  A sufficiently strong long-range Coulomb interaction can induce excitonic
pairing in gapless Dirac semimetals, which generates a finite gap and drives
semimetal-insulator quantum phase transition. This phenomenon is in close
analogy to dynamical chiral symmetry breaking in high energy physics. In most
realistic Dirac semimetals, including suspended graphene, Coulomb interaction
is too weak to open an excitonic gap. The Coulomb interaction plays a more
important role at low energies in a two-dimensional semi-Dirac semimetal, in
which the fermion spectrum is linear in one component of momenta and quadratic
in the other, than a Dirac semimetal, and indeed leads to breakdown of Fermi
liquid theory. We study dynamical excitonic gap generation in a two-dimensional
semi-Dirac semimetal by solving the Dyson-Schwinger equation, and show that a
moderately strong Coulomb interaction suffices to induce excitonic pairing.
Additional short-range four-fermion coupling tends to promote excitonic
pairing. Among the available semi-Dirac semimetals, we find that
TiO$_{2}$/VO$_{2}$ nanostructure provides a promising candidate for the
realization of excitonic insulator. We also apply the renormalziation group
method to analyze the strong coupling between the massless semi-Dirac fermions
and the quantum critical fluctuation of excitonic order parameter at the
semimetal-insulator quantum critical point, and reveal non-Fermi liquid
behaviors of semi-Dirac fermions.
