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arXiv:1101.2885 · Journal of Statistical Mechanics Theory and Experiment

The Jordan structure of two-dimensional loop models

Alexi Morin-Duchesne, Yvan Saint-Aubin

Published Apr 8, 2011• 39 citations
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Abstract

We show how to use the link representation of the transfer matrix D N of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin–Kasteleyn model with various boundary conditions and parameter and, more specifically, partition functions of the corresponding Q -Potts spin models, with Q = β 2 . The braid limit of D N is shown to be a central element F N (β) of the Temperley–Lieb algebra TL N (β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the Wenzl–Jones projector. With any element of this basis is associated a number of defects d , 0 ≤ d ≤ N , and the basis vectors with the same d span a sector. Because components of these eigenvectors are singular when and , the link representations of F N and D N are shown to have Jordan blocks between sectors d and d ′ when d − d ′ < 2 b and ( d > d ′). When a and b do not satisfy the previous constraint, D N is diagonalizable.

DOI
10.1088/1742-5468/2011/04/p04007

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We show how to use the link representation of the transfer matrix D N of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin–Kasteleyn model with various boundary conditions and parameter and, more specifically, partition functions of the corresponding Q -Potts spin models, with Q = β 2 . The braid limit of D N is shown to be a central element F N (β) of the Temperley–Lieb algebra TL N (β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the Wenzl–Jones projector. With any element of this basis is associated a number of defects d , 0 ≤ d ≤ N , and the basis vectors with the same d span a sector. Because components of these eigenvectors are singular when and , the link representations of F N and D N are shown to have Jordan blocks between sectors d and d ′ when d − d ′ < 2 b and ( d > d ′). When a and b do not satisfy the previous constraint, D N is diagonalizable.
Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, T N , is an element of the periodic Temperley–Lieb algebra , where N is the number of sites on a section of the cylinder, and β = − q − q −1 = 2cos λ and α the weights of contractible and non-contractible loops. The thermodynamic limit of T N is believed to describe a conformal field theory of central charge c = 1 − 6λ 2 /(π(λ − π)). The abstract element T N acts naturally on (a sum of) spaces , similar to those upon which the standard modules of the (classical) Temperley–Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a twist parameter v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of T N both between sectors with distinct defect numbers and within a given sector.
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin–Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley–Lieb algebra extended by the shift operators Ω±1. In this enlarged algebra, the non-contractible loop fugacity is α and the contractible loop fugacity is β. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley–Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Trd (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where β=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to α=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.
Abstract Critical site percolation on the triangular lattice is described by the Yang–Baxter solvable dilute A 2 ( 2 ) loop model with crossing parameter specialized to λ = π 3 , corresponding to the contractible loop fugacity β = − 2 cos 4 λ = 1 . We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley–Lieb algebras. The standard modules for these algebras are labelled by the number of defects d and, in the latter case, also by the twist e i γ . Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c = 0 and conformal weights Δ , Δ ˉ . For the ground states, we find Δ = Δ 1 , d + 1 for strip boundary conditions and ( Δ , Δ ˉ ) = ( Δ γ / π , d / 2 , Δ γ / π , − d / 2 ) for periodic boundary conditions, where Δ r , s = 1 24 ( ( 3 r − 2 s ) 2 − 1 ) . We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For d ⩽ 8 , these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 20 or more c
We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/2} (weight of contractible loops) and α(weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.
Abstract We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].