  Topological phase transitions in condensed matters accompany emerging
singularities of the electronic wave function, often manifested by gap-closing
points in the momentum space. In conventional topological insulators in three
dimensions (3D), the low energy theory near the gap-closing point can be
described by relativistic Dirac fermions coupled to the long range Coulomb
interaction, hence the quantum critical point of topological phase transitions
provides a promising platform to test the novel predictions of quantum
electrodynamics. Here we show that a new class of quantum critical phenomena
emanates in topological materials breaking either the inversion symmetry or the
time-reversal symmetry. At the quantum critical point, the theory is described
by the emerging low energy fermions, dubbed the anisotropic Weyl fermions,
which show both the relativistic and Newtonian dynamics simultaneously. The
interplay between the anisotropic dispersion and the Coulomb interaction brings
about a new screening phenomena distinct from the conventional Thomas-Fermi
screening in metals and logarithmic screening in Dirac fermions.
