Dipoles and Anyonic Directional Confinement via Twisted Toric Codes

We introduce a modified 2D toric code Hamiltonian that exhibits explicit anyon confinement along a single spatial direction. By bounding the motion of these confined anyons, we obtain dipolar excitations with restricted mobility. We analyze the resulting logical operators, whose existence depends on the system size, as well as the structure of gapped boundaries and a tensor network representation of the ground state. Furthermore, when confinement is enforced in both directions, fractal-like excitations emerge, resulting in unpaired logical operators. We extend our construction to 3D models, such as the surface code and the X-cube model, leading to novel dipole-loop and dipole-planon excitations that arise from bounding confined excitations. These modifications are implemented through group cohomological twistings–projective representations of finite groups–with most examples based on Z2xZ2.