A Temperley–Lieb quantum chain with two- and three-site interactions

Y Ikhlef¹, J L Jacobsen² and H Saleur^3,4

¹ Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
² Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris, France
³ Institut de Physique Théorique CEA, IPhT, 91191 Gif-sur-Yvette, France
⁴ Department of Physics, University of Southern California, Los Angeles, CA 90089-0484, USA

E-mail: y.ikhlef1@physics.ox.ac.uk

Received 8 June 2009
Published 6 July 2009

1. Introduction

The study of lattice models of self-avoiding loops is a paradigm within two-dimensional (2D) statistical physics. In these models, the Boltzmann weights for loop configurations are encoded in an algebra. A simple example is the dense loop model on the square lattice, associated with the Temperley–Lieb (TL) algebra [1], and for which critical points are present for a loop weight n   [–2, 2]. This algebra is also realized by several 2D statistical models, such as the Ising and Potts model, the restricted solid-on-solid (RSOS) model [2] or the six-vertex (6V) model. At their integrable points and in the very anisotropic limit, these models are equivalent to one-dimensional (1D) quantum Hamiltonians, based on the TL generators. In this communication, we suggest to study TL Hamiltonians directly, with no necessary connection to a 2D statistical model. Various realizations of the TL algebra will lead to distinct applications.

The TL algebra can be realized by spin- variables. For a loop weight n  =  2, TL Hamiltonians describe spin- chains with isotropic exchange interactions . For nearest-neighbor interactions, one gets the Heisenberg chain [3], which is gapless, and has a Bethe Ansatz solution. When frustrating next-nearest neighbor interactions are introduced (with a sufficiently strong coupling constant), the system enters the Majumdar–Ghosh (MG) gapped phase [4], characterized by spontaneous dimerization and incommensurate correlations [5]. One can extend the construction of a Hamiltonian with exact dimerized eigenstates to a TL algebra with general loop weight n [6]. The new exact results we present on this model then apply to anisotropic frustrated spin chains.

Another realization of the TL algebra, with important physical implications, comes from the construction of trial wavefunctions for the fractional quantum Hall effect (FQHE). The principle for this construction is to find an -point function with a prescribed behavior when several points come close to one another. In [7], it was discovered that, for a specific class of wavefunctions, the solution is provided by the correlation functions of the -parafermionic conformal field theories (CFTs), and that the elementary excitations (anyons) above the ground state obey the fusion rules of these CFTs. For k  =  3, the resulting wavefunctions correctly describe the FQHE at filling fraction ν  =  12/5 [8]. It is important to note that the fusion rules for the parafermions are encoded in the RSOS representation of the TL algebra. This fact has been exploited in recent works [9, 10], in which 1D chains of anyons with local TL interactions are studied. In particular, when two- and three-site interactions are present, the phase diagram contains an MG-like gapped phase, separating two critical phases. These anyonic chains are a particular case of the model presented in this communication.

Our approach consists in writing a 1D TL Hamiltonian which combines two- and three-site interactions, and studying its phase diagram. This model contains as special cases: (i) the critical XXZ spin chain, (ii) the q-deformed MG chain [6] and (iii) a new critical, integrable point. The main results we present in this communication are the following. For a loop weight n   [n*, 2], where n* ≊ 1.5, an MG-like gapped phase is present. The critical line separating the XXZ and MG phases is in the universality class of the dilute O(n) model. We locate this transition line and identify the coupling constant in the 1D TL Hamiltonian which plays the role of temperature in the O(n) model. Finally, we identify a new integrable point determining a whole critical phase and solve it by Bethe Ansatz. Since this critical phase is separated from the XXZ phase by a gapped phase, the exact results from Bethe Ansatz are intrinsically non-perturbative.

2. The model

We consider the following Hamiltonian on a chain of 2N sites, based on the TL generators e_j (see definition below):

In the spin- representation, the operator e_j is a nearest-neighbor interaction that projects the spins S_j, S_j + 1 onto the singlet state. In this language, the quadratic terms in (1) are then next-nearest-neighbor interactions (see figure 1). Let us recall the definition of the TL algebra and use the spin- representation to write (1) as a combination of projectors.

Figure 1. Action of the TL operators. Dots represent spin- variables, and operators act upward. Two spins connected by a dotted (resp. full) line are in the same state (resp. form a singlet).

The TL algebra with loop weight n is generated by the operators e_j (j  =  1,  ..., 2N), which satisfy the relations (see figure 2)

The structure of the spin representation is encoded in the U_q[SU(2)] quantum algebra [11], generated by S^z and S^±, with the q-deformed commutation relations,

where [x]_q  =  (q^x – q^–x)/(q – q^–1). For q  =  1, one recovers the usual SU(2) relations. Let us denote by V(ℓ) the spin-ℓ representation. The spin- representation is simply obtained from the Pauli matrices , and the vector space for a spin- chain is the tensor product . Like in SU(2), V(ℓ) follow the decompositions: V(ℓ) ⊗ V(ℓ ')  =  V(|ℓ – ℓ '|) ⊕ · · · ⊕ V(ℓ  +  ℓ '). We call the projector of spins onto V(ℓ) according to these decompositions. It can be shown that the rescaled projectors e_j ≡ (q  +  q^–1)P⁽⁰⁾_j,j + 1 satisfy the TL relations (2) with loop weight n  =  q  +  q^–1. Moreover, P^(3/2)_{j,j + 1,j + 2} can be expressed in terms of e_j:

and thus we can write (1) as

The original parameters are given in terms of θ and n by

As suggested in figure 2, e_j can be represented on a vector space of connectivity diagrams. In this loop representation, each basis state is labeled by a non-intersecting pairing of the 2N sites on the chain, and each pairing represents a U_q[SU(2)] singlet state. The sector with total spin l corresponds to states where 2l sites remain unpaired. We denote by H_loop the Hamiltonian (1) in this representation.

When q is a root of unity (q  =  e^{iπ/(p + 1)}, p ≥ 3 integer), another representation of the TL algebra (2) is available: the RSOS or height representation [2]. The site j carries a height variable h_j   {1,  ..., p}, such that |h_j – h_j + 1|  =  1. The action of e_j on the basis states is defined as

where the sum is on valid height configurations. These operators satisfy the TL relations (2) with loop weight n  =  2cos[π/(p  +  1)]. We denote by H_RSOS the Hamiltonian (1) in this representation. Unlike the spin and loop representations above, e_j in (5) are real symmetric matrices and H_RSOS is Hermitian. For p  =  3 (critical Ising model), additional relations exist [12]: , and hence H_RSOS(θ) is equivalent to ±H_RSOS(0) for any value of θ.

There are three remarkable values of θ. For θ  =  0, we get the XXZ Hamiltonian [3],

where . For θ  =  –π/2, we get the q-deformed MG Hamiltonian [6]:

For the special value θ_int  =  Arctan(1/n) – π, the Hamiltonian is integrable (see section 4).

3. Phase diagram

For a loop weight n greater than n* ≊ 1.5, the Hamiltonian (1) has a gapped phase of the same nature as the MG phase [4]. We describe the phase diagram in the interval θ_int < θ < 0. We use the notation

where p can take real values. We obtain numerically the lowest eigenvalues of H_loop and H_RSOS with the Arnoldi and Lanczos algorithms, respectively. We diagonalize blocks of fixed momentum Q, and 2l unpaired sites (for H_loop), for system sizes up to 2N  =  24 sites. The phase diagram consists of three phases (see figure 3).

Critical phase I (XXZ phase). For θ close enough to 0, the system is in a critical phase, determined by H_XXZ (6). Its continuum limit is [13] a Coulomb gas (CG) critical theory, with central charge and conformal weights:

with 0 ≤ g ≤ 1. The integers e, m are the electric and magnetic charges. The scaling dimension for the sector with 2l unpaired sites (or 2l-leg watermelon exponent) is X_2l  =  2h_0,l. This theory corresponds to the dense phase of the O(n) model. For integer p ≥ 3, H_RSOS(θ  =  0) is [14] a lattice realization of the minimal model of CFT [15].

Gapped phase. For generic n   [n*, 2], H_loop(θ) has a gapped phase around θ  =  –π/2, with the following features (as in the isotropic case [4, 5]): (i) the ground state is doubly degenerate and consists of totally dimerized states; (ii) the elementary excitations (spinons) are domain walls between the two ground-state configurations, and are gapped. In the RSOS representation for p ≥ 3 integer, we can prove that a totally dimerized state , with a   {1,  ..., p}, saturates the lower energy bound of H_MG. Indeed, since H_MG is a sum of Hermitian operators, all energies satisfy E ≥ –2N(n² – 1). The state ψ_a is an eigenstate of H_MG with energy –2N(n² – 1). The height a can take p values, and ψ_a can be translated by one site, so the RSOS ground state of H_MG is 2p-degenerate.

Critical phase II. This phase is determined by H(θ_int), which is solved in section 4.

At the transition point θ_c between the XXZ and gapped phases, H_loop is described by a CG theory ((9)–(11)), with 1 ≤ g ≤ 2, corresponding to the critical dilute O(n) model (see figure 4). So, like the temperature does in the O(n) model [16], θ drives a transition between a non-critical phase and a critical phase, with a higher critical behavior at the transition. Moreover, the scaling exponent for θ is the O(n) thermal exponent y_t  =  4/(p  +  2) (see figure 5). For integer p ≥ 4, the spectrum of H_RSOS(θ_c) coincides with the CFT minimal model [15], as found in [10] for p  =  4.

4. Relations to integrable six-vertex models

From the TL generators for n  =  2cos γ (0 ≤ γ ≤ π/2), one builds an -matrix, acting on sites j, j  +  1: , which satisfies the Yang–Baxter equation: . In the spin- representation, encodes the integrable six-vertex (6V) model. Consider the staggered 6V model on a square lattice of width 2N sites, defined by the spectral parameters (u, u  +  α, u, u  +  α,  ...) in the horizontal direction and (0, α, 0, α,  ...) in the vertical direction. Let T_α(u) be the two-row transfer matrix. In the anisotropic limit u → 0, the Hamiltonian is , where ρ  =  sin(γ – α)sin(γ  +  α). We get

From now on, we focus on the case α  =  π/2: the Hamiltonian (12) then has the form (1), with K₂  =  –K₁/n. The loop model defined by T_π/2(u) is equivalent to the critical antiferromagnetic (AF) Potts model [17], but the sign of is such that its lowest energies correspond to the lowest eigenvalues of T_π/2(u). Therefore, the scaling behavior of is different from the AF Potts model studied in [17, 18]. For periodic boundary conditions, the Bethe Ansatz equations (BAEs) and energy for r particles read

The ground state has N/2 Bethe roots on each of the lines Im α_j  =  0 and Im α_j  =  π. Since each of these two Fermi seas can be excited independently by creating holes, the central charge of the staggered 6V model is c  =  2. Solving the BAE by Fourier transform in the large-N limit, we obtain the spectrum of conformal dimensions

where (resp. ) are integers with the same parity. To achieve the equivalence between the 6V and loop models, we introduce a seam with phase ±πe₀, where e₀  =  γ/π. The effective central charge is then c_eff  =  2 – 6e²₀/g '. The expressions for c_eff and suggest that the field theory for the staggered 6V model consists of one free compact boson and two free Majorana fermions, decoupled in the bulk, but coupled through the boundary conditions.

5. Conclusion

We have defined a Hamiltonian H(θ) based on the Temperley–Lieb algebra, which contains, as particular points, the XXZ and q-deformed Majumdar–Ghosh spin chains. At the transition θ_c between the two corresponding phases, we have given numerical evidence that the spectrum of H(θ_c) is equivalent to that of the dilute O(n) model. For the integrable point θ_int beyond the MG phase, we have used the BAE to find the effective degrees of freedom in the continuum limit. From these results, it is tempting to suppose that H(θ_c) is also the anisotropic limit of some integrable transfer matrix, but this is not proven. A related question is the exact determination of the limiting value n*, where the gapped phase ceases to exist.

Acknowledgments

We thank Fabien Alet for instructive discussions and help on numerics. YI also thanks Fabian Essler for help with the literature. The work of JLJ was supported by the ANR; the work of HS by the ANR and the ESF Network INSTANS.

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