Robust nonergodicity of the ground states in the β ensemble

In various chaotic quantum many-body systems, the ground states show nontrivial athermal behavior despite the bulk states exhibiting thermalization. Such athermal states play a crucial role in quantum information theory and its applications. Moreover, any generic quantum many-body system in the Krylov basis is represented by a tridiagonal Lanczos Hamiltonian, which is analogous to the matrices from the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$, a well-studied random-matrix model with level repulsion tunable via the parameter $\ensuremath{\beta}$. Motivated by this, here we focus on the localization properties of the ground and anti-ground states of the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$. Both analytically and numerically we show that both the edge states demonstrate nonergodic (fractal) properties for $\ensuremath{\beta}\ensuremath{\sim}\mathcal{O}(1)$, while the typical bulk states are ergodic. Surprisingly, the fractal dimension of the edge states remains three times smaller than that of the bulk states irrespective of the global phase of the $\ensuremath{\beta}\phantom{\rule{4.pt}{0ex}}\text{ensemble}$. In addition to the fractal dimensions, we also consider the distribution of the localization centers of the spectral edge states, their mutual separation, as well as the spatial and correlation properties of the first excited states.