New topological quantum order in 2D lattices from non-invertible symmetries

I will introduce an exactly solvable 2D lattice model which reveals a large number of distinct topological phases with non-invertible (generalized) symmetries. In all these topological phases, which have topological ground state degeneracy, a commutative stabilizer monoid of Hermitian operators leave the ground state invariant and can also distinguish *all* local excitations, (These symmetries are indeed symmetry operations.) There exists novel confined fractonic excitations which change the nature of deconfined excitatons profoundly. The fusion rules form an associative but noncommutative. non-Abelian and non-unital category, and are distinct for each of these phases. A class of these phases are adiabatically connected to a limit which can be described in terms of generalized free field theories. I will describe systematic ways to construct such phases. I will also discuss phases which do not have generalized free field limits. These phases have novel forms of non-local entanglement as many of them share the same topological entanglement entropy. They also violate the entanglement bootstrap axioms. When the phases do not have a generalized free field limit, the violation of the entanglement bootstrap axioms can happen for arbitrary large subregions signifying new forms of long-range entanglement.