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Abstract

Weyl semimetals are three-dimensional analogs of graphene with pointlike Fermi surfaces. Their linear electronic dispersion leads to a window in the particle-hole excitation spectrum which allows for undamped propagation of collective excitations. We argue that interactions in Weyl semimetals generically lead to well-defined exciton modes. However, using a minimal model for interactions, we show that the exciton binding energy is exponentially small for weak interactions. This is due to effective two-dimensional character in the space of particle-hole pairs that are available for bound-state formation. This is ultimately a consequence of linear electronic dispersion in three dimensions. Nevertheless, intermediate interaction strengths can lead to sharp excitonic resonances. We demonstrate this in a model Weyl semimetal with broken time-reversal symmetry and Hubbard interactions. Using generalized random phase approximation analysis, we show that excitonic modes here carry spin. Excitons in Weyl semimetals have evoked interest as their condensation could lead to an axionic charge-density-wave order. However, we find that the leading instability corresponds to intravalley spin density wave order which shifts the Weyl points without opening a gap. Our results suggest interesting directions for experimental studies of three-dimensional Dirac systems.