Collective States of Interacting Fibonacci Anyons

Trebst, Simon; Ardonne, Eddy; Feiguin, Adrian; Huse, David A.; Ludwig, Andreas; Troyer, Matthias

doi:10.1103/physrevlett.101.050401

Cited by 84 publications

(98 citation statements)

References 17 publications

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“…19 We then see that the independent D A are in one-to-one correspondence with the (pairs of left and right) Virasoro representations present in the theory (for simplicity, we are assuming here that there is no degeneracy in the spectrum of Virasoro representations). In the next sections we will see that modular covariance of the CFT in the presence of topological defects implies further constraints on the coefficients, AB , but for now we simply note that the D A act like projectors.…”

Section: Defining the Defect Commutator And A Theorem On The Non-trivmentioning

confidence: 99%

“…We will also use the fact that C in contains a set of isomorphisms, called "F-symbols", that allow us to move between different fusion channels of the objects in an associative way. The case C in = Rep [su (2) k ] was studied in [8,9,15,[17][18][19], but we will be more general.…”

Section: Anyonic Chainsmentioning

confidence: 99%

“…In the gapped phase the topological symmetry is broken spontaneously and leads to degeneracy of the ground state as well as to fractionalized anyonic edge excitations, similarly to what happens in the AKLT model. Another natural generalization is the inclusion of the Majumdar-Ghosh type of interaction, which has been studied in spin-1 2 chains [19] (where it also produces a rich phase diagram).…”

Section: The Su(2)mentioning

confidence: 99%

“…After briefly mentioning its generalizations, we move on to discuss anyonic chains [8,9,15,[17][18][19], and we review the construction of the resulting Hilbert spaces, Hamiltonians, and topological symmetries.…”

Section: Anyonic Chainsmentioning

confidence: 99%

“…19 Our notation in (4.3) is somewhat schematic. Indeed, |B stands for a full Virasoro representation-the primary and its descendants.…”

Section: Defining the Defect Commutator And A Theorem On The Non-trivmentioning

confidence: 99%

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Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican

Gromov

2017

Commun. Math. Phys.

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Abstract:Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.

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Section: Defining the Defect Commutator And A Theorem On The Non-trivmentioning

confidence: 99%

Section: Anyonic Chainsmentioning

confidence: 99%

Section: The Su(2)mentioning

confidence: 99%

Section: Anyonic Chainsmentioning

confidence: 99%

“…19 Our notation in (4.3) is somewhat schematic. Indeed, |B stands for a full Virasoro representation-the primary and its descendants.…”

Section: Defining the Defect Commutator And A Theorem On The Non-trivmentioning

confidence: 99%

See 3 more Smart Citations

Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican

Gromov

2017

Commun. Math. Phys.

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Topological field theories and symmetry protected topological phases with fusion category symmetries

Inamura

2021

J. High Energ. Phys.

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Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

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Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Delcamp

2022

J. High Energ. Phys.

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Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group G, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories 2VecG of G-graded 2-vector spaces and 2Rep(G) of 2-representations of G, respectively. In virtue of the Morita equivalence between 2VecG and 2Rep(G) — which we explicitly establish — the topological order can be realised as the Drinfel’d centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case G = ℤ2, we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure ℤ2 gauge theory and the Kramers-Wannier duality of a boundary Ising model.

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Collective States of Interacting Fibonacci Anyons

Cited by 84 publications

References 17 publications

Anyonic Chains, Topological Defects, and Conformal Field Theory

Anyonic Chains, Topological Defects, and Conformal Field Theory

Topological field theories and symmetry protected topological phases with fusion category symmetries

Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

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