Path-dependent Monopole Braiding and Anyonic Dipoles in a Higgsed Tensor Gauge Model

We propose two-dimensional spin models obtained by “Higgsing” the U(1) rank-2 tensor gauge theory with scalar or vector charges on the Lx×Ly square lattice with p degrees of freedom per orbital and three orbitals per unit cell. The resulting spin model is a stabilizer model consisting of three mutually commuting projectors which are, in turn, obtained by Higgsing the mutually commuting Gauss’s law operators and the magnetic field operators in the underlying U(1) lattice gauge theory. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) given by logpGSD = 2 + (1 + δLx mod p,0)(1 + δLy mod p,0) when p is a prime number. Two types of dipoles, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with certain restrictions on monopoles to hop only by p lattice spacing. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, whereas the monopole-dipole braiding is anyonic and depends on the product of the monopole charge and the dipole moment. The braiding phases can be understood as the Aharonov-Bohm phases of some emergent vector potentials.