A topological coupling between a Z_N spin model and a Z_N gauge theory produces 'oblique' phases where zero-form and one-form symmetries break to the same subgroup, and an axion extension gives a lattice model with spontaneously broken non-invertible symmetry.
Senthil, Symmetry-protected topological phases of quantum matter
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Intertwined order of generalized global symmetries
A topological coupling between a Z_N spin model and a Z_N gauge theory produces 'oblique' phases where zero-form and one-form symmetries break to the same subgroup, and an axion extension gives a lattice model with spontaneously broken non-invertible symmetry.