Topological order in interacting 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 non-Abelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, 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 (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and ${\mathrm{QED}}₃$ to $3+1$ dimensions. In addition we discuss a $(3+1)$-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.