  The energy spectra of linearly dispersing gapless spin-3/2 Dirac fermions
display birefringence, featuring two effective Fermi velocities, thus breaking
the space-time Lorentz symmetry. Here, we consider a non-Hermitian (NH)
generalization of this scenario by introducing a masslike anti-Hermitian
birefringent Dirac operator to its Hermitian counterpart. The resulting NH
operator shows real eigenvalue spectra over an extended NH parameter regime,
and a combination of non-spatial and discrete rotational symmetries protects
the gapless nature of such quasiparticles. However, at the brink of dynamic
mass generation, triggered by Hubbardlike local interactions, the birefringent
parameter always vanishes under coarse grain due to Yukawa-type interactions
with scalar bosonic order-parameter fluctuations. The resulting quantum
critical state is, therefore, described by two decoupled copies of spin-1/2
Dirac fermions with a unique terminal Fermi velocity, which is equal to the
bosonic order-parameter velocity, thereby fostering an emergent space-time
Lorentz symmetry. Furthermore, depending on the internal algebra between the
anti-Hermitian birefringent Dirac operator and the candidate mass order, the
system achieves the emergent Yukawa-Lorentz symmetry either by maintaining its
non-Hermiticity or by recovering a full Hermiticity. We discuss the resulting
quantum critical phenomena and possible microscopic realizations of the
proposed scenarios.
