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An interacting quantum system can transition from an ergodic to a many-bodylocalized (MBL) phase under the presence of sufficiently large disorder. Bothphases are radically different in their dynamical properties, which arecharacterized by highly excited eigenstates of the Hamiltonian. Each eigenstatecan be characterized by the set of quantum numbers over the set of (local, inthe MBL phase) integrals of motion of the system. In this work we study theevolution of the eigenstates of the disordered Heisenberg model as the disorderstrength, $W$, is varied adiabatically. We focus on the probability that two`colliding' eigenstates hybridize as a function of both the range $R$ at whichthey differ as well as the strength of their hybridization. We find, in the MBLphase, that the probability of a colliding eigenstate hybridizing strongly atrange $R$ decays as $Pr(R)\propto \exp [-R/\eta]$, with a length scale $\eta(W)= 1 / (B \log(W / W_c) )$ which diverges at the critical disorder strength$W_c$. This leads to range-invariance at the transition, suggesting theformation of resonating cat states at all ranges. This range invariance doesnot survive to the ergodic phase, where hybridization is exponentially morelikely at large range, a fact that can be understood with simple combinatorialarguments. In fact, compensating for these combinatorial effects allows us todefine an additional correlation length $\xi$ in the MBL phase which is inexcellent agreement with previous works and which takes the critical value $1 /\log(2)$ at the transition, found in previous works to destabilize the MBLphase. Finally, we show that deep in the MBL phase hybridization is dominatedby two-level collisions of eigenstates close in energy.

Eigenstates hybridize on all length scales at the many-body localization transition