Long-range spectral statistics of the Rosenzweig-Porter model

TL;DR

This paper investigates the long-range spectral statistics of the Rosenzweig-Porter model, demonstrating that spectral measures like the Thouless time can effectively identify phase transitions between ergodic, fractal, and localized states.

Contribution

It introduces a numerical analysis linking long-range spectral statistics to phase transitions in the Rosenzweig-Porter model, highlighting the use of spectral form factor and Thouless time as probes.

Findings

Thouless time scaling indicates phase transition points.

Spectral statistics can distinguish between ergodic, fractal, and localized phases.

Long-range spectral measures complement short-range level statistics in phase detection.

Abstract

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that supports an ergodic, fractal, and localized phase. The names of these phases refer to the properties of the (midspectrum) eigenstates. This work focuses on the long-range spectral statistics of the recently introduced unitary equivalent of this model. By numerically studying the Thouless time obtained from the spectral form factor, it is argued that long-range spectral statistics can be used to probe the transition between the ergodic and the fractal phases. The scaling of the Thouless time as a function of the model parameters is found to be similar to the scaling of the spreading width of the eigenstates. Provided that the transition between the fractal and the localized phases can be probed through short-range level statistics, such as the average ratio of consecutive level spacings, this work establishes that spectral statistics are sufficient to probe both transitions present in the phase diagram.