The twofold twist defects in the $D({\mathbb{Z}}_{k})$ quantum double model (Abelian topological phase) carry non-Abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one-dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with an even number of twist defects, it is equivalent to the ${\mathbb{Z}}_{k}$ clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the ${\mathbb{Z}}_{k}$ clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the ${\mathbb{Z}}_{k}$ clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for $k=3,4,5$ , and 6. For the even-defect chain, the critical points are the same as the ${\mathbb{Z}}_{k}$ clock conformal field theories (CFTs), while for the odd-defect chain, when $k\ensuremath{\ne}4$, the critical points correspond to orbifolding a ${\mathbb{Z}}_{2}$ symmetry of CFTs of the even-defect chain. For $k=4$ case, we numerically observe some similarity to the ${\mathbb{Z}}_{4}$ twist fields in the $SU{(2)}_{1}/{D}_{4}$ orbifold CFT.The twofold twist defects in the $D({\mathbb{Z}}_{k})$ quantum double model (Abelian topological phase) carry non-Abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one-dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with an even number of twist defects, it is equivalent to the ${\mathbb{Z}}_{k}$ clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the ${\mathbb{Z}}_{k}$ clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the ${\mathbb{Z}}_{k}$ clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for $k=3,4,5$, and 6. For the even-defect chain, the critical points are the same as the ${\mathbb{Z}}_{k}$ clock conformal field theories (CFTs), while for the odd-defect chain, when $k\ensuremath{\ne}4$, the critical points correspond to orbifolding a ${\mathbb{Z}}_{2}$ symmetry of CFTs of the even-defect chain. For $k=4$ case, we numerically observe some similarity to the ${\mathbb{Z}}_{4}$ twist fields in the $SU{(2)}_{1}/{D}_{4}$ orbifold CFT.

arXiv:1709.05560Physical review. B./Physical review. BNov 27, 20173 citations

Disorder and Defects in Topological Phases

Roderich Moessner, Joel E. Moore

Cambridge University Press eBooksApr 9, 20210 citations

Topological phases in one dimension

Antônio Sérgio Teixeira Pires

Institute of Physics eBooksNov 1, 20210 citations

Anyonic Chains, Topological Defects, and Conformal Field Theory

Matthew Buican, Andrey Gromov

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.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.

arXiv:1701.02800Communications in Mathematical PhysicsOct 5, 201786 citations

The twofold twist defects in the

D({\mathbb{Z}}_{k})

quantum double model (Abelian topological phase) carry non-Abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one-dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with an even number of twist defects, it is equivalent to the

{\mathbb{Z}}_{k}

clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the

{\mathbb{Z}}_{k}

clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the

{\mathbb{Z}}_{k}

clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for

k=3,4,5

, and 6. For the even-defect chain, the critical points are the same as the

{\mathbb{Z}}_{k}

clock conformal field theories (CFTs), while for the odd-defect chain, when $k\ensuremath{\ne}4$, the critical points correspond to orbifolding a

{\mathbb{Z}}_{2}

symmetry of CFTs of the even-defect chain. For

k=4

case, we numerically observe some similarity to the

{\mathbb{Z}}_{4}

twist fields in the

SU{(2)}_{1}/{D}_{4}

orbifold CFT.