particle eigenstate spectrum strongly, the value of $\Delta$ also plays an important role. If $\Delta$ is too small $\epsilon^\pm$ vanishes only for regions with higher density of states. If $\Delta$ is too large, smaller features in the single particle spectrum become smeared. This occurs before $\Delta$ exceeds $\Gamma$ so that the system becomes topologically trivial regardless of the values of $\mu$.

Dependence on the direction of the external magnetic field
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![(Color online) Grayscale plot of the value of $\epsilon^+$ for $\hat{{\mbox{\boldmath $B$}}}=\hat{{\mbox{\boldmath $x$}}}\cos\theta + \hat{{\mbox{\boldmath $y$}}}\sin\theta$, $L=200$, $(\Gamma, \alpha, \Delta, V_\mathrm{Q}) = (0.3, 0.3, 0.1, 0.5)$, $g = (\sqrt5-1)/2$, and $-0.6\leq\mu\leq1$, $0 \leq \theta \leq \pi/2$. []{data-label="fig:rotateTheta"}](Full_Fig5){width="8.66cm"}

The existence of the Majorana end fermions depends on the direction of the applied magnetic field. [@Sau2010; @Alicea2010; @Lutchyn2010; @Oreg2010] Namely, the effective magnetic field introduced by the Rashba spin-momentum coupling needs to have perpendicular component to the external magnetic field. In our model the Rashba spin-momentum coupling is in the $y$ direction, so the perpendicular directions lie in the $z$–$x$ plane, Here we ask: can the spin-insensitive quasiperiodic modulation change this situation?

To answer this question we now consider $\hat{{\mbox{\boldmath $B$}}} = \hat{{\mbox{\boldmath $x$}}}\cos\theta + \hat{{\mbox{\boldmath $y$}}}\sin\theta$, with $0\leq \theta \leq 