Nonergodic extended states in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> ensemble
Adway Kumar Das, Anandamohan Ghosh
Abstract
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 $\ensuremath{\beta}$ ensemble, known for its structural simplicity. However, eigenvector properties of the $\ensuremath{\beta}$ ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the $\ensuremath{\beta}$ ensemble and find that the Anderson transition occurs at $\ensuremath{\gamma}=1$ and ergodicity breaks down at $\ensuremath{\gamma}=0$ if we express the repulsion parameter as $\ensuremath{\beta}={N}^{\ensuremath{-}\ensuremath{\gamma}}$. Thus other than the Rosenzweig-Porter ensemble (RPE), the $\ensuremath{\beta}$ ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values $(0<\ensuremath{\gamma}<1)$. We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the $\ensuremath{\beta}$ 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 $\ensuremath{\beta}$ ensemble behave differently than the latter models.