  In various chaotic quantum many-body systems, the ground states show
non-trivial 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 $\beta$ ensemble, a well-studied random
matrix model with level repulsion tunable via the parameter $\beta$. Motivated
by this, here we focus on the localization properties of the ground and
anti-ground states of the $\beta$ ensemble. Both analytically and numerically,
we show that both the edge states demonstrate non-ergodic (fractal) properties
for $\beta\sim\mathcal{O}(1)$ while the typical bulk states are ergodic.
Surprisingly, the fractal dimension of the edge states remain three time
smaller than that of the bulk states irrespective of the global phase of the
$\beta$ 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.
