Probability Distribution of Majorana End-State Energies in Disordered Wires

One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length $L$, these Majorana states combine into fermionic excitations with an energy ${\epsilon }_0$ that is exponentially small in $L$. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of ${\epsilon }_0$ is log normal in the limit of large $L$, whereas the distribution of the lowest-lying bulk energy level ${\epsilon }_1$ has an algebraic tail at small ${\epsilon }_1$. Our findings have implications for the speed at which a topological quantum computer can be operated.

Figure 1

(a) Pairing potential $\Delta (x)$ used for the calculation of the scattering matrix $S(\epsilon ,x^{\prime })$ for $0<x^{\prime }<L$ (solid gray line) and for $x^{\prime }>L$ (dashed). (b) The variable $y$ in the parametrization (5) of the scattering matrix takes values on the real axis $\pm i\pi /4$. The black dotted arrows indicate the boundary conditions at $y=\pm \infty$. In the simplified Langevin process used for the asymptotic analysis (8), the boundary conditions are at $\mathrm{Re}y=\pm \overline{y}$ (full black arrows). The direction of the drift term in topological ($0<x^{\prime }<L$) and nontopological ($x^{\prime }<0$ and $x^{\prime }>L$) regions is indicated by the full red arrow and dashed red arrow, respectively. (c) Typical dependence of the Majorana end-state energy ${\epsilon }_0$ and the energies ${\epsilon }_1,{\epsilon }_2,\dots$, of low-lying bulk states on the system size $L$.

Figure 2

Integrated probability density $P_{ϵ}(0)$ of the Majorana end-state energies ${\epsilon }_0$ (blue dots) and ${\epsilon }_{0,\max }$ (red crosses) in the two models obtained from a numerical solution of the Langevin process, together with the theoretical prediction (solid). Energies are normalized to the median ${ϵ}_m$ of the distribution. Left inset: Logarithm of the integrated probability distribution $P_{\epsilon }(0)+P_{\epsilon }(1)$ of the lowest-lying bulk state energies ${\epsilon }_1$ and ${\epsilon }_{1,\min }$. Right inset: Average of $\log {\epsilon }_0$ and $\log {\epsilon }_{0,\max }$ vs length. In both insets the slope of the solid lines is given by the theoretical predictions of the main text.