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Topological aspects of quantum chaos
P. Leboeuf1, J. Kurchan, M. Feingold
1Division de Physique Theorique,(a)) Institut de Physique Nucleaire, 91406 Orsay Cedex, FranceDipartimento di Fisica, Universita di Roma "La Sapienza," Ple A. Moro 2, Rome, ItalyDepartment of Physics, Ben-Gurion University, Beer-Sheva 84105, IsraelDepartment of Physics, B-019, University of California at San Diego, La Jolla, California 92093.
Chaos (Woodbury, N.Y.)
|January 1, 1992
Summary
This study introduces a topological criterion to understand how quantum eigenfunctions explore phase space in classically chaotic systems. It links eigenfunction delocalization to chaotic dynamics and explores spectral degeneracies in state transitions.
Area of Science:
- Quantum Chaos
- Statistical Mechanics
- Topological Quantum Field Theory
Background:
- Classical chaos in dynamical systems.
- Quantum mechanics of chaotic systems.
- Phase space localization phenomena.
Purpose of the Study:
- To develop a discrete, topological criterion for phase-space localization in quantized chaotic maps.
- To associate an integer invariant with eigenfunctions, analogous to quantized Hall conductivity.
- To investigate the relationship between eigenfunction delocalization and classical chaotic dynamics.
Main Methods:
- Analysis of quantized classically chaotic maps on a toroidal two-dimensional phase space.
- Development of a discrete topological criterion for phase-space localization.
- Association of an integer invariant with eigenfunctions to test phase-space exploration under changing boundary conditions.
Main Results:
- A novel discrete topological criterion for phase-space localization is presented.
- An integer, analogous to quantized Hall conductivity, is associated with each eigenfunction.
- The study discusses the correspondence between delocalization and chaotic classical dynamics.
Conclusions:
- The developed criterion provides insights into phase-space localization in quantum chaotic systems.
- The role of spectral degeneracies in the transition between localized and delocalized states is highlighted.
- The findings are illustrated using a specific model, demonstrating the general applicability of the approach.