Topological superconducting phases in disordered quantum wires with strong spin-orbit coupling

Zeeman fields can drive semiconductor quantum wires with strong spin-orbit coupling and in proximity to $s$-wave superconductors into a topological phase which supports end Majorana fermions and offers an attractive platform for realizing topological quantum information processing. Here, we investigate how potential disorder affects the topological phase by a combination of analytical and numerical approaches. Most prominently, we find that the robustness of the topological phase against disorder depends sensitively and nonmonotonously on the Zeeman field applied to the wire.

Figure 1

Dispersion relations in the absence of disorder. (a) Quantum wire in the normal phase and zero Zeeman field. (b) Quantum wire in the normal phase and finite Zeeman field $B\ll {\varepsilon }_{\mathrm{so}}$, including an illustration (red arrows) of a second-order process contributing to the effective disorder potential in the regime of intermediate Zeeman fields. (c) Quantum wire in the normal phase and strong Zeeman field $B\gg {\varepsilon }_{\mathrm{so}}$. (d) Quasiparticle excitation spectrum near (full line) and at (dashed line) the topological phase transition $\Delta =B$. The red (dashed) arrows illustrate a second-order process to the high-energy subspace near $p=0$, which contributes to the random Zeeman field in the low-energy model.

Figure 2

Phase diagrams of the quantum wire as a function of Zeeman field $B$ and disorder strength $\gamma$ in a log-log plot for (a) fixed density $n$ and (b) fixed chemical potential including numerical results. In (a), the long-dashed line delineates the limit of validity of the low-energy models; at the short-dashed line, $V_{\mathrm{eff}}\left(x\right)$ changes between linear and quadratic dependence on the bare disorder $V\left(x\right)$. In (b), the numerics is based on calculating the determinant of the reflection matrix for the full Hamiltonian [Eq. (1)].[28] ${\gamma }_{cr,ii}$ (dashed line) interpolates between the linear and quadratic contributions to ${\gamma }_{\mathrm{eff}}$ for intermediate Zeeman fields.

Figure 3

Quantum wire of length $L$ (grey shading) described by Hamiltonian $\mathcal{H}$, Eq. (A2), with unperturbed leads (white) described by the Hamiltonian ${\mathcal{H}}_0$. The reflection matrix $r\left(L\right)$ is found by repeatedly concatenating scattering matrices of segments with length $\delta L$.