Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model.

Area of Science:

• Mathematical Physics
• Random Matrix Theory
• Condensed Matter Physics

Background:

• Random matrix ensembles are crucial for modeling complex quantum systems.
• The Muttalib-Borodin ensemble (γ=1) serves as a baseline for studying interactions.
• Understanding eigenvalue distributions is key to characterizing system properties.

Purpose of the Study:

• To analyze the joint probability distribution of γ ensembles, incorporating a two-particle interaction parameter γ.
• To investigate the role of the effective potential in determining eigenvalue density for γ ensembles.
• To explore the impact of γ on potential nonmonotonicity and its relation to conductance in disordered systems.

Main Methods:

• Solving the Riemann-Hilbert problem associated with γ ensembles to define the effective potential.
• Numerical computation of eigenvalue density for γ ensembles across various γ values (γ>0).
• Analysis of the relationship between the effective potential's nonmonotonicity and eigenvalue density changes.

Main Results:

• The effective potential, derived from the Riemann-Hilbert solution, accurately computes eigenvalue density for γ ensembles.
• The interaction parameter γ generates or amplifies nonmonotonicity in the effective single-particle potential.
• Reducing γ can lead to significant nonmonotonicity, altering eigenvalue density and causing decreased conductance in disordered conductors.

Conclusions:

• The γ ensembles provide a flexible framework for studying the effects of interactions on random matrix properties.
• The observed link between potential nonmonotonicity and conductance suggests applications in modeling disordered electronic systems.
• This research offers insights into how interaction parameters influence spectral properties and macroscopic behavior like conductance.