Exact mobility edges in Aubry-Andr\'{e}-Harper models with relative phases

Mobility edge (ME), a critical energy separating localized and extended states in spectrum, is a central concept in understanding the localization physics. However, there are few models with exact MEs. In the paper, we generalize the Aubry-Andr´e-Harper model proposed in [Phys. Rev. Lett. 114 , 146601 (2015)] and recently realized in [Phys. Rev. Lett. 126 , 040603 (2021)], by introducing a relative phase in the quasiperiodic potential. Applying Avila’s global theory we analytically compute localization lengths of all single-particle states and determine the exact expression of ME, which both significantly depend on the relative phase. They are verified by numerical simulations, and a physical perception of the exact expression is also provided. We further demonstrate that the exact expression of ME works for an even broad class of generalized Aubry-Andr´e-Harper models. Moreover, we show that the exact ME is related to the one in the dual model which has long-range hoppings.