Periodic-orbit theory of anderson localization on graphs

Area of Science:

• Quantum physics
• Condensed matter physics
• Mathematical physics

Background:

• Anderson localization describes the suppression of wave function propagation in disordered systems.
• Periodic-orbit theory offers a semiclassical approach to quantum systems.
• The aperiodic Kronig-Penney model is a standard for studying electron behavior in one-dimensional potentials.

Purpose of the Study:

• To present the first quantum system where Anderson localization is fully described by periodic-orbit theory.
• To develop an exact method for calculating the probability of returning to an initial state in a localized system.
• To compare the predictions of periodic-orbit theory with the diagonal approximation for localization.

Main Methods:

• A quantum graph model analogous to the one-dimensional aperiodic Kronig-Penney model was developed.
• The exact expression for the return probability was computed using classical trajectories.
• Families of isometric orbits were identified and analyzed.
• Coherent periodic-orbit sums were performed analytically using combinatorial methods.

Main Results:

• The probability to return to an initially localized state saturates to a finite value, confirming localization.
• The diagonal approximation, in contrast, shows a diffusive decay.
• The study provides an exact description of Anderson localization within the framework of periodic-orbit theory.

Conclusions:

• Periodic-orbit theory provides an accurate and complete description of Anderson localization in the proposed quantum system.
• The analytical approach using combinatorial methods offers a powerful tool for studying quantum localization.
• This work bridges the gap between semiclassical theories and the exact quantum description of localization.