Anyon braiding and the renormalization group

Stottmeister, Alexander

doi:10.48550/arxiv.2201.11562

Cited by 2 publications

(3 citation statements)

References 41 publications

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“…We next observe that both in fig. 44 46, we have summations over ONBs p1 ρ 1 ˆf1 qf 1 1 , p1 ν 1 ˆf2 qf 1 2 , e 11 pe 1 ˆ1ν 1 q, e 12 pe 2 ˆ1ρ 1 q respectively their adjoints. By unitarity of the 6j-symbols we can switch to ONBs of the form pf 1 ˆ1λ 1…”

Section: ) Consider the Operatormentioning

confidence: 99%

Anyonic Chains -- $α$-Induction -- CFT -- Defects -- Subfactors

Hollands¹

2022

Preprint

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Given a unitary fusion category, one can define the Hilbert space of a so-called "anyonic spin-chain" and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to 1 `1-dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bi-partite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of 1`1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called "double triangle algebra" arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on α-induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.

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Section: ) Consider the Operatormentioning

confidence: 99%

Anyonic Chains -- $α$-Induction -- CFT -- Defects -- Subfactors

Hollands¹

2022

Preprint

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“…[19,27]. There is evidence in special models that the algebra generated by the e x can be identified with a product of left-and right-moving Virasoro algebras in a suitable conformal scaling limit of the chain, see [27,44] which uses ideas by [45].…”

Section: Inclusions Of Von Neumann Algebras and Anyonic Chainsmentioning

confidence: 99%

“…4 44. we have a description of (B + ) ∩ B + in terms of the objects A ∈ M X M ; more precisely for each A one can construct in a canonical way a minimal projectionp A ∈ (B + ) ∩ B + .…”

mentioning

confidence: 99%

Anyonic Chains – $$\alpha $$-Induction – CFT – Defects – Subfactors

Hollands

2022

Commun. Math. Phys.

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Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to $$1+1$$ 1 + 1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of $$1+1$$ 1 + 1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on $$\alpha $$ α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.

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Anyon braiding and the renormalization group

Cited by 2 publications

References 41 publications

Anyonic Chains -- $α$-Induction -- CFT -- Defects -- Subfactors

Anyonic Chains -- $α$-Induction -- CFT -- Defects -- Subfactors

Anyonic Chains – $$\alpha $$-Induction – CFT – Defects – Subfactors

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