On classification of modular tensor categories
Eric Rowell, Richard Stong, Zhenghan Wang
TL;DR
This work classifies all unitary modular tensor categories with rank at most four, showing there are 70 UMTCs (35 up to ribbon equivalence) assembled from 10 prime UMTCs via direct products and symmetries. The authors develop a Galois-theoretic framework tied to Verlinde fusion rules to constrain modular data, then exhaustively determine rank 2–4 S-matrices and realize the corresponding fusion rules with explicit F-, braiding, and twist data, including full details for 10 prime UMTCs such as Semion, Fibonacci, Ising, Toric Code, and several SU(2) level theories. They demonstrate that nine of the ten nontrivial primes arise from quantum groups, with the remainder seen in direct product constructions, and provide a comprehensive table of rank-4 modular symbols and realizations. The results support a finiteness conjecture for fixed rank and have implications for topological quantum computation, including universality classifications of braiding. The paper also outlines broader conjectures connecting modular data to topological phases, fault tolerance, and quantum compiling via universal anyons.
Abstract
We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. There are a total of 70 UMTCs of rank $\leq 4$ (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 $S=(\pm1)$. 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 $\leq 4$ 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 $ν=2+\frac{k}{k+2}$ for $k=1,2,3$. For $ν={5/2}$ 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.