The burgeoning field of “fractons,” a class of models where quasiparticles are strictly immobile or display restricted mobility that can be understood through generalized multipolar symmetries and associated conservation laws, is reviewed. With a focus on merely a corner of this fast-growing subject, it is demonstrated how one class of such theories, symmetric tensor and coupled-vector gauge theories, surprisingly emerge from familiar elasticity of a two-dimensional quantum crystal. The disclination and dislocation crystal defects, respectively, map onto charges and dipoles of the fracton gauge theory. This fracton-elasticity duality leads to predictions of fractonic phases and quantum phase transitions to their descendants that are duals of the commensurate crystal, supersolid, smectic, and hexatic liquid crystals, as well as amorphous solids, quasicrystals, and elastic membranes. It is shown how these dual gauge theories provide a field-theoretic description of quantum melting transitions through a generalized Higgs mechanism. It is demonstrated how they can be equivalently constructed as gauged models with global multipole symmetries. Extensions of such gauge-elasticity dualities to generalized elasticity theories are expected to provide a route to the discovery of new fractonic models and their potential experimental realizations.

• Received 31 October 2022

DOI:https://doi.org/10.1103/RevModPhys.96.011001