Band manipulation and spin texture in interacting moiré helical edges

We develop a theory for manipulating the effective band structure of interacting helical edge states realized on the boundary of two-dimensional time-reversal symmetric topological insulators. For a sufficiently strong interaction, an interacting edge band gap develops, spontaneously breaking time-reversal symmetry on the edge. The resulting spin texture, as well as the energy of the time-reversal breaking gaps, can be tuned by an external moiré potential (i.e., a superlattice potential). Remarkably, we establish that by tuning the strength and period of the potential, the interacting gaps can be fully suppressed and interacting Dirac points reemerge. In addition, nearly flat bands can be created by the moiré potential with a sufficiently long period. Our theory provides an unprecedented way to enhance the coherence length of interacting helical edges by suppressing the interacting gap. The implications of this finding for ongoing experiments on helical edge states is discussed.

Figure 1

Setup and band-structure manipulation. (a) Illustration of the setup. A 2D TI edge state is perturbed by an external periodic potential, $V\left(x\right)=wcos\left(Qx\right)$. (b) For a weak repulsive interaction ($1/2<K<1$), the edge band is unaffected by the external potential. (c) For a strong repulsive interaction ($K<1/2$), interaction-induced gaps can open at both zero energy and finite energies. In addition, for particular $w$ and $Q$, the interaction-induced gap may vanish, resulting in a reemergent gapless edge (middle figure). The gaps are labeled by $m$ corresponding to $E=mvQ/4$.

Figure 2

Phase diagram and spin texture of the time-reversal breaking order. (a) The phase diagram for $\left|Q\right|>4\delta Q_c^{\left(m\right)}$ based on bosonization analysis. The yellow region denotes the metallic phase. For $K<1/2$, time-reversal breaking gaps (green regions) can be induced by the interaction. The energy of the gap is $E=mvQ/4$ for an integer $m$. The half width of the gap is $v\delta Q_c^{\left(m\right)}$, as discussed in the main text. (b), (c) Normalized $s_y\left(x\right)$ [Eq. (6)] as a function of $Qx$. The phase shift ${\zeta }_{m,N}$ is set to zero for convenience. (b) $m=0$. (c) $m=1$. We define the dimensionless parameter $\tilde{\omega }=\frac{2Kw}{vQ}$. In both figures, $\tilde{\omega }=0,1,2,3$ correspond to the black dashed lines, blue lines, green lines, and red lines respectively.

Figure 3

Luther-Emery energy bands for $vQ=3M$, varying $\tilde{\omega }=Kw/\left(vQ\right)$ (with $K=1/4$). (a) $\tilde{\omega }=0.4$, (b) $\tilde{\omega }=1$, (c) $\tilde{\omega }=1.25$, (d) $\tilde{\omega }=2$. The Luther-Emery bands contain gaps due to spontaneous time-reversal symmetry breaking. The Dirac points at $k=0$ and $k=\pm Q/2$ (circled by red dashed lines) are recovered for particular values of $\tilde{\omega }$ in (c) and (d).

Figure 4

Luther-Emery energy bands for $3vQ=M$, varying $\tilde{\omega }=Kw/\left(vQ\right)$ (with $K=1/4$). (a) $\tilde{\omega }=0.5$, (b) $\tilde{\omega }=1$, (c) $\tilde{\omega }=3.5$, (d) $\tilde{\omega }=4.05$. The energy bands are nearly flat. The gap size at $E=0$ depends on $\tilde{w}$. Nearly doubly degenerate zero-energy bands emerge in (c). Similar nearly degenerate bands can also happen at finite energies as shown in (d).