"Self-Dual" Quantum Critical Point on the surface of $3d$ Topological Insulator

Understand

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.