Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition

A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the $Z_2$ topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an $N=1-ϵ$ expansion in the number of replicas, where for small $ϵ$ the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Figure 1

(a) A 2D TI ($m<0$) in region $S$ with boundary $C$ is surrounded by trivial insulator ($m>0$). The sign of $Z_{\mathrm{eff}}$ depends on the number of vortices in $S$. (b) For $m=0$ the eigenvalues of ${\mathcal{H}}^{\mathrm{eff}}$ in (10) exhibit a linear zero crossing, which leads to a vanishing vortex fugacity.

Figure 2

(a) and (c) RG flow diagrams based on (12). The stable fixed point at $(t,v)=(0,0)$ is the symplectic metal (SM). The unstable fixed points at ($t^{*}$, $\pm v^{*}$) approach the KT transition for $ϵ=1-N\ll 1$ and for $N\to 0$ are identified with the Anderson transition. (c) includes a third fixed point at ($t_m$), 0 along with a fixed point at ($t_s$), 0 describing a direct transition between TI and I. (b) and (d) are phase diagrams corresponding to (a) and (c).