Spectral statistics and dynamics of Lévy matrices

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Random Matrix Theory

Background:

• Lévy matrices, characterized by power-law tailed distributions (parameter mu), extend beyond Gaussian universality.
• These matrices model diverse physical systems, including spin glasses and electronic systems with long-range interactions.

Purpose of the Study:

• Investigate spectral statistics and dynamics of Lévy matrices.
• Extend understanding of the sparse matrix limit (mu --> 0).
• Analyze level dynamics and phase transitions in Lévy matrix models.

Main Methods:

• Analysis of spectral statistics and dynamics for Lévy matrices.
• Extension of previous work to the sparse matrix limit.
• Application of the delta3 statistic to map phase diagrams.
• Computation of conductance using the Thouless formula.

Main Results:

• Geometrical level repulsion in 2x2 Lévy matrices is generally unaffected by distribution broadness.
• Essential singularities in Lévy distributions break geometrical repulsion, making it mu-dependent.
• The delta3 statistic successfully maps the phase diagram, revealing new insights into level dynamics.
• The mixed phase separating conducting and insulating states exhibits unique conductance properties.

Conclusions:

• Lévy matrices exhibit distinct spectral properties compared to Gaussian ensembles.
• The study clarifies phase transitions and conductance behavior in systems modeled by Lévy matrices.
• Findings contribute to understanding complex physical phenomena like the metal-insulator transition.