Emergent strength-dependent scale-free mobility edge in a non-reciprocal long-range Aubry-Andr\'e-Harper model

We investigate the properties of mobility edge in an Aubry-Andr\'e-Harper model with non-reciprocal long-range hopping. The results reveal that there can be a new type of mobility edge featuring both strength-dependent and scale-free properties. By calculating the fractal dimension, we find that the positions of mobility edges are robust to the strength of non-reciprocal long-range hopping. Furthermore, through scale analysis of the observables such as fractal dimension, eigenenergy and eigenstate, etc., we show that four different specific mobility edges can be observed in the system. This paper extends the family tree of mobility edges and hopefully it will shed more light on the related theory and experiment.