LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g<sub>2</sub>

Macfarlane, Alan

doi:10.1142/s0217751x01004335

Cited by 15 publications

(12 citation statements)

References 31 publications

(65 reference statements)

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“…In this appendix we discuss the G 2 symmetry of the spin-3 Hamiltonian, H 3 = − i P 3 . G 2 is the smallest exceptional semi-simple Lie group and also the automorphism group of the octonion algebra [55]. We will focus on its corresponding Lie algebra g 2 .…”

Section: Appendix B: G2 Symmetry Of the Spin-3 Modelmentioning

confidence: 99%

The case of SU$(3)$ criticality in spin-2 chains

Li,

Quito,

Miranda

et al. 2021

Preprint

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It was proposed in [Chen et al., Phys. Rev. Lett. 114, 145301 (2015)] that spin-2 chains display an extended critical phase with enhanced SU(3) symmetry. This hypothesis is highly unexpected for a spin-2 system and, as we argue, would imply an unconventional mechanism for symmetry emergence. Yet, the absence of convenient critical points for renormalization group perturbative expansions, allied with the usual difficulty in the convergence of numerical methods in critical or small-gapped phases, renders the verification of this hypothetical SU(3)-symmetric phase a non-trivial matter. By tracing parallels with the well-understood phase diagram of spin-1 chains and searching for signatures robust against finite-size effects, we draw criticism on the existence of this phase. We perform non-Abelian density matrix renormalization group studies of multipolar static correlation function, energy spectrum scaling, single-mode approximation, and entanglement spectrum to shed light on the problem. We determine that the hypothetical SU(3) spin-2 phase is, in fact, dominated by ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like behavior in correlation functions that suggests that is perhaps not critical. We further construct an infinite family of spin-S systems with similar ferro-octupolar-dominated quasi-SU(3)-like phenomenology; curiously, we note that the spin-3 version of the problem is located in a subspace of exact G2 symmetry, making this a point of interest for search of Fibonacci topological properties in magnetic systems.

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Section: Appendix B: G2 Symmetry Of the Spin-3 Modelmentioning

confidence: 99%

The case of SU$(3)$ criticality in spin-2 chains

Li,

Quito,

Miranda

et al. 2021

Preprint

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“…Exceptional cases Conserved currents can also be constructed for categories built on exceptional Lie algebras. One nice example comes from the (G 2 ) k category by taking ρ to be the 7-dimensional vector representation (treating G 2 as a subalgebra of so( 7) [69]). Its fusion is…”

Section: Tensor-product Graphs With Cyclesmentioning

confidence: 99%

Integrability and braided tensor categories

Fendley

2020

Preprint

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Many integrable statistical mechanical models possess a fractional-spin conserved current. Such currents have been constructed by utilising quantum-group algebras and ideas from "discrete holomorphicity". I find them naturally and much more generally using a braided tensor category, a topological structure arising in knot invariants, anyons and conformal field theory. I derive a simple constraint on the Boltzmann weights admitting a conserved current, generalising one found using quantum-group algebras. The resulting trigonometric weights are typically those of a critical integrable lattice model, so the method here gives a linear way of "Baxterising", i.e. building a solution of the Yang-Baxter equation out of topological data. It also illuminates why many models do not admit a solution. I discuss many examples in geometric and local models, including (perhaps) a new solution.

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“…Here we are concerned only with the derivation of G 2 . For a systematic discussion of G 2 invariants (in tensorial notation) we refer the reader to Macfarlane [221].…”

Section: Chapter Sixteen G 2 Family Of Invariance Groupsmentioning

confidence: 99%

Group Theory

2011

Problems and Solutions in Mathematics

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LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g₂

Cited by 15 publications

References 31 publications

The case of SU$(3)$ criticality in spin-2 chains

The case of SU$(3)$ criticality in spin-2 chains

Integrability and braided tensor categories

Group Theory

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LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g2

Cited by 15 publications

References 31 publications

The case of SU$(3)$ criticality in spin-2 chains

The case of SU$(3)$ criticality in spin-2 chains

Integrability and braided tensor categories

Group Theory

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Product

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LIE ALGEBRA AND INVARIANT TENSOR TECHNOLOGY FOR g₂