TL chain in Appendix A.

2. Transfer matrix

We begin this section by describing in more detail the construction of the Hamiltonian (1.2).

We then construct the corresponding transfer matrix, which is the generating function of the

Hamiltonian and the higher local conserved commuting quantities, and we review some of its

important properties.

We consider the TL open quantum spin chain corresponding to the spin-s representation of

sl(2). The X

(i)

appearing in the Hamiltonian (1.2) are operators on



2s+1



⊗N

deﬁned by

(i)

i,i+1

, (2.1)

where X is a (2s + 1)

by (2s + 1)

matrix (an endomorphism of C

2s+1

⊗ C

2s+1

) with the

following matrix elements [2]

m

|X|m



=(−1)

−m



, (2.2)

where m

, m



, m



=−s, −s +1, ..., s, and s =

, 1,

, .... In other words, X

(i)

is an oper-

ator on N copies of C

2s+1

, which acts as X on copies i and i + 1, and otherwise as the identity

operator,

(i)

⊗(i−1)

⊗X ⊗I

⊗(N−i−1)

, (2.3)

where I is the identity operator on C

2s+1

. These operators satisfy the TL algebra (1.1), where c

is given by

c =

[

2s + 1

]

2s+1

−Q

−2s−1

Q −Q

−1



k=−s

≡−



q +



. (2.4)

We assume throughout this paper that Q has a generic value.

The Hamiltonian (1.2) is inte

grable for any value of spin s. In the notation of [12], the corre-

sponding R-matrix is given by [28]

R(u) =



uq −



P +



u −



PX, (2.5)

where P is the permutation matrix on C

2s+1

⊗C

2s+1

. Indeed, the Yang–Baxter equation

) = R

) (2.6)

is satisﬁed. This R-matrix has the unitarity property

(u)R

−1

) = ζ(u)I

⊗2

,ζ(u)=ω(uq

−1

)ω(u

−1

), (2.7)

where R

, and ω(u) is deﬁned as

ω(u) =u −

. (2.8)

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