Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

We describe a method for computing the anomaly of any finite unitary symmetry group G acting by finite-depth quantum circuits on a two-dimensional lattice system. The anomaly is characterized by an index valued in the cohomology group H⁴(G,U(1)), which generalizes the Else-Nayak index for locality preserving symmetries of quantum spin chains. We show that a nontrivial index precludes the existence of a trivially gapped symmetric Hamiltonian; it is also an obstruction to ``onsiteability" of the symmetry action. We illustrate our method via a simple example with G=Z₂₂₂₂. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.