Odd-even effect of the mosaic modulation period of quasiperiodic hopping on the Anderson localization in a one-dimensional lattice model

In this study, we investigate Anderson localization in a one-dimensional lattice with mosaic off-diagonal quasiperiodic hopping. Our findings reveal that the localization behavior of zero-energy states is highly dependent on the parity of the mosaic modulation period, denoted as $\ensuremath{\kappa}$. Specifically, when $\ensuremath{\kappa}$ is an odd integer, there is no Anderson localization transition even for large quasiperiodic hopping strengths, and the zero-energy state remains in a critical state. On the other hand, for an even $\ensuremath{\kappa}$ and generic quasiperiodic hopping, the zero-energy state becomes a localized edge state at either the left or right end of the system. Additionally, we observe that the geometric mean value of the energy spectrum is equal to the constant hopping for an even $\ensuremath{\kappa}$, while for an odd $\ensuremath{\kappa}$, it is equal to the geometric mean value of the hopping. This odd-even effect of the mosaic period also extends to other eigenstates near zero energy. More specifically, for an odd $\ensuremath{\kappa}$, an energy window in which the eigenstates remain critical even for strong quasiperiodic hopping exists. In contrast, for an even $\ensuremath{\kappa}$, an Anderson localization transition occurs as the hopping strength increases. Furthermore, we are able to accurately determine the Lyapunov exponent $\ensuremath{\gamma}(E)$ and the mobility edges ${E}_{c}$. By analyzing the Lyapunov exponent, we identify critical regions in the hopping-energy parameter planes. Additionally, as the energy approaches the mobility edges, we observe a critical index of the localization length of $\ensuremath{\nu}=1$. Finally, we demonstrate that different systems can be characterized by their Lyapunov exponent $\ensuremath{\gamma}(E)$ and Avila's acceleration $\ensuremath{\omega}(E)$.