\pi/2$. We plot the value of $\epsilon^+$ in a color-coded plot in Fig. \[fig:rotateTheta\]. While the region with vanishing $\epsilon^+$ persists up to $\theta \lesssim \pi/4$, which corresponds to $\Gamma_x\equiv \Gamma\cos\theta
\gtrsim 0.2$, for larger $\theta$ with $\Gamma_x \lesssim 0.2$ we no longer have a vanishing eigenvalue of the BdG equation and the system is topologically nontrivial.

Therefore, the introduction of the site level modulation still does not lift the limitation in the direction of ${\mbox{\boldmath $B$}}$ for TS states to be observed. Because $\Delta = 0.1$ still does not exceed $\Gamma_x$, the result above indicates that the $y$ component of the applied magnetic field is rather detrimental for the realization of a TS in our model.

Dependence on the phase of the modulation potential
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![(Color online) Top figure: Single particle eigenstates of eq. (\[eqn:Hamiltonian\_gen\]) color-coded according to the value of $S_z$ for $g=\sqrt5-2$, $\hat{{\mbox{\boldmath $B$}}}=\hat{{\mbox{\boldmath $z$}}}$, $L=200$, $(\Gamma, \alpha, V_\mathrm{Q}) = (0.3, 0.3, 0.5)$ and $-0.6\leq\mu\leq1$, $0 \leq \phi_0 \leq \pi$. Lower figures: Grayscale plots of $\epsilon^+$ for the same set of parameters except that $\Delta = 0.1$ is introduced and $L=50, 100, 200$.[]{data-label="fig:f-L200lambda0.5"}](Full_Fig6_200){width="8.66cm"}

It has been known (see *e.g.* [@Mei2012; @Ganeshan1301.5639]) for systems with $\Gamma = \alpha 