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Weyl and Dirac (semi)metals in three dimensions have robust gaplesselectronic band structures. Their massless single-body energy spectra areprotected by symmetries such as lattice translation, (screw) rotation and timereversal. In this manuscript, we discuss many-body interactions in thesesystems. We focus on strong interactions that preserve symmetries and areoutside the single-body mean-field regime. By mapping a Dirac (semi)metal to amodel based on a three dimensional array of coupled Dirac wires, we show (1)the Dirac (semi)metal can acquire a many-body excitation energy gap withoutbreaking the relevant symmetries, and (2) interaction can enable an anomalousWeyl (semi)metallic phase that is otherwise forbidden by symmetries in thesingle-body setting and can only be present holographically on the boundary ofa four dimensional weak topological insulator. Both of these topological statessupport fractional gapped (gapless) bulk (resp. boundary) quasiparticleexcitations.

From Dirac semimetals to topological phases in three dimensions: a coupled wire construction