Generalized Eilenberger theory for Majorana zero-mode-carrying disordered $p$-wave superconductors

Disorder is known to suppress the gap of a topological superconducting state that would support non-Abelian Majorana zero modes. In this paper, we study using the self-consistent Born approximation the robustness of the Majorana modes to disorder within a suitably extended Eilenberger theory, in which the spatial dependence of the localized Majorana wave functions is included. We find that the Majorana mode becomes delocalized with increasing disorder strength as the topological superconducting gap is suppressed. However, surprisingly, the zero bias peak seems to survive even for disorder strength exceeding the critical value necessary for closing the superconducting gap within the Born approximation.

Figure 1

(a) DOS for a semi-infinite $s$-wave superconducting wire. Note the result is position-independent, and is not affected by disorder. (b)–(d) The DOS of a semi-infinite $p$-wave superconducting wire, from the clean limit ${\tau }^{-1}=0$ to a heavily disordered case ${\tau }^{-1}=2\Delta$ at (b) $x\to \infty$ (in the bulk), (c) $x={\xi }_0$, and (d) $x=0$, respectively. For all plots, the energy spectra are broadened by $\eta =0.01\Delta$.

Figure 2

DOS $\nu (x,\omega ,\eta )={\nu }_0\mathrm{Re}\left[g_{33}(x,\omega +i\eta )\right]$ plotted as a function of position $x$ (in units of ${\xi }_0=v_F/\Delta$) and energy $\omega$ (in units of $\Delta$), where $x$ is measured from the end of the wire and $\eta =0.01\Delta$ is the broadening parameter. The four panels correspond to disorder strengths (a) ${\tau }^{-1}=0$, (b) ${\tau }^{-1}=0.5\Delta$, (c) ${\tau }^{-1}=\Delta$, and (d) ${\tau }^{-1}=2\Delta$. When the system is clean, the salient features are the zero-energy peak localized at the end and a pristine bulk gap. As disorder is introduced, the bulk gap shrinks and the singularity is smeared out, homogenizing the DOS of the whole system, but the zero-energy peak at the end of the wire is still visible even at strong disorder.

Figure 3

Log-linear plot of zero-energy DOS $\nu (x,\omega =0)$ as a function of distance $x$ measured from the end of a $p$-wave superconducting wire, with oscillations of length scale $k_F^{-1}$ ignored. The steepest line corresponds to clean case ${\tau }^{-1}=0$ where the MM is most localized. The least steep line corresponds to the critical disorder strength ${\tau }^{-1}=\Delta$ where the bulk gap closes. The intermediate lines are sampled at equally spaced ${\tau }^{-1}$ with a step size of $\delta \left({\tau }^{-1}\right)=0.1\Delta$. The inset shows the last curve corresponding to ${\tau }^{-1}=\Delta$ in log-log scale. Its slope is approximately $-2$.

Figure 4

Plot of the localization length of the MM as a function of disorder strength. The black solid line shows the numerical values extracted from Fig. 3 by fitting the tails of the curves (at $log\frac{\nu \left(x\right)}{\nu (x=0)}<-6$) with straight lines. Note that the result is meaningful only for weak disorder $({\tau }^{-1}\lesssim \Delta )$ where the corresponding curve in Fig. 3 is approximately linear. The red dashed line is the best-fit line of a power-law form of $\frac{\xi }{{\xi }_0}={[1-{\left(\Delta \tau \right)}^{-1}]}^{-0.84}$.

Figure 5

Plot of the spectral weight $Z_{\tau }$ (defined in the text) of the zero-energy end mode against disorder strength. The dots show the values obtained from the numerical solution of Eqs. (14) for a range of disorder strengths. The solid line plots the empirical formula Eq. (34).