Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles

TL;DR

This paper introduces a two-parameter random matrix ensemble to study the transition from chaos to integrability, analyzing spectral correlations and dynamical fidelity to identify phase transitions and crossover behavior.

Contribution

It develops the rpe\, a generalized 2x2 matrix ensemble, and provides analytical insights into spectral and dynamical signatures of chaos-integrability crossover.

Findings

Spectral correlations quantify level repulsion crossover.

Dynamical fidelity reveals chaos to integrability transition.

Second order phase transition at b3=2 for large matrices.

Abstract

We introduce a two-parameter ensemble of generalized $2\times 2$ real symmetric random matrices called the $\beta$-Rosenzweig-Porter ensemble (\brpe), parameterized by $\beta$, a fictitious inverse temperature of the analogous Coulomb gas model, and $\gamma$, controlling the relative strength of disorder. \brpe\ encompasses RPE from all of the Dyson's threefold symmetry classes: orthogonal, unitary and symplectic for $\beta=1,2,4$. Firstly, we study the energy correlations by calculating the density and 2nd moment of the Nearest Neighbor Spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of \brpe\ in the $\gamma$-$\beta$ plane. Our results are in qualitative agreement with numerically computed fidelity for $N\gg2$ matrix ensembles. Furthermore, we observe that for large $N$ the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at $\gamma=2$.