  Some interacting disordered many-body systems are unable to thermalize when
the quenched disorder becomes larger than a threshold value. Although several
properties of nonzero energy density eigenstates (in the middle of the
many-body spectrum) exhibit a qualitative change across this many-body
localization (MBL) transition, many of the commonly-used diagnostics only do so
over a broad transition regime. Here, we provide evidence that the transition
can be located precisely even at modest system sizes by sharply-defined changes
in the distribution of extremal eigenvalues of the reduced density matrix of
subsystems. In particular, our results suggest that $p* = \lim_{\lambda_2
\rightarrow \ln(2)^{+}}P_2(\lambda_2)$, where $P_2(\lambda_2)$ is the
probability distribution of the second lowest entanglement eigenvalue
$\lambda_2$, behaves as an ''order-parameter'' for the MBL phase: $p*> 0$ in
the MBL phase, while $p* = 0$ in the ergodic phase with thermalization. Thus,
in the MBL phase, there is a nonzero probability that a subsystem is entangled
with the rest of the system only via the entanglement of one subsystem qubit
with degrees of freedom outside the region. In contrast, this probability
vanishes in the thermal phase.
