Mobility edge (ME), a critical energy separating localized and extended states in spectrum, is a central concept in understanding localization physics. However, there are few models with exact MEs, and their presences are fragile against perturbations. In the paper, we generalize the Aubry-André-Harper model proposed in (Ganeshanet al2015Phys. Rev. Lett.114146601) and recently realized in (Anet al2021Phys. Rev. Lett.126040603), 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 physical perception of the exact expression is also provided. We show that old exact MEs, guaranteed by the delicate self-duality which is broken by the relative phase, are special ones in a series controlled by the phase. Furthermore, we demonstrate that out of expectation, exact MEs are invariant against a shift in the quasiperiodic potential, although the shift changes the spectrum and localization properties. Finally, we show that the exact ME is related to the one in the dual model which has long-range hoppings.
Keywords: mobility edge; one dimension; quantum localization; quasiperiodic system.
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