parameters with $\Delta = 0.05, 0.1, 0.15, 0.2$.[]{data-label="fig:L400lambda0.2"}](Full_Fig3_200){width="8.66cm"}

![(Color online) Top figure: Single particle eigenstates of eq. (\[eqn:Hamiltonian\_gen\]) color-coded according to the value of $S_z$ for $\hat{{\mbox{\boldmath $B$}}}=\hat{{\mbox{\boldmath $z$}}}$, $L=400$, $(\Gamma, \alpha, V_\mathrm{Q}) = (0.3, 0.3, 0.5)$ and $-0.6\leq\mu\leq1$, $0 < \kappa < \pi$. Lower figures: Grayscale plots of $\epsilon^+$ for the same parameters with $\Delta = 0.05, 0.1, 0.15, 0.2$.[]{data-label="fig:L400lambda0.5"}](Full_Fig4_200){width="8.66cm"}

Dependence on the lattice modulation wavenumber
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In Ref. [@Tezuka2012] we fixed the wave vector $\kappa$ of the quasiperiodic lattice modulation. However, as we have observed in Fig. \[fig:DoubleHofstadter\], we have effectively single-band regions of the chemical potential for a wide range of the value of $\kappa$. Therefore it is interesting to study the dependence of the appearance of topologically non-trivial superfluid with end Majorana fermions on the value of $\kappa$. Especially it is intriguing whether a $\kappa$ such that $2\pi/\kappa$ is an integer has any difference.

In Figs. \[fig:L400lambda0.2\] and \[fig:L400lambda0.5\] we plot the value of $\epsilon^+$ in a color code, along with the single particle state energies obtained from eq. (\[eqn:Hamiltonian\_gen\]). For smaller values of $\Delta$, we notice that vanishing eigenvalue corresponds to regions covered by a single band of single particle eigenstates, regardless of the value 