Effective field theory of dipolar braiding statistics in two dimensions

A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022) the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.