We reveal the connection between two-dimensional subsystem symmetry-protected topological (SSPT) states and two-dimensional topological orders via a self-dual frustrated toric code model. This model, an enrichment of the toric code (TC) with its dual interactions, can be mapped to a model defined on the dual lattice with subsystem symmetries and subextensive ground state degeneracy. The map connects exactly the frustrated TC to two copies of the topological plaquette Ising model (TPIM), as a strong SSPT model with linear subsystem symmetries. The membrane order parameter of the TPIM is exactly mapped to dual TC stabilizers as the order parameter of the frustrated TC model, SSPT gapless edge states of the TPIM are mapped to zero-energy dangling operators under open boundaries, and the transition from the SSPT-ordered TPIM to the trivial paramagnetic phase is mapped to the transition between two distinct topological orders. We also demonstrate that this mapping can be used to elucidate the structure of other SSPT models, reflecting the subtle linkage between SSPT order and topological order in two dimensions.
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.