Localization-driven correlated states of two isolated interacting helical edges

We study the localization-driven correlated states among two isolated dirty interacting helical edges as realized at the boundaries of two-dimensional ${\mathbb{Z}}_2$ topological insulators. We show that an interplay of time-reversal symmetric disorder and strong interedge interactions generically drives the entire system to a gapless localized state, preempting all other intraedge instabilities. For weaker interactions, an antisymmetric interlocked fluid, causing a negative perfect drag, can emerge from dirty edges with different densities. We also find that the interlocked fluid states of helical edges are stable against the leading intraedge perturbation down to zero temperature. The corresponding experimental signatures including zero-temperature and finite-temperature transport are discussed.

Figure 1

Zero-temperature phase diagram of two dirty TI edges with different densities. We assume $1>K_{-}>K_{+}$ due to the repulsive interactions. For $K_{-}>K_{+}>3/4$, the two helical Luttinger liquids are decoupled. The symmetric interedge mode is localized when $K_{-}>3/4>K_{+}$. An antisymmetric interlocked fluid is developed. For $3/4>K_{-}>K_{+}$, the a gapless localized insulator is predicted.

Figure 2

(a) The proposed experimental setup [39] (“edge gear”) for studying the interedge correlated states. The top TI is attached to two external electrodes which result in two separated edges carrying current $I_1$ and $I_1^{\prime }$; the bottom TI forms a close edge loop with a current $I_2$ (but without a voltage drop). The two proximate edge states (carrying currents $I_1$ and $I_2$) interact via interedge Coulomb interactions. As discussed in the main text, the two terminal conductance on the top TI encodes the information of the interedge correlated states. (b) The standard Coulomb drag experiment in the lateral geometry as a comparison.

Figure 3

The two terminal conductance at zero temperature as a function of interaction in the edge gear setup [Fig. 2 (a)]. For sufficiently weak interedge interactions, the conductance (in the unit of $e^2/h$) is 2 as the absence of the bottom close loop TI. In the perfect drag regime, the conductance follows Eq. (8) with an upper bound $3/2$ and a lower bound $14/11$ (red dotted lines). These bounds guarantee discontinuities of the conductance. For sufficiently strong interactions, two TI edge states become localized insulators. The conductance becomes to 1.

Figure 4

The sketched temperature dependence of drag conductance in various regimes. $G_{\text{drag}}=G-\frac{e^2}{h}$ where $G$ is the two-terminal conductance of edge gear setup. The yellow solid line indicates the two helical liquids regime ($K_{+},K_{-}>3/4$); the blue dot-dashed line indicates the negative drag ($K_{+}<3/4<K_{-}$); the red dashed line indicates the localized regime ($K_{+},K_{-}<3/4$). The detail features of each curve are explained qualitatively in the main text.