Classification of fermionic topological orders from congruence representations

Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the Γθ subgroup of the modular group SL2(Z). We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod 12) via a novel method, which does not require the explicit computation of modular extensions.11Nsciescopu