\phi_0),
\label{eqn:sitePotential}$$ in which $V_\mathrm{Q}\geq 0$, $l_\textrm{c} \equiv (L-1)/2$, $\phi_0$ is the phase of the potential at the center of the system and $\kappa = 2 \pi g$, in which $g$ is a real number such that $0<g<1$. The potential is periodic for a rational $g$, while it is quasiperiodic for an irrational $g$. In the following we choose $\phi_0 = 0$, except when we study the effect of $\phi_0$ and when we study the effect of phase jumps in the system.

In a finite length system with $L$ lattice sites, we numerically obtain the set of single particle level energies. For $\alpha = \Gamma = 0$, each single body wavefunction is extended for $V_\mathrm{Q}<t$ and localized for $V_\mathrm{Q}>t$ in the $L\rightarrow\infty$ limit. [@Kohmoto1983] The self-similar structure of the two-dimensional spectrum plotted against $\kappa$ for $V_\mathrm{Q} = t$ is called the Hofstadter butterfly. [@Hofstadter1976; @Kohmoto1983]

When the $2L$ energy levels obtained are plotted against various values of $\kappa$, for $V_\mathrm{Q} \sim t$, the spectrum shows a self-similar structure resembling two Hofstadter butterflies shifted in energy and braided together, as shown in Fig. \[fig:DoubleHofstadter\]. We call this structure the double Hofstadter butterfly. [@Tezuka2012]

The spin-orbit coupling $\alpha$ mixes the spin-up states 