https://doi.org/10.1140/epjb/s10051-022-00477-9
Regular Article - Statistical and Nonlinear Physics
Statistical properties related to angle variables in Hamiltonian map approach for one-dimensional tight-binding models with localization
College of science and New energy technology Engineering of Jiangsu province, Nanjing University of Posts and Telecommunications, 210003, Nanjing, China
Received:
17
August
2022
Accepted:
30
December
2022
Published online:
21
January
2023
The Hamiltonian map approach transforms a one-dimensional (1D) discrete Schrödinger equation to a classical two-dimensional (2D) iterative equation with action-angle variables 
at the nth iterative step. The corresponding Hamiltonian describes a linear parametric oscillator with time-dependent linear periodic delta kicks. We use a 
-related order parameter R to measure the degree of instability of trajectory 
. 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.
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