Rank-2 toric code in two dimensions

We study a two-dimensional spin model obtained by "Higgsing" the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on L_x * L_y square under periodic boundary condition (PBC). There are p degrees of freedom per orbital and three orbitals unit cell in model. The resulting is stabilizer code consisting mutually commuting projectors that are, turn, Higgsing Gauss's law operators magnetic field underlying LGT. thus exactly solvable, ground state degeneracy (GSD) D given log_p D=2+(1+delta_{L_x mod p,0})(1+delta_{L_y p,0}) when prime number. Two types dipole excitations, pristine emergent, identified. Both monopoles dipoles free to move, restrictions hop only spacing along certain directions. monopole-monopole braiding phase depends separation x y coordinates initial monopole positions, making it distinct from ordinary anyon statistics. monopole-dipole obeys usual anyonic Despite oddity, can be understood as Aharonov-Bohm some emergent potentials.