Abstract: Exactly solvable spin models such as toric codes and X-cube model have heightened our understanding of spin liquids and topological matter in two and three dimensions. Their exact solvability, it turns out, is rooted in the existence of commuting generators in their parent lattice gauge theory (LGT). We can understand the toric codes as Higgsed descendants of the rank-1 U(1) LGT in two and three dimensions, and the X-cube model as that of rank-2 U(1) LGT in three dimensions. Furthermore, the transformation properties of the gauge fields in the respective LGT is responsible for, and nearly determines the structure of the effective field theory (EFT) of the accompanying matter fields. We show how to construct the EFT of e and m particles in the toric codes and of fractons and lineons in the X-cube model by following such an idea. Recently we proposed some stabilizer Hamiltonians termed rank-2 toric code (R2TC) and F3 model (3D). We will explain what they are, and construct their EFTs using the gauge principle as guidance. The resulting field theory of the matter fields are usually highly interacting and exhibit unusual conservation laws. Especially for the R2TC, we demonstrate the existence of what we call the “dipolar braiding statistics” and outline the accompanying field theory which differs from the usual BF field theory of anyon braiding.
 “Model for fractions, fluxons, and free verte excitations”, JT Kim, JH Han, Phys. Rev. B 104, 115128 (2021)
 “Rank-2 toric code in two dimensions”, YT Oh, JT Kim, EG Moon, JH Han, Phys. Rev. B 105, 045128 (2022)
 “Effective field theory for the exactly solvable stabilizer spin models”, JT Kim, YT Oh, JH Han, in preparation.
 “Effective field theory of dipolar braiding statistics in two dimensions”, YT Oh, JT Kim, JH Han, in preparation.