Classifying Subsystem Symmetry Protected Topological Phases

We discuss symmetry protected topological (SPT) phases in 2D systems with subsystem symmetries: symmetries which act on rigid subsystems, such as along straight lines or fractals.
The total symmetry group of such systems grows with system size and is infinitely large in the thermodynamic limit.
Systems with linear subsystem symmetries exhibit a phenomenon wherein two states from different phases may differ only along a subsystem --- leading to an infinitude of possible phases.
To address this issue we identify and classify the ``intrinsic'' information of phase which can be measured locally.
This may be likened to equivalence classes of phases which differ by only transformations along subsystems.
We show that for linear subsystem SPTs, despite there being infinitely many phases, the classification of this intrinsic information is finite and depends only on the on-site symmetry group.
For fractal subsystem symmetries, this is not necessary: locality is enough to enforce a certain translation symmetry which leads to a number of possible phases.
We give an upper bound for the number of phases in such models where Hamiltonian terms are supported within L-by-L boxes.
These phases showcase how subsystem symmetries can lead to intriguing new physics even in simple cases