Statistical properties related to angle variables in Hamiltonian map approach for one-dimensional tight-binding models with localization

The Hamiltonian map approach transforms a one-dimensional (1D) discrete Schrödinger equation to a classical two-dimensional (2D) iterative equation with action-angle variables $$(r_n,\theta _n)$$ ( r n , θ n ) at the nth iterative step. The corresponding Hamiltonian describes a linear parametric oscillator with time-dependent linear periodic delta kicks. We use a $$\theta $$ θ -related order parameter R to measure the degree of instability of trajectory $$\{(r_n,\theta _n)\}$$ { ( r n , θ n ) } . Two prototypical models, i.e., the 1D Anderson model and the 1D slowly varying incommensurate potential model, are as examples. All states are localized in the former model, and states may be extended, localized and critical in the latter model. In the two models, we find R increases with the Lyapunov exponent and the inverse localization tensor (they are inversely proportional to localization length), so the instability of trajectory relates to Anderson localization. Graphic abstract