On classification of modular tensor categories

Abstract

We classify all unitary modular tensor categories (UMTCs) of rank $\le 4$. There are a total of 70 UMTCs of rank $\le 4$ (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs distinguished by the modular $S$ matrix $S=(\pm 1)$. Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. UMTCs encode topological properties of anyonic quantum systems and can be used to build fault-tolerant quantum computers. We conjecture that there are only finitely many equivalence classes of MTCs for any given rank, and a UMTC is universal for anyonic quantum computation if and only if its global quantum dimension $D^2$ is \emph{not} an integer. Discovery of non-abelain anyons in Nature will be a landmark in condensed matter physics. The non-abelian anyons in UMTCs of rank $\le 4$ are the simplest, and, therefore, are most likely to be found. G. Moore and N. Read proposed that non-abelian statistics could occur in fractional quantum Hall (FQH) liquids. The Read-Rezayi conjecture predicts the existence of anyons related to $SU(2{)}_k$ in FQH liquids at filling fractions $\nu =2+\frac{k}{k+2}$ for $k=1,2,3$. For $\nu =5/2$ and $SU(2{)}_2$, there is a numerical proof and experimental evidence for this conjecture. The Ising anyons might exist in chiral superconductors (strontium ruthenate). There are theoretical designs for the toric code MTC using Josephson junction array, for the $Fib\times Fib$ MTC using optical lattice, and for the $SU(2{)}_k$ using cold trapped bosonic atoms.

Several references added. Minor revisions of the abstract, introduction and Section 2. Error in one R-matrix eigenvalue for (A_1,2) in the published version corrected

References in corpus (16)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• String-net condensation: A physical mechanism for topological phases
• Topological orders and Edge excitations in FQH states
• Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
• Interacting anyons in topological quantum liquids: The golden chain
• A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State
• The kernel of the modular representation and the Galois action in RCFT
• Conformal invariance of chiral edge theories
• Topological properties of Abelian and non-Abelian quantum Hall states from the pattern of zeros
• Large Fourier transforms never exactly realized by braiding conformal blocks
• The N-eigenvalue Problem and Two Applications
• Hopf Algebras and Congruence Subgroups
• Some non-braided fusion categories of rank 3
• Integral modular data and congruences

Cited by in corpus (12)

• Condensate induced transitions between topologically ordered phases
• Physics, Topology, Logic and Computation: A Rosetta Stone
• Quantum computation with Turaev-Viro codes
• Classification of Rank 5 Premodular Categories
• Holomorphic modular bootstrap revisited
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Hecke Relations, Cosets and the Classification of 2d RCFTs
• Rank 4 Premodular Categories
• Tensor categories: A selective guided tour
• Drinfeld centers of fusion categories arising from generalized Haagerup subfactors
• On the Central Charge of a Factorizable Hopf Algebra
• A finiteness property for braided fusion categories