Non-ergodic extended states in $β$-ensemble

Abstract

Matrix models showing chaotic-integrable transition in the spectral statistics are important for understanding Many Body Localization (MBL) in physical systems. One such example is the βensemble, known for its structural simplicity. However, eigenvector properties of β-ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of β-ensemble and identify the ergodic transition point (γET = 0) and localization transition point (γAT = 1) where we express the repulsion parameter as β = N −γ . Thus other than Rosenzweig-Porter ensemble (RPE), β-ensemble is another example where Non-Ergodic Extended (NEE) states are observed over a finite interval of parameter values (0 < γ < 1). We find that the chaotic-integrable transition coincides with the ergodic transition unlike the RPE or the 1-D disordered spin-1/2 Heisenberg model where this coincidence occurs at the localization transition. As a result, the dynamical time-scales in the NEE regime of β-ensemble behave differently than the former models.