Stark localization near Aubry-Andr\'{e} criticality

In this work we investigate the Stark localization near the Aubry-Andr\'{e} (AA) critical point. We study system-dependent parameters, such as localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited state, for characterizing the localization-delocalization transition. We show that the scaling exponents possessed by these key descriptors of localization are quite different from that of a pure AA model or Stark model. Near the critical point of the AA model, inducing Stark field of strength h, the localization length ζ scales as ζ∝ h-ν with ν≈0.29 which is different than both the pure AA model (ν=1) and Stark model (ν≈0.33). The IPR scales as IPR ∝ hˢ with s≈0.096 which is again significantly different than both the pure AA model (s≈0.33) and Stark model (s≈0.33). The energy gap, Δ, scales as E∝ hν z, where z≈2.37 which is however same as the pure AA model.