Spacetime Symmetries of the Quantum Hall Effect

TL;DR

<p>The fractional quantum Hall problem is approached from a symmetry-centric, non-relativistic geometric perspective. By introducing sources for energy density and energy current and promoting non-relativistic diffeomorphism invariance to full spacetime diffeomorphism invariance, the authors derive comprehensive Ward identities and recast them covariantly using Newton–Cartan geometry with torsion. A massless lowest Landau level limit is achieved by a special choice of spin and gyromagnetic ratio (g=2, s=1), enabling exact integration of higher Landau levels and a consistent LLL theory with enhanced symmetry. The framework yields explicit viscosity–conductivity relations, a tensorial decomposition of response functions, and a covariant action for currents, providing a robust foundation for analyzing Hall transport and related phenomena in FQH systems.

Abstract

We study the symmetries of non-relativistic systems with an emphasis on applications to the fractional quantum Hall effect. A source for the energy current of a Galilean system is introduced and the non-relativistic diffeomorphism invariance studied in previous work is enhanced to a full spacetime symmetry, allowing us to derive a number of Ward identities. These symmetries are smooth in the massless limit of the lowest Landau level. We develop a formalism for Newton-Cartan geometry with torsion to write these Ward identities in a covariant form. Previous results on the connection between Hall viscosity and Hall conductivity are reproduced.