Classification of Non-Unitary MTC and Topological Phases

Unitary modular tensor categories (MTCs) can characterize 2+1D topological phases with their modular data. Due to the rank-finiteness for MTCs, there have been many efforts to classify MTCs by the rank. Recent studies classified MTCs (both unitary and non-unitary) completely up to rank 5, and partially for higher ranks. Meanwhile, modular data of non-unitary MTCs characterize non-unitary conformal field theories, which are relevant for the Yang-Lee singularity or Gaffnian quantum Hall states. We classify non-unitary MTCs with rank greater than 5 computationally by checking consistency conditions of category theory and present the table of non-unitary MTCs for low ranks. Importantly, the consistency conditions for non-unitary MTCs can be obtained from that of unitary MTCs by allowing negative quantum dimensions, or equivalently allowing negative first row entries of the S matrices. Our classification table contains the fusion rules, the central charges, the quantum dimensions, and the modular data of each non-unitary MTC. We believe that our table would be a useful reference for researchers who study category theory or topological phases.