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Correlated random band matrices: localization-delocalization transitions

Janssen¹, Pracz

¹Institut fur Theoretische Physik III, Ruhr-Universitat Bochum, 44780 Bochum, Germany.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

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Correlated random band matrices exhibit complex behavior, including quantum Hall-type localization-delocalization transitions. These transitions occur under specific conditions of matrix bandwidth and element correlation.

Area of Science:

Statistical mechanics
Condensed matter physics
Quantum chaos

Background:

Random matrix theory is crucial for understanding complex quantum systems.
Correlations between matrix elements significantly alter system properties.
Previous models often neglected these crucial correlations.

Purpose of the Study:

Investigate eigenvector statistics in correlated random band matrices.
Characterize the influence of bandwidth and correlation parameters.
Identify novel phase transitions in these models.

Main Methods:

Analysis of Hermitian NxN matrices with tunable bandwidth B(N) and correlation C(N).
Examination of eigenvector statistics under varying scaling conditions.
Theoretical modeling of correlated band matrix properties.

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Main Results:

Correlated matrices display richer phenomenology than uncorrelated ones.
A localization-delocalization transition, akin to the quantum Hall effect, emerges.
This transition is observed when bandwidth and correlation parameters scale as approximately sqrt(N).

Conclusions:

Correlations are essential for understanding complex matrix behaviors.
The quantum Hall-type transition highlights a new universality class.
This research provides insights into disordered quantum systems.