Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map

Area of Science:

• Quantum Chaos
• Statistical Physics
• Condensed Matter Theory

Background:

• Extends previous work on Anderson localization in monochromatically perturbed kicked quantum maps.
• Investigates delocalization-localization transitions in quantum maps subjected to multiple harmonic perturbations.
• Relates the number of perturbations (M) to the spatial dimension (d=M+1) of disordered lattices.

Purpose of the Study:

• To analyze the dependency of critical phenomena in polychromatically perturbed quantum maps on the number of perturbations (M).
• To compare numerical results with predictions from the self-consistent theory (SCT) of localization.
• To investigate the critical perturbation strength and localization length exponent.

Main Methods:

• Analysis based on the self-consistent theory (SCT) of localization.
• Application of a hypothesis for the mean-free-path parameter successfully used in monochromatic cases.
• Numerical simulations with a large increase in the number of perturbations (M) for standard and Anderson maps.

Main Results:

• The critical subdiffusion index (α) from numerical results agrees with one-parameter scaling theory (α=2/(M+1)).
• Significant deviation observed between numerically obtained critical exponent of localization length and SCT predictions.
• Critical perturbation strength is found to be 1/(M-1) times smaller than SCT predictions, indicating strong cooperativity among perturbations.

Conclusions:

• While scaling theory accurately predicts subdiffusion behavior, the SCT overestimates localization length exponents in polychromatically perturbed quantum maps.
• The critical perturbation strength is substantially underestimated by SCT, suggesting cooperative effects among multiple harmonic perturbations.
• Findings highlight the limitations of current SCT for complex perturbation regimes and point to cooperative phenomena in quantum localization.