We define an index for invertible phases of two-dimensional fermionic and bosonic quantum spin systems without any additional symmetry. Conjecturally, it provides a microscopic definition of an invariant related to the chiral central charge of the boundary modes $c_- \bmod 24$ when the effective conformal field theory description is valid. Using this index, we prove that free fermionic systems with Chern number $\nu \bmod 48 \neq 0$ are in a non-trivial invertible phase.
Categories: math-ph cond-mat.str-el math.MP quant-ph