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

TL;DR

This paper investigates a novel self-dual quantum critical point on the surface of a 3D topological insulator, characterized by a specific field theory involving Dirac fermions and scalar bosons coupled to a gauge field, with universal properties calculable via a large-k expansion.

Contribution

It introduces a new self-dual quantum critical point on 3D topological insulator surfaces and develops a systematic method to compute universal quantities using a 1/k^2 expansion.

Findings

Quantum critical point exhibits quasi self-duality.

Universal quantities like conductivity can be calculated systematically.

Large-k limit corresponds to a 3D XY transition.

Abstract

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.