Classification of subsystem symmetry-protected topological phases

We consider symmetry-protected topological (SPT) phases in two dimensions protected by linear subsystem symmetries, i.e., those that act along rigid lines. There is a distinction between a ``strong'' subsystem SPT phase and a ``weak'' one, which is composed of decoupled one-dimensional SPTs with global symmetries. We propose a natural definition for strong equivalence of such phases, in terms of a linearly symmetric local unitary transformation, under which a weak subsystem SPT is equivalent to the trivial phase. This leads to a number of distinct equivalence classes of strong subsystem SPTs, which we show are in one-to-one correspondence with elements of the group C[Gₛ]=H²[Gₛ²,U(1)]/H²[Gₛ,U(1)]³, where Gₛ is the finite Abelian on-site symmetry group. We also show that strong subsystem SPTs by our classification necessarily exhibit a spurious topological entanglement entropy on a cylinder.