  In the last few years a lot of exotic and anomalous topological phases were
constructed by proliferating the vortex like topological defects on the surface
of the $3d$ topological insulator (TI). In this work, rather than considering
topological phases at the boundary, we will study quantum critical points
driven by vortex like topological defects. In general we will discuss a
$(2+1)d$ quantum phase transition described by the following field theory:
$\mathcal{L} = \bar{\psi}\gamma_\mu (\partial_\mu - i a_\mu) \psi +
|(\partial_\mu - i k a_\mu)\phi|^2 + r |\phi|^2 + g |\phi|^4$, with tuning
parameter $r$, arbitrary integer $k$, Dirac fermion $\psi$ and complex scalar
bosonic field $\phi$ which both couple to the same $(2+1)d$ dynamical
noncompact U(1) gauge field $a_\mu$. The physical meaning of these
quantities/fields will be explained in the text. We demonstrate that this
quantum critical point has a quasi self-dual nature. And at this quantum
critical point, various universal quantities such as the electrical
conductivity, and scaling dimension of gauge invariant operators can be
calculated systematically through a $1/k^2$ expansion, based on the observation
that the limit $k \rightarrow + \infty$ corresponds to an ordinary $3d$ XY
transition.
