In and around Abelian anyon models

Abstract

Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern-Simons theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a $K$-matrix in Chern-Simons theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.

substantial revisions with new introduction and an added summary section

References in corpus (3)

• Direct observation of anyonic braiding statistics at the $\nu$=1/3 fractional quantum Hall state
• Fractional statistics in anyon collisions
• On classification of extremal non-holomorphic conformal field theories

Cited by in corpus (13)

• Pauli stabilizer models of twisted quantum doubles
• Pauli topological subsystem codes from Abelian anyon theories
• Fractonic order in infinite-component Chern-Simons gauge theories
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Galois Orbits of TQFTs: Symmetries and Unitarity
• XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
• Generalized toric codes on twisted tori for quantum error correction
• Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases
• Fermionic CFTs from topological boundaries in abelian Chern-Simons theories
• Generalized Statistics on Lattices
• Ordering the topological order in the fractional quantum Hall effect
• Topological stabilizer models on continuous variables
• Partons from stabilizer codes