Temperley–Lieb Research Articles

The ghost algebra and the dilute ghost algebra

We introduce the ghost algebra, a two-boundary generalisation of the Temperley–Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang–Baxter equations, given existing solutions to the Yang–Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang–Baxter integrable.

The XY Spin Chain and the Topological Basis Realization

Temperley-Lieb (T-L) algebra plays an important role in quantum computation, quantum teleportation, knot theory, statistical physics and topological quantum field theory. At the same time, a large number of models are represented by the generators of T-L algebra. In this paper, it is shown that the XY model can be constructed from the linear combination of the T-L algebra generators.We construct three new topological basis states, then investigate the particular properties of the topological basis in this system. For instance the energy ground state of the system falls on one of the topological basis states.

The dilute Temperley–Lieb O(n = 1) loop model on a semi infinite strip: the ground state

We consider the integrable dilute Temperley–Lieb (dTL) O(n = 1) loop model on a semi-infinite strip of finite width L. In the analogy with the Temperley–Lieb (TL) O(n = 1) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of q-difference equations, sometimes called the qKZ equations. We compute some ground state components of the transfer matrix of the dTL model, and show that all ground state components can be recovered for arbitrary L using the qKZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.

Universal Bethe ansatz solution for the Temperley–Lieb spin chain

We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated 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.

Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras

In previous work with Scullard, we have defined a graph polynomial PB(q, T) that gives access to the critical temperature Tc of the q-state Potts model defined on a general two-dimensional lattice It depends on a basis B, containing n × m unit cells of and the relevant root Tc(n, m) of PB(q, T) was observed to converge quickly to Tc in the limit Moreover, in exactly solvable cases there is no finite-size dependence at all. In this paper we show how to reformulate this method as an eigenvalue problem within the periodic Temperley–Lieb (TL) algebra. This corresponds to taking first, so that the bases B are semi-infinite cylinders of circumference n. The limit implies faster convergence in n, while maintaining the n-independence in exactly solvable cases. In this setup, Tc(n) is determined by equating the largest eigenvalues of two topologically distinct sectors of the transfer matrix. Crucially, these two sectors determine the same critical exponent in the continuum limit, and the observed fast convergence is thus corroborated by results of conformal field theory. We obtain similar results for the dense and dilute phases of the O(N) loop model, using now a transfer matrix within the dilute periodic TL algebra. Compared with our previous study, the eigenvalue formulation allows us to double the size n for which Tc(n) can be obtained, using the same computational effort. We study in details three significant cases: (i) bond percolation on the kagome lattice, up to nmax = 14; (ii) site percolation on the square lattice, to nmax = 21; and (iii) self-avoiding polygons on the square lattice, to nmax = 19. Convergence properties of Tc(n) and extrapolation schemes are studied in details for the first two cases. This leads to rather accurate values for the percolation thresholds: pc = 0.524 404 999 167 439(4) for bond percolation on the kagome lattice, and pc = 0.592 746 050 792 10(2) for site percolation on the square lattice.

The fusion rules for the Temperley–Lieb algebra and its dilute generalization

The Temperley–Lieb (TL) family of algebras is well known for its role in building integrable lattice models. Even though a proof is still missing, it is agreed that these models should go to conformal field theories in the thermodynamic limit and that the limiting vector space should carry a representation of the Virasoro algebra. The fusion rules are a notable feature of the Virasoro algebra. One would hope that there is an analogous construction for the TL family. Such a construction was proposed by Read and Saleur (2007 Nucl. Phys. B 777 316) and partially computed by Gainutdinov and Vasseur (2013 Nucl. Phys. B 868 223–70) using the bimodule structure over the TL algebras and the quantum group Uq (sl2).We use their definition for the dilute Temperley–Lieb (dTL) family, a generalization of the original TL family. We develop a new way of computing fusion by using induction and show its power by obtaining fusion rules for both dTL and TL. We recover those computed by Gainutdivov and Vasseur and new ones that were beyond their scope. In particular, we identify a set of irreducible TL- or dTL-representations whose behavior under fusion is that of some irreducibles of the minimal models of conformal field theory.

The pop-switch planar algebra and the Jones–Wenzl idempotents

The Jones–Wenzl idempotents are elements of the Temperley–Lieb (TL) planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra (PSPA), which contains the TL planar algebra. It is motivated by Jones' idea of the graph planar algebra (GPA) of type An. In the tensor category of idempotents of the PSPA, the nth Jones–Wenzl idempotent is isomorphic to a direct sum of n + 1 diagrams consisting of only vertical strands.

Multi-state Asymmetric Simple Exclusion Processes

It is known that the Markov matrix of the asymmetric simple exclusion process (ASEP) is invariant under the Uq(sl2) algebra. This is the result of the fact that the Markov matrix of the ASEP coincides with the generator of the Temperley-Lieb (TL) algebra, the dual algebra of the Uq(sl2) algebra. Various types of algebraic extensions have been considered for the ASEP. In this paper, we considered the multi-state extension of the ASEP, by allowing more than two particles to occupy the same box. We constructed the Markov matrix by dimensionally extending the TL generators and derived explicit forms of the particle densities and the currents on the steady states. Then we showed how decay lengths differ from the original two-state ASEP under the closed boundary conditions.

Fusion hierarchies, T-systems, and Y-systems of logarithmic minimal models

A Temperley–Lieb (TL) loop model is a Yang–Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width , the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TLN(β) with loop fugacity β = 2cosλ, . Similarly, on a cylinder, the single-row transfer tangle T(u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models comprise a subfamily of the TL loop models for which the crossing parameter λ = (p′− p)π/p′ is a rational multiple of π parameterized by coprime integers 1 ≤ p < p′. For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers , with β = 0, D(u) and T(u) satisfy inversion identities that have led to the exact determination of the eigenvalues in any representation and for arbitrary finite system size N. The generalization for p′ > 2 takes the form of functional relations for D(u) and T(u) of polynomial degree p′. These derive from fusion hierarchies of commuting transfer tangles Dm, n(u) and Tm, n(u), where D(u) = D1,1(u) and T(u) = T1,1(u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl–Jones projectors Pk on k = m or k = n nodes. Some projectors Pk are singular for k ≥ p′, but we argue that Dm, n(u) and Tm, n(u) are nonsingular for every in certain cabled link state representations. For generic λ, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p′ translates into functional relations of polynomial degree p′ for Dm, 1(u) and Tm, 1(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability, and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

Continuum limit and symmetries of the periodic [formula omitted] spin chain

This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley–Lieb (TL) algebra and the quantum group Uqsℓ(2). The extension of this analysis in the bulk case involves considerable difficulties, since the Uqsℓ(2) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones–Temperley–Lieb — JTL — algebra). Even the simplest case of the gℓ(1|1) spin chain — corresponding to the c=−2 symplectic fermions theory in the continuum limit — presents very rich aspects, which we will discuss in several papers.In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the gℓ(1|1) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of Uqsℓ(2) at q=i that we dub Uqoddsℓ(2). We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress–energy tensor, we identify families of JTL elements going over to the Virasoro generators Ln,L¯n in the continuum limit. We then discuss the sℓ(2) symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view.The analysis of the spin chain as a bimodule over Uqoddsℓ(2) and JTLN is discussed in the second paper of this series.

Lattice fusion rules and logarithmic operator product expansions

The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley–Lieb (TL) fusion functor introduced in physics by Read and Saleur [N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules at roots of unity cases. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the “lattice” fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator product expansions of quantum fields.

YANG–BAXTER EQUATIONS IN QUANTUM INFORMATION

The connection between Yang–Baxter system and quantum information has been discussed. Based on the topological basis for both Temperley–Lieb (TL) algebra and Birman–Wenzl (BW) algebra the representations of N by N braiding matrices associated with the corresponding N2 by N2 ones are obtained. Some of physical consequences of the braiding matrices connecting with quantum information are shown.

Knot theory related to generalized and cyclotomic Hecke algebras of type ℬ

In [12] is established that knot isotopy in a 3-manifold may be interpreted in terms of Markov braid equivalence and, also, that the braids related to the 3-manifold form algebraic structures. Moreover, the sets of braids related to the solid torus or to the lens spaces L(p, 1) form groups, which are in fact the Artin braid groups of type B. As a consequence, in [12, 13] appeared the first construction of a Jones-type invariant using Hecke algebras of type B, and this had a natural interpretation as an isotopy invariant for oriented knots in a solid torus. In a further ‘horizontal’ development and using a different technique we constructed in [8] all such solid torus knot invariants derived from the Hecke algebras of type B. Furthermore, in [7] all Markov traces related to the Hecke algebras of type D were consequently constructed.