Remarks on invertible phases with non-onsite symmetry

We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic ${\mathbb{Z}}_4^F$ symmetry in (2+1) dimensions. For an onsite ${\mathbb{Z}}_4^F$ symmetry, symmetric invertible phases have integer chiral central charge $c_{-}\in \mathbb{Z}$, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local ${\mathbb{Z}}_4^F$ symmetry of a $p+ip$ superconductor with $c_{-}=1\mathrm{/}2$. We compute its lattice anomaly, and find that the symmetry indeed forbids $c_{-}\in \mathbb{Z}$ invertible states. The allowed invertible phases are consequently shifted to $c_{-}\in \mathbb{Z}+1\mathrm{/}2$. By gauging the fermion parity of the $p+ip$ superconductor, one obtains an Ising topological order enriched by a ${\mathbb{Z}}_4$ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the $p+ip$ state admits an enlarged non-onsite $U(1{)}^f$ symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite $U(1{)}^f$ symmetry again forbids Chern insulators carrying $c_{-}\in \mathbb{Z}$, while being compatible with invertible states with $c_{-}\in \mathbb{Z}+1\mathrm{/}2$. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.