Half dualization and non-invertible particle-vortex duality defect on lattice

We introduce half dualization as a general principle to construct non-invertible duality defects. The construction performs a duality transformation only in a subregion of spacetime, leaving an interface between the original theory and its dual. A non-invertible duality defect can be hosted on this interface. Half dualization is applicable whenever a local duality transformation is available, and does not depend on the spacetime dimension. We demonstrate the construction procedure in the (2+1)-dimensional Villainized charge-$n$ XY model on a cubic lattice. Half dualization yields a non-invertible particle-vortex duality defect whose fusion with its orientation reverse produces a (1+1)-dimensional $\mathbb{Z}_n$ gauge theory on the fusion surface. We then apply half dualization to the $\mathbb{Z}_n$-gauged XY model relevant to the 3D XY$^\ast$ transition. In the gauged model, the half dualization interface hosts a gauge covariant duality wall, rather than a genuine gauge invariant duality defect. It flows to an invertible duality defect in infrared when the $\mathbb{Z}_n$ gauge theory is confined or Higgsed.