  Lorentz spacetime symmetry represents a unifying feature of the fundamental
forces, typically manifest at sufficiently high energies, while in quantum
materials it emerges in the deep low-energy regime. However, its fate in
quantum materials coupled to an environment thus far remained unexplored. We
here introduce a general framework of constructing symmetry-protected Lorentz
invariant non-Hermitian (NH) Dirac semimetals (DSMs), realized by invoking
masslike anti-Hermitian Dirac operators to its Hermitian counterpart. Such NH
DSMs feature purely real or imaginary isotropic linear band dispersion,
yielding a vanishing density of states. Dynamic mass orderings in NH DSMs thus
take place for strong Hubbardlike local interactions through a quantum phase
transition, hosting a non-Fermi liquid, beyond which the system becomes an
insulator. We show that depending on the internal Clifford algebra between the
NH Dirac operator and candidate mass order-parameter, the resulting
quantum-critical fluid either remains coupled with the environment or recovers
full Hermiticity by decoupling from the bath, while always enjoying an emergent
Yukawa-Lorentz symmetry in terms of a unique terminal velocity. We showcase the
competition between such mass orderings, their hallmarks on quasiparticle
spectra in the ordered phases, and the relevance of our findings for correlated
designer NH Dirac materials.
