Majorana Bound States in Two-Channel Time-Reversal-Symmetric Nanowire Systems

We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to a conventional $s$-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that the necessary conditions are (1) the determinant of the pairing matrix in channel space must be negative, (2) inversion symmetry must be broken, and (3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical potentials of the channels. For the case of collinear spin-orbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the low-energy sector, we solve for the zero modes explicitly and study the details of the gap closing, which in the general case happens at finite momenta.

Figure 1

Top: Sketch of the geometry of the two-channel (or two-wire) superconducting system. Bottom: General structure of the low-energy bands at the gapless transition point. Because the blocks in the block-diagonal Hamiltonian have chiral symmetry, the bands that cross at zero energy are related by $\mathcal{C}$, which means $E_2(p)=-E_3(p)$, while the bands that cross at $p=0$ are related by TRS, $\mathcal{T}:E_1(p)=E_2(-p)$. Finally, particle-hole symmetry implies $\mathcal{P}:E_2(p)=-E_4(-p)$. This plot is generated using the model in Eq. (1), which assumes collinear spin-orbit coupling. However, it should be emphasized that the general structure is preserved even without this assumption.

Figure 2

(a)–(c) Spectrum of the two-wire model for three values of the potential difference $V$. In (a), the system is in the trivial state, $V<V_{c1}$, (b) is at the transition point, $V=V_{c1}$, while (c) shows the gapped spectrum in the topological phase. (d)–(f) The contour followed by the complex determinants $Z_p$ from negative to positive values of $p$. In (d) the system is in the trivial state, which means $V$ smaller than the lowest critical value $V_{c1}$ or larger than the largest critical value $V_{c2}$. In (e) the contours go through the origin, which means that the gap closes at both $V=V_{c1}$ and $V=V_{c2}$, while in (f) the contour encircles the origin, signifying the topological state. The constant parameters in the plots are $\alpha =\gamma =0$, ${\mathrm{\Delta }}_1=10m{\beta }^2$, ${\mathrm{\Delta }}_0=5m{\beta }^2$, $t=0$, and $\mu =10m{\beta }^2$, which gives $V_{c1}=4.19m{\beta }^2$ and $V_{c2}=13.13m{\beta }^2$. (g)–(i) Examples of the topological phase space. In (g) the real part of the $Z_p$ determinant at $p=p_1$ and $p=p_2$ is shown as a function of $V$. For the topological criterion to be fulfilled, the product of the two function must be negative, which occurs for $V$ between $V_{c1}$ and $V_{c2}$. (h) Phase diagram when varying the potential $V$ and the interwire tunneling $t$ (same constant parameters as above), and (i) a phase diagram in the $V\text{−}\mu$ space (same as above except ${\mathrm{\Delta }}_1=5m{\beta }^2$ and ${\mathrm{\Delta }}_0=4m{\beta }^2$).

Figure 3

Topological phase diagram in the ${\mathrm{\Delta }}_1\text{−}{\mathrm{\Delta }}_3$ plane for ${\mathrm{\Delta }}_0=2.5m{\beta }^2$, $\alpha =2\beta$, $\gamma =0$, $V=3m{\beta }^2$, and $\mu =t=10m{\beta }^2$. The light gray region fulfills the condition that $\det \mathrm{\Delta }<0$, while the orange (dark gray) region corresponds the nontrivial topological phase.

Figure 4

Topological phase diagram for the low-energy model [Eq. (11)] in the presence of a magnetic field, assumed to be orthogonal to the $z$ axis, and for $p_c=2vm$. Orange (dark gray) regions have single localized MBS, while the light gray region has MBS doublets.