Massive higher-spin fields in the fractional quantum Hall effect

Incompressibility plays a key role in the geometric description of fractional quantum Hall fluids. It is naturally related to quantum area-preserving diffeomorphisms and the underlying Girvin-MacDonald-Plazman algebra, which gives rise to an emergent nonrelativistic massive spin-2 mode propagating in the bulk. The corresponding metric tensor can be identified with a nematic order parameter for the bulk states. In the linearized regime with a flat background, it has been shown that this mode can be described by a spin-2 Schr\"odinger action. However, quantum area-preserving diffeomorphisms also suggest the existence of higher-spin modes that cannot be described through nematic fractional quantum Hall states. Here, we consider p-atic Hall phases, in which the corresponding p-atic order parameters are related to higher-rank symmetric tensors. We then show that in this framework, nonrelativistic massive chiral higher-spin fields naturally emerge and that their dynamics is described by higher-spin Schr\"odinger actions. We finally show that these effective actions can be derived from relativistic massive higher-spin theories in $2+1$ dimensions after taking a nonrelativistic limit.