Thermodynamic and dynamical signatures of a quantum spin Hall insulator to superconductor transition

The thermodynamic and dynamical properties of a model of Dirac fermions with a deconfined quantum critical point (DQCP) separating an interaction-generated quantum spin Hall insulator from an $s$-wave superconductor [Liu et al., Nat. Commun. 10, 2658 (2019)] are studied using quantum Monte Carlo simulations. Inside the deconfined quantum critical region bound by the single-particle gap, spinons and spinless charge-$2e$ skyrmions emerge. Since the model conserves total spin and charge, and has a single length scale, these excitations lead to a characteristic linear temperature dependence of the uniform spin and charge susceptibilities. At the DQCP, the order parameter dynamic structure factors show remarkable similarities that support emergent Lorentz symmetry. Above a critical temperature, superconductivity is destroyed by the proliferation of spin-1/2 vortices.

Figure 1

(a) Notation in Eq. (1). (b) Phase diagram of Hamiltonian (1). Two quantum critical points at ${\lambda }_{c1}=0.0187\left(2\right)$ [Dirac semimetal (DSM) to quantum spin Hall insulator (QSHI)] and ${\lambda }_{c2}=0.0332\left(2\right)$ [QSHI to $s$-wave superconductor (SSC)] were established in Refs. [7, 8]. The DQCP quantum critical region is shown above $T=5.35|\lambda -{\lambda }_{c2}{|}^{z\nu }$ [red dashed lines; $z=1, \nu =0.58$, the prefactor is obtained using $T_c(\lambda =0.04)\approx 1/6.5$] and to the right of the crossover temperature scale $30.7|\lambda -{\lambda }_{c1}{|}^{z\nu }$ associated with the DSM-QSHI QCP (black dashed lines; $z=1, \nu =1/1.14$ [7], the prefactor is from ${\mathrm{\Delta }}_{\text{sp}}(\lambda =0.026)\approx 0.41$). The curve for $T_c\left(\lambda \right)$ corresponds to $5.35|\lambda -{\lambda }_{c2}{|}^{z\nu }$ for $\lambda \le 0.04$, and to a straight line for $\lambda >0.04$.

Figure 2

(a)–(c) QSH and (d)–(f) SSC susceptibilities, revealing long-range QSH order at $T=0$ for $\lambda =0.026$ [(a) and (d)] and $s$-wave superconductivity at $T<T_c$ for $\lambda =0.050$ [(b) and (e)]. Data collapse at the DQCP using $\lambda ={\lambda }_{c2}=0.0332, {\eta }_{\text{QSH}}=0.21, {\eta }_{\text{SSC}}=0.22$, and $z=1$ [(c) and (f)].

Figure 3

(a) Scaling analysis of the SSC susceptibility using $\eta =0.25$. (b) Extraction of $T_c$ from finite-size extrapolation of the crossing points. Here, $\lambda =0.050$ and $T_c=0.0246\left(1\right)$.

Figure 4

(a)–(c) Temperature dependence of the uniform spin and charge susceptibilities. (d) Rescaled susceptibilities for selected parameters (see text), revealing linear behavior at low temperatures. The key applies to all panels.

Figure 5

Notation used in the context of the extended zone scheme and illustration for (a) local operators and (b) nonlocal QSH operators. (c) Path in the Brillouin zone.

Figure 6

(a)–(c) Zero-temperature single-particle spectral function for $L=24$ and (d) finite-size scaling of the single-particle gap. The path in the Brillouin zone is shown in Fig. 5.

Figure 7

Zero-temperature dynamical (a), (c), and (e) QSH and (b), (d), and (f) $s$-wave pairing structure factors at different values of $\lambda$ for $L=24$. The dashed lines in (e) and (f) correspond to $\omega =a(\mathit{q}-\mathrm{\Gamma })$ with the same $a$. The path in the Brillouin zone is shown in Fig. 5.

Figure 8

Zero-temperature (a), (c), and (e) spin and (b), (d), and (f) charge dynamic structure factors for $L=24$. The path in the Brillouin zone is shown in Fig. 5.