Gapped and gapless topological order in interacting Weyl semimetals

It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with

the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions. Here we point out that a second distinct fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first (and as the noninteracting Weyl semimetal), but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization of the known duality between a noninteracting two-dimensional Dirac fermion and QED$_3$ to $(3+1)$ dimensions.