  Understanding the correlation effects in unconventional topological
materials, in which the fermion excitations take unusual dispersion, is an
important topic in recent condensed matter physics. We study the influence of
short-range four-fermion interactions on three-dimensional semi-Dirac semimetal
with an unusual fermion dispersion, that is linear along two directions but
quadratic along the third one. Based on renormalization group theory, we find
all of 11 unstable fixed points including 5 quantum critical points, 5
bicritical points, and one tricritical point. The physical essences of the
quantum critical points are determined by analyzing the susceptibility
exponents for all of the source terms in particle-hole and particle-particle
channels. We also verify phase diagram of the system in the parameter space
carefully through numerically studying the flows of the four-fermion coupling
parameters and behaviors of the susceptibility exponents. These results are
helpful for us to understand the physical properties of candidate materials for
three-dimensional semi-Dirac semimetal such as ZrTe$_{5}$.

Strongly Correlated Electrons;Materials Science