Anomalous wave function statistics on a one-dimensional lattice with power-law disorder

Area of Science:

• Condensed matter physics
• Disordered systems
• Quantum mechanics

Background:

• Anderson localization describes the suppression of quantum wave functions in disordered systems.
• The Lloyd model provides a theoretical framework for studying Anderson localization.
• Understanding wave function statistics is crucial for predicting transport properties.

Purpose of the Study:

• To investigate wave function statistics in the Lloyd model of Anderson localization.
• To analyze the impact of Cauchy-distributed random on-site potentials on a 1D lattice.
• To identify anomalies in probability distributions at specific energy levels.

Main Methods:

• Developed a general theoretical framework for wave function statistics.
• Analyzed the Lloyd model with a Cauchy distribution for random potentials.
• Calculated probability distributions for wave functions, conductance, and local density of states.

Main Results:

• Discovered a hierarchy of anomalies in probability distributions at energies with rational wavelength-to-lattice constant ratios.
• Observed power-law tails dominating short-distance wave function statistics.
• Identified deviations from log-normal behavior in the disorder strength's leading order.

Conclusions:

• The study reveals universal features in wave function statistics for Anderson localization.
• Power-law tails are a significant characteristic of short-distance wave functions in this model.
• Anomalies at rational energies provide insights into the interplay between disorder and quantum transport.