Nonergodic extended states in the β ensemble

Matrix models showing a 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 the β ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the 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 the Rosenzweig-Porter ensemble (RPE), the β ensemble is another example where nonergodic 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 the β ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the β ensemble behave differently than the latter models.