Exact complex mobility edges and flagellate-like spectra for non-Hermitian quasicrystals with exponential hoppings

We propose a class of general non-Hermitian quasiperiodic lattice models with exponential hoppings and analytically determine the genuine complex mobility edges by solving its dual counterpart exactly utilizing Avila's global theory. Our analytical formula unveils that the complex mobility edges usually form a loop structure in the complex energy plane, i.e, the mobility ring. By shifting the eigenenergy a constant $t$, the complex mobility edges of the family of models with different hopping parameter $t$ can be described by a unified formula, formally independent of $t$. By scanning the hopping parameter, we demonstrate the existence of a type of intriguing flagellate-like spectra in the complex energy plane, in which the localized states and extended states are well separated by the complex mobility edges. Our result provides a firm ground for understanding the complex mobility edges in non-Hermitian quasiperiodic lattices.