Subsystem symmetry protected topological order

In this work, we introduce a new type of topological order that is protected by subsystem symmetries that 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 nonlocal 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.