Paramagnetically induced gapful topological superconductors

We propose a generic scenario for realizing gapful topological superconductors (TSCs) from gapless spin-singlet superconductors (SCs). Noncentrosymmetric nodal SCs in two dimensions are shown to be gapful under a Zeeman field, as a result of the cooperation of inversion-symmetry breaking and time-reversal-symmetry breaking. In particular, non-$s$-wave SCs acquire a large excitation gap. Such paramagnetically induced gapful SCs may be classified into TSCs in the symmetry class D specified by the Chern number. We show nontrivial Chern numbers over a wide parameter range for spin-singlet SCs. A variety of the paramagnetically induced gapful TSCs are demonstrated, including $D+p$-wave TSC, extended $S+p$-wave TSC, $p+D+f$-wave TSC, and $s+P$-wave TSC. Natural extension toward three-dimensional Weyl SCs is also discussed.

Figure 1

(a), (b) Bulk energy spectrum around $E\simeq 0$. The energy band $E_n\left(\mathit{k}\right)$ is calculated by numerically diagonalizing the BdG Hamiltonian $H_{\mathrm{BdG}}\left(\mathit{k}\right)$. We take $t=1,t^{\prime }=0.2,\alpha =0.3,\mu =-0.7,{\psi }_0=0.4$, and $d_0/{\psi }_0=0.2$. (a) ${\mu }_{\mathrm{B}}H=0$ and (b) ${\mu }_{\mathrm{B}}H={\psi }_0/5$. (c) Edge-state spectrum for the parameters in (b). Edge states localized at a $\left(010\right)$ edge are highlighted by the red lines, while edge states on the opposite edge are shown by the green lines. (d), (e) Illustrations for counting the Chern number of $D+p$-wave TSCs: zeros of $E_{\pm }$ (solid red line), $\psi \pm \mathit{d}·\widehat{g}$ (dashed blue line), and ${\mu }_{\mathrm{B}}\mathit{H}·\mathit{g}\times \mathit{d}$ (dotted black line) are shown. In the shaded region $(\psi \pm \mathit{d}·\widehat{g}){\mu }_{\mathrm{B}}\mathit{H}·\mathit{g}\times \mathit{d}>0$. The left panels in (d) and (e) are illustrated for the estimation of ${\nu }_{+}$, while the right panels for ${\nu }_{-}$. We assume (d) $\mu =-0.7$ and (e) $\mu =-3.1$. The other parameters are the same as Fig. 1.

Figure 2

(a) Illustrations for counting the Chern number of extended $S+p$-wave TSCs: zeros of $E_{\pm }$ (solid red line), $\psi \pm \mathit{d}·\widehat{g}$ (dashed blue line), and ${\mu }_{\mathrm{B}}\mathit{H}·\mathit{g}\times \mathit{d}$ (dotted black line) are shown. In the shaded region, $(\psi \pm \mathit{d}·\widehat{g}){\mu }_{\mathrm{B}}\mathit{H}·\mathit{g}\times \mathit{d}>0$. The left panel is illustrated for the estimation of ${\nu }_{+}$, while the right panel for ${\nu }_{-}$. We take $t=1,t^{\prime }=0.2,\alpha =0.3,\mu =-0.7,{\psi }_0=0.05,d_0/{\psi }_0=0.2$, and ${\delta }_1=0$. (b) Edge-state spectrum. Edge states localized at a $\left(010\right)$ edge are highlighted by the red lines, while edge states on the opposite edge are shown by the green lines. We assume ${\psi }_0=0.4$ and ${\mu }_{\mathrm{B}}H={\psi }_0/5$. Other parameters are the same as Fig. 2. (c), (d) Chern number of the extended $S+p$-wave state. In (c) the parameters are the same as Fig. 2, while ${\delta }_1=-0.1$ in (d). The Chern number is numerically calculated by using the method developed by Ref. [69]. The black region is trivial, while $\nu =4$ in the red region and $\nu =8$ in the yellow region. The white region with $\nu =6$ is an exceptional case where Eq. (39) is not valid.

Figure 3

Schematic picture for adiabatic change of chiral edge states in Fig. 2. Edge modes around $k_x=\pi$ are topologically equivalent to two positive velocity modes. Taking other two edge states around $k_x=0$ into account, edge modes in Fig. 2 are equivalent to four net chiral edge states with positive velocity, in accordance with bulk-edge correspondence.

Figure 4

$k_z$-dependent Chern number $\nu$ of the extended $S+p$-wave Weyl SC. We take $t=1,t^{\prime }=0.475$, and $\tilde{t}=0.3$ in accordance with Ref. [75]. Chemical potential $\mu =-0.9$ is assumed so that the charge density is near half-filling. In-plane $\mathit{k}$ dependence of the spin-singlet order parameter is parametrized by ${\delta }_2=0.3$. The other parameters are $\alpha =0.3,{\psi }_0=0.05$, and $d_0/{\psi }_0=0.2$.

Figure 5

(a) Illustrations for counting the Chern number of $p+D+f$-wave SCs. The left panel is for the estimation of ${\nu }_{+}$ at $k_z=0.3$, while the right panel for ${\nu }_{-}$. We adopt the 2D dispersion relation in Sec. 4a and take $t=1,t^{\prime }=0.2,\alpha =0.3,\mu =-0.6,\beta =-0.2,\psi =0.05$, and $d_0/{\psi }_0=\frac{1}{3}$. (b) Chern number as a function of the chemical potential. The black region is trivial, while $\nu =2$ in the pink region and $\nu =-2$ in the light-blue region. Other parameters are the same as Fig. 5.

Figure 6

(a) Illustrations for counting the Chern number of $s+P$-wave TSCs. The left panel is for the estimation of ${\nu }_{+}$, while the right panel for ${\nu }_{-}$. We take $t=1,t^{\prime }=0.2,\alpha =0.3,d_0=0.05,{\psi }_0/d_0=0.5$, and $\mu =-3.315$. (b) Chern number of the paramagnetically induced gapful $s+P$-wave SC. The Chern number is trivial in large black regions, while we obtain $\nu =4$ in tiny red regions. Other parameters are the same as (a).