Engineering mobility in quasiperiodic lattices with exact mobility edges

We investigate the effect of an additional modulation parameter $\ensuremath{\delta}$ on the mobility properties of quasiperiodic lattices described by a generalized Ganeshan-Pixley-Das Sarma model with two onsite modulation parameters. For the case with bounded quasiperiodic potential, we unveil the existence of self-duality relation, independent of $\ensuremath{\delta}$. By applying Avila's global theory, we analytically derive Lyapunov exponents in the whole parameter space, which enables us to determine mobility edges or anomalous mobility edges exactly. Our analytical results indicate that the mobility edge equation is described by two curves and their intersection with the spectrum gives the true mobility edge. Tuning the strength parameter $\ensuremath{\delta}$ can change the spectrum of the quasiperiodic lattice, and thus engineers the mobility of quasiperiodic systems, giving rise to completely extended, partially localized, and completely localized regions. For the case with unbounded quasiperiodic potential, we also obtain the analytical expression of the anomalous mobility edge, which separates localized states from critical states. By increasing the strength parameter $\ensuremath{\delta}$, we find that the critical states can be destroyed gradually and finally vanish.