Fate and remnants of Majorana zero modes in a quantum wire array

Experimental signatures of Majorana zero modes in a single superconducting quantum wire with spin-orbit coupling have been reported as zero-bias peaks in tunneling spectroscopy. We study whether these zero modes can persist in an array of coupled wires, and if not, what their remnant might be. The bulk exhibits topologically distinct gapped phases and an intervening gapless phase. Even though the bulk pairing structure is topological, the interaction between Majorana zero modes and superfluid phases leads to spontaneous time-reversal symmetry breaking. Consequently, edge supercurrent loops emerge and edge Majorana fermions are in general gapped out except when the number of chains is odd, in which case one Majorana fermion survives.

Figure 1

Bulk phase diagram of the 2D Hamiltonian Eqs. (1) and (2) in the $\mu$-$B$ plane with $B>0$; that with $B<0$ is symmetric with respect to the axis of $B=0$. The parameter values are $t=1$, $t_{\perp }=0.5$, $\lambda =2$, $\Delta =0.5$. (a) The white solid lines enclose the gapless phase and separate the rest into two topologically trivial gapped phases and two nontrivial phases, respectively. Inside the gapless phase, states in the diamond enclosed by the white dashed lines exhibit edge modes associated with opposite winding numbers, and those outside the diamond exhibit only edge modes associated with the same winding number. The color scale encodes the momentum-averaged winding number $r$ defined in Eq. (20). The gapless phase is suppressed as $t_{\perp }$ decreases, and it is compressed into the black dashed line at $t_{\perp }=0$ (the single-chain limit). Points I–IV are used in Fig. 2. (b) $W_{k_y}$ vs $k_y$ and $\mu$ are shown along the lines of $L_1–L_5$ in ($\text{a}$), respectively. The white solid and dashed boundaries of the regions of $W_{k_y}=\pm 1$ represent that the gap closing points are located at $(0,k_y)$ and $(\pi ,k_y)$, respectively.

Figure 2

Edge spectra with open and periodical boundary conditions along the $x$ and $y$ directions, respectively. ($\text{a}$), ($\text{b}$), ($\text{c}$), and ($\text{d}$) correspond to points I, II, III, and IV marked in Fig. 1, respectively. ($\text{a}$) Gapped trivial phase; ($\text{b}$) gapless phase with edge modes associated with the same winding number; ($\text{c}$) gapped weak topological phase; ($\text{d}$) gapless phase with edge modes associated with opposite winding numbers. Parameters used are the same as those in Fig. 1$\left(\text{a}\right)$.

Figure 3

Self-consistent solutions for ${\Delta }_{\mathbf{r}}$ ($\text{a}$) and supercurrent $J_{\mathbf{r}{\mathbf{r}}^{\prime }}$ ($\text{b}$). Parameter values are $L_x=120$, $L_y=8$, $B=1.25$, $t=1$, $t_{\perp }=0.5$, $\mu =-2$, $\lambda =2$, $g=5$. Open and periodical boundary conditions are used along the $x$ direction (vertical) and $y$ direction (horizontal), respectively. Only the first ten sites from the upper edge are plotted. The distributions of ${\Delta }_{\mathbf{r}}$ and $J_{\mathbf{r}{\mathbf{r}}^{\prime }}$ are reflection symmetric with respect to the center line of the system. ($\text{a}$) The direction and length of each arrow represent the phase and amplitude of ${\Delta }_{\mathbf{r}}$ on site $\mathbf{r}$. Its distribution is nearly uniform in the bulk but exhibits spatial variations near the edge. ($\text{b}$) Each arrow represents $J_{\mathbf{r}{\mathbf{r}}^{\prime }}$ on bond $\mathbf{r}{\mathbf{r}}^{\prime }$, which is prominent near the edge but vanishes in the bulk.

Figure 4

Self-consistent solutions for coupled chains with varying $B$ field. Open and periodical boundary conditions are used along the $x$ and $y$ directions, respectively. Parameters are $L_x=120$, $L_y=8$, $t=1$, $t_{\perp }=0.5$, $\mu =-2$, $\lambda =2$, and $g=5$. In both ($\text{a}$) and ($\text{b}$), the bulk gapless phase is marked as the shaded region, which separates the topologically trivial (on its left) and nontrivial (on its right) gapped phases. ($\text{a}$) The bulk pairing ${\Delta }_{\mathrm{bulk}}$ and the characteristic edge current magnitude $J_{\max }$ extracted as the maximal current in the system. ($\text{b}$) The energy spectra close to $E=0$. The inset of ($\text{b}$) is for the case of $L_y=7$. TR symmetry is spontaneously broken between the two dashed red lines as evidenced by $J_{\max }\ne 0$. Please note that at large values of $B$, the edge current vanishes, which is an artifact due to the finite length of $L_x$. The decaying lengths of edge Majorana modes are on the order of the superconducting coherence length which is long due to the suppression of the pairing gap. As a result, Majorana modes on opposite edges can hybridize and are gapped out without breaking TR symmetry.