Multiband $s$-wave topological superconductors: Role of dimensionality and magnetic field response

We further investigate a class of time-reversal-invariant two-band $s$-wave topological superconductors introduced earlier [Deng, Viola, and Ortiz, Phys. Rev. Lett. 108, 036803 (2012)]. Provided that a sign reversal between the two superconducting pairing gaps is realized, the topological phase diagram can be determined exactly (within mean field) in one and two dimensions as well as in three dimensions upon restricting to the excitation spectrum of time-reversal-invariant momentum modes. We show how, in the presence of time-reversal symmetry, ${\mathbb{Z}}_2$ invariants that distinguish between trivial and nontrivial quantum phases can be constructed by considering only one of the Kramers’ sectors in which the Hamiltonian decouples into. We find that the main features identified in our original two-dimensional setting remain qualitatively unchanged, with nontrivial topological superconducting phases supporting an odd number of Kramers’ pairs of helical Majorana modes on each boundary, as long as the required $\pi$-phase difference between gaps is maintained. We also analyze the consequences of time-reversal-symmetry breaking either due to the presence of an applied or impurity magnetic field or to a deviation from the intended phase matching between the superconducting gaps. We demonstrate how the relevant notion of topological invariance must be modified when time-reversal symmetry is broken, and how both the persistence of gapless Majorana modes and their robustness properties depend in general upon the way in which the original Hamiltonian is perturbed. Interestingly, a topological quantum phase transition between helical and chiral superconducting phases can be induced by suitably tuning a Zeeman field in conjunction with a phase mismatch between the gaps. Recent experiments in doped semiconducting crystals, of potential relevance to the proposed model, and possible candidate material realizations in superconductors with $s_{\pm }$ pairing symmetry are discussed.

Figure 1

Phase diagram of the Hamiltonian $H_D$ in 1D as a function of $u_{cd}$ and $\Delta$, with $t=1=\lambda$, for representative chemical potential $\mu =0$. As noted, the phase diagram is independent upon $\lambda$ provided that $\lambda \ne 0$. The horizontal black dotted line at $\Delta =0$ represents an insulator or metal phase, depending on the filling. The topological response is characterized via the partial Berry-phase sum parity $P_B$, given in square brackets. Notice that the phase diagram is symmetric under $\Delta \mapsto -\Delta$. System size: $N_x=100$.

Figure 2

Phase diagrams of the Hamiltonian $H_D$ in 2D (top panels, $\mu =0$ and $-1$, respectively) and 3D (bottom panels, $\mu =0$ and $-1$, respectively) as a function of $u_{cd}$ and $\Delta$. The topological response is characterized in terms of the partial CN sum $C_{+}$ in 2D, and in terms of the partial CN sum of the modes $k_z=k_{z,c}$$(C_{+}^0,C_{+}^{\pi })$ in 3D, the respective values being given in parentheses. The corresponding ${\mathbb{Z}}_2$ invariant is the parity of the partial CN sum, Eq. (18). System size: $N_x=N_y=N_z=100$.

Figure 3

Excitation spectrum of the 3D Hamiltonian $H_D$ with OBC along the $y$ direction, for $\mu =0,t=1,\lambda =1=\Delta$. Top panels: $u_{cd}=2$, with $C_{+}^0=0$ for $k_z=0$ in panel (a), and $C_{+}^{\pi }=1$ for $k_z=\pi$ in panel (b). Bottom panels: $u_{cd}=4$, with $C_{+}^0=2$ for $k_z=0$ in panel (c), and $C_{+}^{\pi }=0$ for $k_z=\pi$ in panel (d). Note that the bulk gap scales as $\min (\lambda ,\Delta )$, indicating that the edge modes are more stable for stronger SO and superconductivity. System size: $(N_x,N_y,N_z)=(40,100,40)$.

Figure 4

Excitation spectrum of the 3D Hamiltonian $H_D$ with OBC in the $y$ direction and in the presence of the boundary perturbation $H_p$ given in Eq. (20). The relevant parameters are $\mu =0,t=1=\lambda =\Delta$, and $u_{cd}=4$, with perturbation strength $u_p=0.1$. The gapless surface modes clearly become gapped under $H_p$. System size: $(N_x,N_y,N_z)=(40,100,40)$.

Figure 5

Topological phase diagram of the 2D Hamiltonian $H_D+H_M$, with $H_M\equiv H_M^{\left(z\right)}$ being a longitudinal Zeeman field with strength $h_z$. The remaining parameters are $t=1$, $u_{cd}=1$, $\mu =-1$ for arbitrary $\lambda \ne 0$. The topological response is characterized via the partial CN sum $(C_{+,+},C_{+,-})$ from the occupied bands of ${\widehat{H}}_{+,\mathbf{k}}^{\prime }$ and ${\widehat{H}}_{-,\mathbf{k}}^{\prime }$, respectively. The horizontal black dotted line has the same meaning of the zero-field case. The vertical black dotted line indicates that a different classification of phases applies along the TR-invariant line $h_z=0$ (see text).

Figure 6

Excitation spectrum of the 2D Hamiltonian $H_D+H_M$ on a cylinder, with $H_M=H_M^{\left(z\right)}$ being a longitudinal Zeeman field with strength $h_z=1$, and $\mu =-1,t=1=\lambda$. Top panel: $u_{cd}=4,\Delta =2$, ${\tilde{P}}_C=1$. Two (helical) Majorana edge states exist on each boundary, corresponding to a trivial phase. Bottom panel: $u_{cd}=5,\Delta =1$, ${\tilde{P}}_C=-1$. One (chiral) Majorana edge state exists on each boundary, corresponding to a TS phase. System size: $(N_x,N_y)=(40,100)$.

Figure 7

Excitation spectrum of the 2D Hamiltonian $H_D+H_M+H_b$ on a cylinder, with $H_M=H_M^{\left(z\right)}$ being a longitudinal Zeeman field with strength $h_z$ and $H_b$ a backscattering potential with strength $u_1=0.2$, respectively, for $\mu =-1,t=1=\lambda ,u_{cd}=4,\Delta =2$. Top panel: TR-invariant case, $h_z=0$. The two Majorana modes on each edge form a Kramers’ pair and remain gapless under $H_b$. Bottom panel: TR-broken case, $h_z=1$. The two Majorana modes on each edge form a quasi-TR pair; note the appearance of a gap in the edge spectrum. System size: $(N_x,N_y)=(40,100)$.

Figure 8

Excitation spectrum of the 2D Hamiltonian $H_D+H_M$ on a cylinder, with $H_M=H_M^{\left(\nu \right)}$ being a uniform transverse field, for $\mu =-1,t=1=\lambda ,u_{cd}=4,\Delta =1$. Top panel: $x$ field, with strength $h_x=0.5$. The edge spectrum remains gapless. Bottom panel: $y$ field, with strength $h_y=0.2$. The edge spectrum becomes gapped. System size: $(N_x,N_y)=(40,100)$.

Figure 9

Single-particle excitation spectrum of the 2D Hamiltonian $H_D+H_M$ on a cylinder, with $H_M=H_M^{\left(z\right)}$ representing a longitudinal impurity magnetic field, for $\mu =-1,t=1=\lambda ,u_{cd}=4,\Delta =2$. The blue triangle corresponds to case (i), with a single magnetic impurity with strength $h_z=0.5$ on each boundary, whereas the red dots correspond to uniformly distributed random magnetic impurities on each boundary (averaged over 10 different realizations). System size: $(N_x,N_y)=(16,16)$.

Figure 10

Minimum gap ${\Delta }_{\mathrm{edge}}$ of the edge spectrum of the 2D Hamiltonian $H_D$ on a cylinder, away from the symmetry point ${\Delta }_c=-{\Delta }_d$. Top panel: effect of a TR-preserving amplitude mismatch in the absence ($\epsilon =0$) or in the presence ($\epsilon =\pi$) of a concomitant phase mismatch. Bottom panel: effect of a TR-breaking phase mismatch perturbation. System size: $(N_x,N_y)=(40,100)$.

Figure 11

Effect of a longitudinal Zeeman field in restoring gapless Majorana excitations in the presence of a phase mismatch $\epsilon \ne 0$, for two different sets of parameters in 2D. In the bottom panel, the minimum magnetic field strength required for restoring gapless Majorana modes depends more strongly upon $\epsilon$ as a result of the relatively large amplitude of $\left|\Delta \right|$. System size: $(N_x,N_y)=(40,100)$.

Figure 12

Excitation spectrum for the 2D Hamiltonian $H_D+H_M^{\left(z\right)}$ on the cylinder, for $\mu =0$, $u_{cd}=4$, $|{\Delta }_c|=|{\Delta }_d|=1$. Only the excitation spectrum of the edge modes located on one boundary is plotted for clarity in all cases. (a) $h_z=0,\epsilon =0$. A pair of gapless Majorana modes exist, with TR partners counterpropagating along the boundary. (b) $h_z=0,\epsilon =0.2$. Due to the phase mismatch, the helical edge modes become gapped [compare to Fig. 11(top)]. (c) $h_z=0.2,\epsilon =0.2$. Gapped helical edge modes still exist with asymmetric bulk excitation spectrum for the counterpropagating modes. The bulk gap closes at $\epsilon =0.2,h_z\approx 1.0$. (d) $h_z=1.5,\epsilon =0.2$. A pair of gapless Majorana modes is restored, with both members propagating along the same direction along the boundary. System size: $(N_x,N_y)=(80,100)$.

Figure 13

Density of Majorana modes on the boundary [Eq. (27)] for the 2D (main panel) and 3D (inset) Hamiltonian $H_D+H_M^{\left(z\right)}$ as a function of the applied magnetic field strength $h_z$, and $\mu =-1$. System size: $(N_x,N_y)=(40,100)$ for 2D and $(N_x,N_y,N_z)=(40,100,40)$ for 3D.

Figure 14

Local density of states on the boundary [$j_y=1$, Eq. (28)] for the 2D Hamiltonian $H_D+H_M^{\left(z\right)}$ for different values of the applied magnetic field strength $h_z$. The remaining parameters are the same as in Fig. 13, except the system size is now $(N_x,N_y)=(400,80)$.