Gapless layered three-dimensional fractional quantum Hall states

Using the parton construction, we build a three-dimensional (3D) multilayer fractional quantum Hall state with average filling ν=1/3 per layer that is qualitatively distinct from a stacking of weakly coupled Laughlin states. The state supports gapped charge $e/3$ fermionic quasiparticles that can propagate both within and between the layers, in contrast to the quasiparticles in a multilayer Laughlin state which are confined within each layer. Moreover, the state has gapless neutral collective modes, a manifestation of an emergent ``photon,'' which is minimally coupled to the fermionic quasiparticles. The surface sheath of the multilayer state resembles a chiral analog of the Halperin-Lee-Read state, which is protected against gap-forming instabilities by the topological character of the bulk 3D phase. We propose that this state might be present in multilayer systems in the ``intermediate tunneling regime,'' where the interlayer tunneling strength is on the same order as the Coulomb energy scale. We also find that the parton construction leads to a candidate state for a bilayer ν=1/3 system in the intermediate tunneling regime. The candidate state is distinct from both a bilayer of ν=1/3 Laughlin states and the single layer ν=2/3 state but is, nonetheless, a fully gapped fractional quantum Hall state with charge $e/3$ anyonic quasiparticles.