The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) provide a quite complete way to characterize 2D topological order.The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) provide a quite complete way to characterize 2D topological order.

arXiv:1212.5121Dec 20, 201212 citations

Exact realization of integer and fractional quantum Hall phases in models in

Scott Geraedts, Olexei I. Motrunich

arXiv:1302.1436Annals of PhysicsApr 17, 201341 citations

Modular transformation and anyonic statistics of multicomponent fractional quantum Hall states

Liangdong Hu, Zhao Liu, W. Zhu

We investigate the response to modular transformations and the fractional statistics of Abelian multicomponent fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using the anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the $K$ matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and $(2+1)\mathrm{D}$ topological orders for multicomponent FQH states.We investigate the response to modular transformations and the fractional statistics of Abelian multicomponent fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using the anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the $K$ matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and $(2+1)\mathrm{D}$ topological orders for multicomponent FQH states.

arXiv:2301.06427Physical review. B./Physical review. BDec 5, 20231 citation

Theory of four-dimensional fractional quantum Hall states

Chyh-Hong Chern

arXiv:cond-mat/0606434Annals of PhysicsJan 5, 20073 citations

Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states

Jeffrey C. Y. Teo, Abhishek Roy, Xiao Chen

We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic $(mmn)$ FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations ( $F$ symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global ${\mathbb{Z}}_{2}$ bilayer symmetry.We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic $(mmn)$ FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations ($F$ symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global ${\mathbb{Z}}_{2}$ bilayer symmetry.

arXiv:1308.5984Physical Review BOct 8, 201428 citations

K

matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and

(2+1)\mathrm{D}

topological orders for multicomponent FQH states.

We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic

(mmn)

FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations (

F

symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global

{\mathbb{Z}}_{2}

bilayer symmetry.