Theory of a quantum critical phenomenon in a topological insulator: (3+1)-dimensional quantum electrodynamics in solids

Abstract

We study theoretically the quantum critical phenomenon of the phase transition between the trivial insulator and the topological insulator in (3+1) dimensions, which is described by a Dirac fermion coupled to the electromagnetic field. The renormalization group (RG) equations for the running coupling constant $\ensuremath{\alpha}$, the speed of light $c,$ and electron $v$ are derived. The almost exact analytic solutions to these RG equations are obtained to reveal that (i) $c$ and $v$ approach to the common value with combination ${c}^{2}v$ being almost unrenormalized, (ii) the RG flow of $\ensuremath{\alpha}$ is the same as that of usual QED with ${c}^{3}$ being replaced by ${c}^{2}v$, and (iii) there are two crossover momentum/energy scales separating three regions of different scaling behaviors. The dielectric and magnetic susceptibilities, angle-resolved photoemission spectroscopy (ARPES), and the behavior of the gap are discussed from this viewpoint.