Potts Models and Related Problems in Statistical Mechanics

Martin, Paul

doi:10.1142/0983

Cited by 370 publications

(617 citation statements)

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“…By construction, this action preserves the number of defects. It is linearly extended to any tangle in T L N (β) and any element of V d N , and it is well known that it generates a representation of T L N (β), see [50,51], for example. The matrix representative of c ∈ T L N (β) on the standard module with d defects is denoted by ρ d N (c).…”

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

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%

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

2014

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“…There is a known bijection between Dyck paths with L + 1 sites and non-intersecting link patterns (SU(2) singlets) for a one-dimensional spin 1/2 system with L sites [17]. In Fig.…”

Section: Estimators Of Shared Information For Dyck Paths Configurationsmentioning

confidence: 99%

Shared information in stationary states at criticality

Alcaraz¹,

Rittenberg²

2010

J. Stat. Mech.

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We consider bipartitions of one-dimensional extended systems whose probability distribution functions describe stationary states of stochastic models. We define estimators of the shared information between the two subsystems. If the correlation length is finite, the estimators stay finite for large system sizes. If the correlation length diverges, so do the estimators. The definition of the estimators is inspired by information theory. We look at several models and compare the behavior of the estimators in the finite-size scaling limit. Analytical and numerical methods as well as Monte Carlo simulations are used. We show how the finite-size scaling functions change for various phase transitions, including the case where one has conformal invariance.

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“…The projectors are linear combinations of I and e i 's, but do not introduce crossings either. This construction has been known for some time [4,5,40].…”

Section: The Potts S-matrix In Terms Of Tl Generatorsmentioning

confidence: 99%

“…In fact, following old work in lattice models [4,5], we construct the SO(3) version of the latter within the former. As the TL algebra is of central importance, we introduce it now.…”

Section: Introductionmentioning

confidence: 99%

ExactS-matrices for supersymmetric sigma models and the Potts model

Fendley

Read

2002

J. Phys. A: Math. Gen.

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We study the algebraic formulation of exact factorizable S-matrices for integrable twodimensional field theories. We show that different formulations of the S-matrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the Temperley-Lieb algebra, in various representations. This enables us to construct the S-matrices for certain nonlinear sigma models that are invariant under the Lie "supersymmetry" algebras sl(m + n|n) (m = 1, 2, n > 0), both for the bulk and for the boundary, simply by using another representation of the same algebra. These Smatrices represent the perturbation of the conformal theory at θ = π by a small change in the topological angle θ. The m = 1, n = 1 theory has applications to the spin quantum Hall transition in disordered fermion systems. We also find S-matrices describing the flow from weak to strong coupling, both for θ = 0 and θ = π, in certain other supersymmetric sigma models.

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Potts Models and Related Problems in Statistical Mechanics

Cited by 370 publications

References 0 publications

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

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

Shared information in stationary states at criticality

ExactS-matrices for supersymmetric sigma models and the Potts model

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