  We propose a new approach to probing ergodicity and its breakdown in quantum
many-body systems based on their response to a local perturbation. We study the
distribution of matrix elements of a local operator between the system's
eigenstates, finding a qualitatively different behaviour in the many-body
localized (MBL) and ergodic phases. To characterize how strongly a local
perturbation modifies the eigenstates, we introduce the parameter ${\cal
G}(L)=\langle \ln (V_{nm}/\delta) \rangle$, which represents a
disorder-averaged ratio of a typical matrix element of a local operator $V$ to
the energy level spacing, $\delta$; this parameter is reminiscent of the
Thouless conductance in the single-particle localization. We show that the
parameter ${\cal G}(L)$ decreases with system size $L$ in the MBL phase, and
grows in the ergodic phase. We surmise that the delocalization transition
occurs when ${\cal G}(L)$ is independent of system size, ${\cal G}(L)={\cal
G}_c\sim 1$. We illustrate our approach by studying the many-body localization
transition and resolving the many-body mobility edge in a disordered 1D XXZ
spin-1/2 chain using exact diagonalization and time-evolving block decimation
methods. Our criterion for the MBL transition gives insights into microscopic
details of transition. Its direct physical consequences, in particular
logarithmically slow transport at the transition, and extensive entanglement
entropy of the eigenstates, are consistent with recent renormalization group
predictions.
