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arXiv:1402.5990 · Journal of Physics A Mathematical and Theoretical

Cyclic representations of the periodic Temperley–Lieb algebra, complex Virasoro representations and stochastic processes

Francisco Castilho Alcaraz, Arun Ram, V. Rittenberg

Published May 9, 2014• 0 citations
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Abstract

An $N$ ${L} \choose {L/2}$-dimensional representation of the periodic Temperley-Lieb algebra $TL_L(x)$ is presented. It is also a representation of the cyclic group $Z_N$. We choose $x = 1$ and define a Hamiltonian as a sum of the generators of the algebra acting in this representation. This Hamiltonian gives the time evolution operator of a stochastic process. In the finite-size scaling limit, the spectrum of the Hamiltonian contains representations of the Virasoro algebra with complex highest weights. The $N = 3$ case is discussed in detail. One discusses shortly the consequences of the existence of complex Virasoro representations on the physical properties of the systems.

DOI
10.1088/1751-8113/47/21/212003

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In a previous letter (Alcaraz F C et al 2014 J. Phys. A: Math. Theor. 47 212003 ) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley–Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a Z N symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For N even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.
In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O(n) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.
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We discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.
Abstract The Hamiltonian of the N -state clock model is written in terms of a coupled Temperley–Lieb (TL) algebra defined by N − 1 types of TL generators. This generalizes a previous result for N = 3 obtained by Fjelstad and Månsson (2012 J. Phys. A 45 155208). The Z N -symmetric clock chain Hamiltonian expressed in terms of the coupled TL algebra generalizes the well known correspondence between the N -state Potts model and the TL algebra. The algebra admits a pictorial description in terms of a planar algebra involving parafermionic operators attached to n strands. A key ingredient in the resolution of diagrams is the string Fourier transform. The pictorial presentation also allows a description of the Hilbert space. We also give a pictorial description of the representation related to the staggered XX spin chain. Just as the pictorial representation of the TL algebra has proven to be particularly useful in providing a visual and intuitive way to understand and manipulate algebraic expressions, it is anticipated that the pictorial representation of the coupled TL algebra may lead to further progress in understanding various aspects of the Z N clock model, including the superintegrable chiral Potts model.
Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.