Bandwidth-controlled quantum phase transition between an easy-plane quantum spin Hall state and an $s$-wave superconductor

The quantum spin Hall state can be understood in terms of spontaneous O(3) symmetry breaking. Topological skyrmion configurations of the O(3) order parameter vector carry a charge 2e, and as shown previously, when they condense, a superconducting state is generated. We show that this topological route to superconductivity survives easy-plane anisotropy. Upon reducing the O(3) symmetry to $\mathrm{O}\left(2\right)\times {\mathrm{Z}}_2$, skyrmions give way to merons that carry a unit charge. On the basis of large-scale auxiliary field quantum Monte Carlo simulations, we show that at the particle-hole symmetric point, we can trigger a continuous and direct transition between the quantum spin Hall state and $s$-wave superconductor by condensing pairs of merons. This statement is valid in both strong and weak anisotropy limits. Our results can be interpreted in terms of an easy-plane deconfined quantum critical point. However, in contrast to the previous studies in quantum spin models, our realization of this quantum critical point conserves $\text{U}\left(1\right)$ charge such that skyrmions are conserved.

Figure 1

Honeycomb plaquette illustrating (a) the spin-orbit coupling operator ${\widehat{\mathit{J}}}_{\mathit{r}+{\mathit{\delta }}_n,\mathit{r}+{\mathit{\eta }}_n}$ and (b) the pairing operator ${\widehat{\eta }}_{\mathit{r},{\tilde{\mathbf{\delta }}}_{\mathit{a}}}^{+}$.

Figure 2

Mean-field phase diagram at zero temperature as a function of $\mathrm{\Delta }$ and $\lambda$.

Figure 3

Ground-state phase diagram in the $(\mathrm{\Delta },\lambda )$ plane which covers the range of $\mathrm{\Delta }\in [0.1,1]$. The DSM-QSH and QSH-SSC phase boundaries at $\mathrm{\Delta }=0.1,0.5$, and 0.75 are estimated from PQMC simulations in this paper. The phase boundaries of $\text{SU}\left(2\right)$ symmetric Hamiltonian (the overline of $\mathrm{\Delta }=1$) are from Ref. [18].

Figure 4

Equal-time correlation ratio $R^{\mathrm{SSC}}$ (a) and $R^{\mathrm{QSH}}$ (b) as a function of $\lambda$ for $\mathrm{\Delta }=0.1$.

Figure 5

Same as Fig. 4 for $\mathrm{\Delta }=0.5$.

Figure 6

Same as Fig. 4 for $\mathrm{\Delta }=0.75$.

Figure 7

QSH and SSC order parameter as a function of $\lambda$ for $\mathrm{\Delta }=0.1$ (a, b), 0.5 (c, d) and 0.75 (e, f).

Figure 8

Single-particle gap ${\mathrm{\Delta }}_{sp}$ across the QSH-SSC transition for $\mathrm{\Delta }=0.1$ (a), $\mathrm{\Delta }=0.5$ (b), and $\mathrm{\Delta }=0.75$ (c), respectively.

Figure 9

Correlation length (a) and scaled correlation length (b) in disordered phase for $\mathrm{\Delta }=0.1$.

Figure 10

Same as Fig. 9 for $\mathrm{\Delta }=0.5$.

Figure 11

Same as Fig. 9 for $\mathrm{\Delta }=0.75$.

Figure 12

Free-energy derivative $\partial f/\partial \lambda$ as a function of $\lambda$ in the vicinity of the QSH-SSC transition point for $\mathrm{\Delta }=0.1$ (a), $\mathrm{\Delta }=0.5$ (b), and $\mathrm{\Delta }=0.75$ (c), respectively.

Figure 13

Equal-time correlation ratio $R^{{\mathrm{QSH}}_{\mathrm{z}}}$ for $\mathrm{\Delta }=0.1$ (a), $\mathrm{\Delta }=0.5$ (b), and $\mathrm{\Delta }=0.75$ (c), respectively.

Figure 14

QSH (a), SSC (b), and ${\mathrm{QSH}}_{\mathrm{z}}$ (c) dynamical spectrum at the QSH-SSC critical point ($\lambda =0.038$) for $\mathrm{\Delta }=0.75$. Here, $L=18$. Blue lines in (a) and (b) are the momentum dependence of the extrapolated excitation gap of ${\widehat{J}}^{XY}$ and ${\widehat{\eta }}^{+}\left({\widehat{\eta }}^{-}\right)$ operators. The blue line in (c) is copied from the SSC dispersion relation in (b) to guide the eye.

Figure 15

Mean-field solution for the QSH and SSC order parameters as a function of $\lambda$ for $\mathrm{\Delta }=0.1$ (a), $\mathrm{\Delta }=0.5$ (b), and $\mathrm{\Delta }=0.75$ (c), respectively.

Figure 16

Time-displaced correlation ratio $R_{\chi }^{\mathrm{SSC}}$ (a) and $R_{\chi }^{\mathrm{QSH}}$ (b) as a function of $\lambda$ for $\mathrm{\Delta }=0.1$.

Figure 17

Same as Fig. 16 for $\mathrm{\Delta }=0.5$.

Figure 18

Same as Fig. 16 for $\mathrm{\Delta }=0.75$.

Figure 19

Time-displaced correlation ratio $R_{\chi }^{{\mathrm{QSH}}_{\mathrm{z}}}$ for $\mathrm{\Delta }=0.1$ (a), $\mathrm{\Delta }=0.5$ (b), and $\mathrm{\Delta }=0.75$ (c), respectively.

Figure 20

QSH, SSC, ${\mathrm{QSH}}_{\mathrm{z}}$, and single-particle spectrum inside two (QSH, SSC) phases and near the critical point, for $\mathrm{\Delta }=0.1$. Blue lines are the momentum dependence of the extrapolated excitation gap of $\widehat{J}$ and ${\widehat{\eta }}^{+}\left({\widehat{\eta }}^{-}\right)$ operators. We took $L=18$.

Figure 21

Same as Fig. 20 for $\mathrm{\Delta }=0.5$.

Figure 22

Same as Fig. 20 for $\mathrm{\Delta }=0.75$.

Figure 23

$S_{\mathrm{\Omega }}^{xy}\left(\mathit{i}\right)$ (a) and $S_{\mathrm{\Omega }}^{\text{charge}}\left(\mathit{i}\right)$ (b) distributions on a $L=12$ honeycomb lattice. The simulation is performed deep inside the QSH state ($\mathrm{\Delta }=0.1,\lambda =0.045$). Here, $\beta =24$.