Fractional Quantum Hall Effect Based on Weyl Orbits

The fractional quantum Hall effect is a well-known demonstration of strongly correlated topological phases in two dimensions. However, the extension of this phenomenon into a three-dimensional

context has yet to be achieved. Recently, the three-dimensional integer quantum Hall effect based

on Weyl orbits has been experimentally observed in a topological semimetallic material under a

magnetic field. This motivates us to ask whether the Weyl orbits can give rise to the fractional

quantum Hall effect when their Landau level is partially filled in the presence of interactions. Here

we theoretically demonstrate that the fractional quantum Hall states based on Weyl orbits can

emerge in a Weyl semimetal when a Landau level is one-third filled. Using concrete models for Weyl

semimetals in magnetic fields, we project the Coulomb interaction onto a single Landau level from

the Weyl orbit and find that the ground state of the many-body Hamiltoian is triply degenerate. We

further show that the ground states exhibit the many-body Chern number of 1/3 and the uniform

occupation of electrons in both momentum and real space, implying that they are the fractional

quantum Hall states. In contrast to the two-dimensional case, the states are spatially localized on

two surfaces hosting Fermi arcs. Additionally, our findings suggest that the excitation properties of

these states resemble those of the Laughlin state in two dimensions, as inferred from the particle

entanglement spectrum of the ground states.