Interplay of Exciton Condensation and the Quantum Spin Hall Effect in $\mathrm{InAs}/\mathrm{GaSb}$ Bilayers

We study the phase diagram of the inverted $\mathrm{InAs}/\mathrm{GaSb}$ bilayer quantum wells. For a small tunneling amplitude between the layers, we find that the system is prone to the formation of an $s$-wave exciton condensate phase, where the spin structure of the order parameter is uniquely determined by the small spin-orbit coupling arising from the bulk inversion asymmetry. The phase is topologically trivial and does not support edge transport. On the contrary, for a large tunneling amplitude, we obtain a topologically nontrivial quantum spin Hall insulator phase with a $p$-wave exciton order parameter, which enhances the hybridization gap. These topologically distinct insulators are separated by an insulating phase with spontaneously broken time-reversal symmetry. Close to the phase transition between the quantum spin Hall and time-reversal broken phases, the edge transport shows quantized conductance in small samples, whereas in long samples the mean free path associated with the backscattering at the edge is temperature independent, in agreement with recent experiments.

Figure 1

(a) Parity of order parameter $\mathcal{P}$ as a function of $E_G^R$ and $A$ for ${\mathrm{\Delta }}_z/E_0=0.02$, $m_e/m_h=1$, $W_{1,2}/d_0=0.8$, and $\mu =d={\mathrm{\Delta }}_e={\mathrm{\Delta }}_h={\xi }_e=0$. (b) The time-reversal symmetry-breaking order parameter ${\mathcal{T}}^{\text{br}}$ as a function of $E_G^R$ and $A$ for the same parameters. (c) Line cut showing $\mathcal{P}$ and ${\mathcal{T}}^{\text{br}}$ as a function of $E_G^R$ for $A/E_0{d}_0=0.06$. The insulating phase with broken time-reversal symmetry is separated from the trivial and QSH insulators by second-order phase transitions.

Figure 2

Band structures for (a) $E_G^R=0.3E_0$, (b) $E_G^R=0.78E_0$, (c) $E_G^R=0.83E_0$, and (d) $E_G^R=1.11E_0$ and the other parameters same as in Fig. 1. Protected helical edge states appear only in the QSH phase. By decreasing $E_G^R$, the time-reversal symmetry breaking opens a gap in the edge state spectrum, and finally the edge states disappear when one approaches the trivial phase.

Figure 3

Disorder-averaged differential conductance $⟨G⟩$ as a function of $V_{\text{dis}}$ for a device with length $L=100d_0$ and different values of $E_G^R$ and voltage $eV$. In the QSH regime $E_G^R=1.11E_0$, the conductance is quantized to $G=2G_0$ ($G_0=e^2/h$). In the regime of weakly broken time-reversal symmetry $E_G^R=0.83E_0$, numerically calculated $⟨G⟩$ (thick lines) are in agreement with $G=2G_0(1-L/\ell )$ (dashed lines), where $\ell$ is obtained from Born approximation with a fitting parameter $a=3.1$. The other parameters are the same as in Fig. 1.