Reentrant topological phase transitions in a disordered spinless superconducting wire

In a one-dimensional spinless $p$-wave superconductor with coherence length $\xi$, disorder induces a phase transition between a topologically nontrivial phase and a trivial insulating phase at the critical mean-free path $l=\xi /2$. Here, we show that a multichannel spinless $p$-wave superconductor goes through an alternation of topologically trivial and nontrivial phases upon increasing the disorder strength, the number of phase transitions being equal to the channel number $N$. The last phase transition, from a nontrivial phase into the trivial phase, takes place at a mean-free path $l=\xi /(N+1)$, parametrically smaller than the critical mean-free path in one dimension. Our result is valid in the limit that the wire width $W$ is much smaller than the superconducting coherence length $\xi$.

Figure 1

Topological number $Q_{\mathrm{chiral}}$ for a wire with width $2W/{\lambda }_F=9.5$ such that the channel number is $N=9$, as a function of the ratio $\xi /l$ of disorder strength and induced superconductivity. Data shown are for a single disorder realization with ${\lambda }_{\mathrm{F}}/l=0.011$ and wire length $L/l\sim 2100$. The red curve shows the analytical prediction (15) and the blue one the numerical data. Inset: Schematic picture of a disordered superconducting wire with two ideal normal-metal leads.

Figure 2

Phase diagrams showing the topological number $Q=\det \left(r\right)$ of a wire with $2W/{\lambda }_F=9.5$ as a function of the ratio $l/\xi$ and the transverse coupling ${\Delta }_y^{\prime }$ in a spinless superconducting wire with $N=9$ channels in the limit of (a) weak and (b) strong disorder. The disorder strength is (a) ${\lambda }_{\mathrm{F}}/l=0.011$ and (b) ${\lambda }_{\mathrm{F}}/l=0.43$. The data were obtained for a wire length (a) $L/l=150$ and (b) $L/l\sim 770$.