Doping-Induced Quantum Spin Hall Insulator to Superconductor Transition

A quantum spin Hall insulating state that arises from spontaneous symmetry breaking has remarkable properties: skyrmion textures of the SO(3) order parameter carry charge $2e$. Doping this state of matter opens a new route to superconductivity via the condensation of skyrmions. We define a model amenable to large-scale negative sign free quantum Monte Carlo simulations that allows us to study this transition. Our results support a direct and continuous doping-induced transition between the quantum spin Hall insulator and an $s$-wave superconductor. We can resolve dopings away from half-filling down to $\delta =0.0017$. Such routes to superconductivity have been put forward in the realm of twisted bilayer graphene.

Figure 1

Phase diagram of the model of Eq. (1) in the interaction strength $\lambda$ versus chemical potential plane. The data at half-filling are reproduced from Ref. [8]. The critical chemical potential ${\mu }_c$ for the transition from the quantum spin Hall (QSH) to the $s$-wave superconductor (SSC) is computed by measuring the pairing gap at half-filling [see Ref. [9] and Fig. 3].

Figure 2

Mean-field ground-state phase diagram. The blue and purple (green) lines correspond to continuous (first-order) transitions.

Figure 3

Momentum dependence of (a) the pairing gap and (b) the QSH gap for the half-filled case, in the vicinity of the $\mathrm{\Gamma }$ point along the direction toward the $M$ point in the Brillouin zone of the honeycomb lattice. The inset of (a) shows the $1/L$ dependence of the single-particle gap ${\mathrm{\Delta }}_{\mathrm{SP}}$ and half of the $s$-wave pairing gap ${\mathrm{\Delta }}_{\eta }/2$. The inset of (b) shows the QSH gap, ${\mathrm{\Delta }}_{\mathrm{QSH}}$, versus $1/L$.

Figure 4

Doping factor $\delta$ as a function of chemical potential $\mu \equiv {\mathrm{\Delta }}_{{\eta }^{-}}/2$ for sizes $L=9$, 12, 15, 18, and 21. The red dashed line is the critical chemical potential from the extrapolated pairing gap ${\mathrm{\Delta }}_{\eta }/2$ shown in Fig. 3.

Figure 5

Correlation ratios for (a) SSC and (b) QSH orders as a function of doping $\delta$. The system sizes are $L=9$, 12, 15, 18, 21, and 24. The inset of (b) shows the $\delta$ dependence of the finite-size correlation length for the QSH order parameter.

Figure 6

Real-space equal-time correlation function of the QSH order parameter around a pinned hole pair at the origin. Hamiltonian (1) was supplemented with a pinning potential ${\widehat{H}}_{\mathrm{pin}}\equiv C{\sum }_{\mathit{r}}{\sum }_{\tilde{\mathit{\delta }}}{e}^{-|(\mathit{r}+\tilde{\mathit{\delta }})|/\xi }{\widehat{c}}_{\mathit{r}+\tilde{\mathit{\delta }}}^{†}{\widehat{c}}_{\mathit{r}+\tilde{\mathit{\delta }}}$ with $C=1$ and $\xi =1$. We consider $L=15$, at (a) $\lambda =0.026$ and (b) $\lambda =0.05$.