  We study disorder-induced ergodicity breaking transition in high-energy
eigenstates of interacting spin-1/2 chains. Using exact diagonalization we
introduce a cost function approach to quantitatively compare different
scenarios for the eigenstate transition. We study ergodicity indicators such as
the eigenstate entanglement entropy and the spectral level spacing ratio, and
we consistently find that an (infinite-order) Kosterlitz-Thouless transition
yields a lower cost function when compared to a finite-order transition.
Interestingly, we observe that the transition point in finite systems exhibits
nearly thermal properties, i.e., ergodicity indicators at the transition are
close to the random matrix theory predictions.
