Solvable critical dense polymers on the cylinder

Pearce, Paul A.; Rasmussen, Jørgen; Villani, Simon

doi:10.1088/1742-5468/2010/02/p02010

Cited by 39 publications

(142 citation statements)

References 41 publications

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“…The elements of the planar TL algebra are called tangles [38,39] and are diagrammatic objects formed by adding or linking together a number of elementary face operators. Noting that 5) we stress that individual connectivity diagrams are themselves tangles. Tangles are linear combinations of planar connectivity diagrams with an even number of free nodes connected by non-intersecting loop segments.…”

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

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Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N ∈ N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL N (β) with loop fugacity β = 2 cos λ, λ ∈ R. 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 LM(p, p ′ ) comprise a subfamily of the TL loop models for which the crossing parameter λ = (p ′ − p)π/p ′ is a rational multiple of π parameterised 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 LM(1, 2), 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 generalisation 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 D m,n (u) and T m,n (u) where D(u) = D 1,1 (u) and T (u) = T 1,1 (u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl-Jones projectors P k on k = m or k = n nodes. Some projectors P k are singular for k ≥ p ′ , but we argue that D m,n (u) and T m,n (u) are nonsingular for every m, n ∈ N 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 D m,1 (u) and T m,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.

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Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

“…The case (p, p ′ ) = (1, 2), for which θ = 1 2 N mπ, was investigated in [5] for m = 1 corresponding to critical dense polymers LM(1, 2). For critical percolation LM (2, 3), the functional relation for the single-row transfer tangle T = T 1,1 on the cylinder can be written as…”

Section: Closure Of the Fusion Hierarchy On The Cylindermentioning

confidence: 99%

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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“…Using the representation in terms of link paths of the PTL, this Hamiltonian is equal up to a factor to the Hamiltonian H obtained from the transfer matrix of the O(n) models [8]. This factor is equal to the sound velocity v s = π γ sin(γ ): In the spin representation of the PTL, using a similarity transformation, the Hamiltonian H can be written as…”

Section: The Density Of Noncontractible Loops Obtained From the Xmentioning

confidence: 99%

“…Pearce, Rasmussen, and Villani [8] have solved the dense polymer model on a cylinder using the single-row transfer matrix and the inverse identity for the transfer matrix. They solved the inverse identity for different boundary conditions, including the case in which noncontractible loops are allowed.…”

Section: Introductionmentioning

confidence: 99%

Noncontractible loops in the denseO(n)loop model on the cylinder

Alcaraz

Brankov

Priezzhev³

et al. 2014

Phys. Rev. E

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A lattice model of critical dense polymers O(n) is considered for finite cylinder geometry. Due to the presence of noncontractible loops with a fixed fugacity ξ , the model at n = 0 is a generalization of the critical dense polymers solved by Pearce, Rasmussen, and Villani. We found the free energy for any height N and circumference L of the cylinder. The density ρ of noncontractible loops is obtained for N → ∞ and large L. The results are compared with those found for the anisotropic quantum chain with twisted boundary conditions. Using the latter method, we derived ρ for any O(n) model and an arbitrary fugacity.

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“…In a logarithmic CFT setting, critical dense polymers LM (1,2) with Robin boundary conditions is described by the Z 4 sector of symplectic fermions [55][56][57][58]. In particular, the characters of the representations with conformal weights (1.3) and half-integer Kac labels are irreducible and associated with Virasoro Verma modules.…”

Section: Introductionmentioning

confidence: 99%

Critical dense polymers with Robin boundary conditions, half-integer Kac labels andZ4fermions

2014

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For general Temperley-Lieb loop models, including the logarithmic minimal models LM(p, p ) with p, p coprime integers, we construct an infinite family of Robin boundary conditions on the strip as linear combinations of Neumann and Dirichlet boundary conditions. These boundary conditions are Yang-Baxter integrable and allow loop segments to terminate on the boundary. Algebraically, the Robin boundary conditions are described by the one-boundary Temperley-Lieb algebra. Solvable critical dense polymers is the first member LM(1, 2) of the family of logarithmic minimal models and has loop fugacity β = 0 and central charge c = −2. Specialising to LM(1, 2) with our Robin boundary conditions, we solve the model exactly on strips of arbitrary finite size N and extract the finite-size conformal corrections using an EulerMaclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double row transfer matrices. This inversion identity is established directly in the Temperley-Lieb algebra. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitised characters related to spaces of coinvariants of Z 4 fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights Δ r,s− 1 2 = 1 32 (L 2 − 4) where L = 2s − 1 − 4r, r ∈ Z, s ∈ N. These conformal weights populate a Kac table with half-integer Kac labels. Fusion of the corresponding modules with the generators of the Kac fusion algebra is examined and general fusion rules are proposed.

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Solvable critical dense polymers on the cylinder

Cited by 39 publications

References 41 publications

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Noncontractible loops in the denseO(n)loop model on the cylinder

Critical dense polymers with Robin boundary conditions, half-integer Kac labels andZ4fermions

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