Exact<i>S</i>-matrices for supersymmetric sigma models and the Potts model

Fendley, Paul; Read, N.

doi:10.1088/0305-4470/35/50/301

Cited by 31 publications

(65 citation statements)

References 53 publications

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“…(The fusion index m should not be confused with the minimal model label m used above.) The explicit decomposition coefficients generalise results of [44,45] for (n, n)-fusion and their computation is deferred to Appendix A. Planar identities similar to the ones in the non-fused case are obtained for the fused faces. The generalisation of the boundary Yang-Baxter equation to the fused setting is nontrivial and a proof is presented in Appendix B, following ideas developed in [29].…”

Section: Lm(pmentioning

confidence: 74%

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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: Lm(pmentioning

confidence: 74%

“…Along with the corresponding coefficients α 2,2 a , diagrams similar to (3.8) were introduced in [44] and later generalised to the n × n case in [45]. The (2, 2)-fused loop model and its conformal properties are investigated in [35].…”

Section: Fused Face Operatorsmentioning

confidence: 99%

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

2014

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“…In the anyonic version, the truncated su(2) k representations play a central role. For a related discussion in the general context of loop models, we refer to 51,52 .…”

Section: Discussionmentioning

confidence: 99%

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

et al. 2013

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There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2) k anyons, well-known deformations of SU (2), in which only the first k + 1 angular momenta of SU(2) occur. In this manuscript, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S = 1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2) k anyonic theories with k ≥ 5, as well as for the special case of the su(2)4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

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“…E i E i+1 E i = E i ) are straightforward to work out using the Temperley-Lieb algebra; they can be found for example in Ref. 42. Most become fairly obvious after drawing the appropriate picture.…”

Section: B the So(3) Theorymentioning

confidence: 99%

“…(4.4). 42,57,59,60 However, when Q is an integer (k = 2, 4, ∞), this S matrix is diagonal, so the braiding which follows from it is Abelian. For non-Abelian statistics, we need to use the Potts model for Q not an integer.…”

Section: B the So(3) Theorymentioning

confidence: 99%

Realizing non-Abelian statistics in time-reversal-invariant systems

2005

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We construct a series of 2+1-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of 1+1-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical twodimensional Potts model. We discuss the transition from a topological phase to a conventionallyordered phase, showing in some cases there is a quantum critical point.

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ExactS-matrices for supersymmetric sigma models and the Potts model

Cited by 31 publications

References 53 publications

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

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

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

Realizing non-Abelian statistics in time-reversal-invariant systems

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