Anomalous localization at the boundary of an interacting topological insulator

The boundary of a topological insulator (TI) hosts an anomaly restricting its possible phases: e.g., three-dimensional (3D) strong and weak TIs maintain surface conductivity at any disorder if symmetry is preserved on average at least when electron interactions on the surface are weak. However, the interplay of strong interactions and disorder with the boundary anomaly has not yet been theoretically addressed. Here we study this combination for the edge of a two-dimensional TI and the surface of a 3D weak TI, showing how it can lead to an “Anomalous Many Body Localized” (AMBL) phase that preserves the anomaly. We discuss how the anomalous Kramers parity switching with $\pi$ flux arises in the bosonized theory of the localized helical state. The anomaly can be probed in localized boundaries by electrostatically sensing nonlinear hopping transport with $e/2$ shot noise. Our AMBL construction in 3D weak TIs fails for 3D strong TIs, which suggests that their anomaly restrictions are distinguished by strong interactions.

Figure 1

Three-dimensional weak TI and its AMBL surface. The weak TI is pictured as a stack of 2D TIs with a translation vector (dashed arrow) defining its $Z_2$ symmetry. Helical Luttinger liquids (LLs; thick black lines) are bound to dislocations [17] (gray partial plane) and are also bound to domain walls between $Z_2$-translation-broken hybridization domains (blue and red, shown schematically upon coarse graining) on side surfaces. Weak $Z_2$-symmetry-breaking disorder that preserves $Z_2$ on average produces a helical network with a spanning helical mode and sparse junctions. Stronger symmetry-preserving disorder then allows this spanning helical LL to localize as in Fig. 2.

Figure 2

Localized spin-glass state on the edge of a 2D TI and its time-reversal transformation. (a) Spatial profile of the local phonon-type field $\delta \theta \left(x\right)$. (b) Density profile of the Luther-Emery fermions, appearing as $T$-breaking-domain walls carrying charge $\pm e/2$. (c) Boson zero mode of the overall shift ${\theta }_0$; arrows denote its transformation under time reversal $T:\theta \to \theta -\pi /2$. Panels (a) and (b) are adapted from Ref. [13].

Figure 3

Flux insertion anomaly in the localized state. The antiperiodic boundary conditions for electrons when $\pi$ flux is threaded through the 1D edge can be captured by inserting two Luther-Emery fermions, each carrying charge $e/2$ and $T^2=+1$. Tracking many-body $T^2$ via the ${\theta }_0$ “clock” variables shows that they are modified by the $\pi$ flux in such a way as to preserve $T^2$; so far, the total charge $n_f$ has changed (say, by coupling to a metal) but $T^2$ has not, giving the relative factor $\nu =-1$ in Eq. (1). Maintaining physical charge neutrality entails formally removing an electron while inserting the $e/2$ charges; ${\theta }_0$ changes to account for the electron's $T^2=-1$, again giving the relative factor $\nu =-1$ in Eq. (1).

Figure 4

Nonlinear transport and $e/2$ shot noise. Crossover in Eq. (13) of the nonlinear IV hopping transport from heterogeneous nucleation theory. Inset: Illustration of the setup for probing the $e/2$ quantized shot noise. Transport proceeds via thermally activated or quantum tunneling events wherein a droplet of localized $e/2$ charges rearranges and shifts in an applied voltage. An electrostatic probe such as a Kelvin probe (green disk) captures the electric field lines of the carriers via their image charge. The resulting current shot noise shows $e/2$ quantization.