Exotic massive fermionic systems with huge vacuum degeneracy at boundaries

We investigate a massive non-relativistic fermionic system exhibiting exotic features. When the mass parameter is set to zero, the system acquires the fermionic subsystem symmetry. Introducing the mass term explicitly breaks this symmetry, resulting in a trivially gapped system in the absence of boundaries. We demonstrate that with the introduction of boundaries, the system remains gapped, but it has a huge vacuum degeneracy. The residual entropy is proportional to the area of the boundary.