Gapless insulating edges of dirty interacting topological insulators

We demonstrate that a combination of disorder and interactions in a two-dimensional bulk topological insulator can generically drive its helical edge insulating. We establish this within the framework of helical Luttinger liquid theory and exact Emery-Luther mapping. The gapless glassy edge state spontaneously breaks time-reversal symmetry in a “spin glass” fashion, and may be viewed as a localized state of solitons, which carry half-integer charge. Such a qualitatively distinct edge state provides a simple explanation for heretofore puzzling experimental observations. This phase exhibits a striking nonmonotonicity, with the edge growing less localized in both the weak and strong disorder limits.

Figure 1

The phase diagram of a disordered, interacting edge state of a 2D topological insulator. The horizontal axis is the Luttinger parameter $K$, which encodes the interaction. $K=1$ is the noninteracting limit and $K=1/4$ is the strongly repulsive, exactly solvable Luther-Emery point. The vertical axis is the measure of the incommensuration $\delta Q=(4k_F-Q)$ between electron and ion densities, with $Q$ the reciprocal lattice vector. The red solid curve denotes the phase boundary between the gapless insulating edge (green region) and the helical ballistic edge. The gapless edge insulator corresponds to the quantum glass edge state identified in this paper, which is a Bose-glass phase [10] of helical edge bosons. A commensurate gapped insulating edge state (white shaded region), which is stable for weak bounded disorder, is rendered gapless by Lifshitz tails for strong or unbounded disorder. In the latter case, there are only two phases, a glassy insulating edge and the helical Luttinger liquid edge, with a crossover inside the glassy phase (blue dashed line) which is further illustrated in Fig. 4.

Figure 2

Domain wall proliferation in the presence of disorder. (a) The configuration of $\theta \left(x\right)$. (b) The associated density profile. (c) The disorder potential. The value of $\theta$ is restricted to $\theta =(2N+1)\pi /4$ (integer $N$) for minimizing the ${\mathcal{S}}_U$ [given by Eq. (5c)] in the clean case. In the presence of disorder, frozen-in domain walls appear, corresponding to solitons (red shaded density bump) and antisoliton (blue shaded density depletion), and lead to spatial wandering of $\theta$ between equivalent minima. These excitations carry half of an electron charge [41]. The solitons and the antisolitons are “pinned” by disorder potential $V\left(x\right)$ [sketched in (c)], leading to the quantum “spin-glass” edge state, which may be viewed as a localized state of these half-charge solitons.

Figure 3

The inverse localization length as a function of disorder. The localization length $l_U$ shows nonmonotonic behavior as a function of the dimensionless parameter, $\tilde{\mathrm{\Delta }}=8K^2\mathrm{\Delta }/\left(v^2\delta Q\right)$, which measures disorder strength. Here, $l_U$ is extracted via RG analysis, $l_U\sim \alpha {\mathrm{\Delta }}_U^{-1/(3-8K)}$, and $l_U^{*}$ is the smallest value of the localization length with fixed values of $\delta Q$ and $\tilde{U}$. Different values of the Luttinger parameters are plotted in black ($K=0.374$), orange ($K=0.35$), green ($K=0.3$), blue ($K=0.25$), and red ($K=0.15$) curves. Regardless of the interaction, the localization length diverges both in the zero disorder limit and in the infinite disorder limit.

Figure 4

The disorder-averaged density of states of Luther-Emery fermions with Hamiltonian (9) based on the analytic solution in Ref. [51], for a range of variances, $\tilde{\mathrm{\Delta }}/\left(Mv\right)=0.1$, 0.5, 1 (red, green, blue respectively) of Gaussian disorder potential. The black dashed line is the disorder-free density of states. For unbounded disorder (e.g., Gaussian) distributions, the density of states is nonzero for arbitrarily small $\mathrm{\Delta }$, due to exponentially small subgap Lifshitz tails, a relic of a Mott insulator in the clean edge.

Figure 5

Schematic illustration of the modification of the wave function in the presence of external magnetic field. (a) The sketched wave functions for $K>3/8$. The yellow curve represents a delocalized wave function which corresponds to ballistic transport. In the presence of magnetic field, the wave function becomes localized (blue dash curve). (b) The sketched wave function for $K<3/8$. In the absence of magnetic field, the ground state is a quantum glass (green curve), which is a gapless TR breaking insulator. The state remains localized upon turning on the magnetic field (blue dash curve).