Subsystem symmetry protected topological order

In this work, we introduce a new type of topological order which is protected by subsystem symmetries which act on lower dimensional subsets of lattice many-body system, e.g. along lines or planes in a three dimensional system. The symmetry groups for such systems exhibit a macroscopic number of generators in the infinite volume limit. We construct a set of exactly solvable models in $2d$ and $3d$ which exhibit such subsystem SPT (SSPT) phases with one dimensional subsystem symmetries. These phases exhibit analogs of phenomena seen in SPTs protected by global symmetries: gapless edge modes, projective realizations of the symmetries at the edge and non-local order parameters. Such SSPT phases are proximate, in theory space, to previously studied phases that break the subsystem symmetries and phases with fracton order which result upon gauging them.