3D quantum Hall effect in a topological nodal-ring semimetal

A quantized Hall conductance (not conductivity) in three dimensions has been searched for more than 30 years. Here we explore it in 3D topological nodal-ring semimetals, by employing a minimal model describing the essential physics. In particular, the bulk topology can be captured by a momentum-dependent winding number, which confines the drumhead surface states in a specific momentum region. This confinement leads to a surface quantum Hall conductance in a specific energy window in this 3D system. The winding number for the drumhead surface states and Chern number for their quantum Hall effect form a two-fold topological hierarchy. We demonstrate the one-to-one correspondence between the momentum-dependent winding number and wavefunction of the drumhead surface states. More importantly, we stress that breaking chiral symmetry is necessary for the quantum Hall effect of the drumhead surface states. The analytic theory can be verified numerically by the Kubo formula for the Hall conductance. We propose an experimental setup to distinguish the surface and bulk quantum Hall effects. The theory will be useful for ongoing explorations on nodal-ring semimetals.