and spin-down states differently at each value of the quasimomentum $k$ of the resulting RZ bands. Therefore the spin-independent site potential, which has components with $|k|=\pm \kappa$ and $|k| = \pm(2\pi-\kappa)$, further mixes the upper and lower RZ bands. Most of the states in the resulting double Hofstadter butterfly do not have a completely polarized spin. We may, however, obtain the expectation value of the $z$ component of the spin, $\langle S_z\rangle$, for each of the $2L$ eigenstate. The energy levels plotted in Fig. \[fig:DoubleHofstadter\] have been color-coded according to the value of $\langle S_z\rangle$.

We find that, for a fixed value of $\kappa$, the sets of states from two Hofstadter-butterfly-like structures with separated values of spin polarizations overlap within some energy ranges. In some regions in energy there are no single particle states, even inside the range of $|\epsilon - t|\leq t + \Gamma$, which was occupied by states of eq. (\[eqn:RZbands\]) before the introduction of the site potential modulation eq. (\[eqn:sitePotential\]). Other regions are occupied by states in only one of the Hofstadter-butterfly-like structure. We study the consequences of the site potential modulation on the realization of TS states for the many-body states with a finite chemical potential 