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Non-ergodic extended states in $β$-ensemble

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 $β$-ensemble, 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 find that the Anderson transition occurs at $γ= 1$ and ergodicity breaks down at $γ= 0$ if 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 breaking of ergodicity in $β$-ensemble but with the localization transition in 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 later models.