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

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 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 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.