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Supersymmetry (SUSY) interchanges bosons and fermions but no direct evidencesof it have been revealed in nature yet. In this letter, we observe thatfluctuating pair density waves (PDW) consist of two complex order parameterswhich can be superpartners of the unavoidably-doubled Weyl fermions inthree-dimensional lattice models. We construct explicit fermionic latticemodels featuring 3D Weyl fermions and show that PDW is the leading instabilityvia a continuous phase transition as short-range interactions exceed a criticalvalue. Using renormalization group, we theoretically show that N=2 space-timeSUSY emerges at the continuous PDW transitions in 3D Weyl semimetals, which webelieve is the first realization of emergent (3+1)D space-time SUSY inmicroscopic lattice models. We further discuss possible routes to realize suchlattice models and experimental signatures of emergent SUSY at the PDWcriticality.

Emergent space-time supersymmetry in 3D Weyl and 2D Dirac semimetals