Probing the many-body localization transition with matrix elements of local operators

We propose the statistics of matrix elements of local operators as a new probe of the many-body localized (MBL) phase. Matrix elements of a given local operator $V$ encode many physical properties, such as the response of the system to a local perturbation induced by the action of $V$, spectral functions, and dynamics of the system. The distribution of matrix elements of a local operator between system's eigenstates exhibits qualitatively different behavior in the many-body localized and ergodic phases, allowing for an accurate determination of the two phases. To characterize this distribution, for a given system size $L$, we introduce a parameter $g(L)=\langle\log\frac{V_{i,i+1}}{\Delta}\rangle$, which is a disorder-averaged ratio of the matrix element of operator $V$ between adjacent eigenstates, and $\Delta$ is the level spacing. We find that $g(L)$ decreases with $L$ in the MBL phase, and grows in the ergodic phase. We propose that at the MBL-delocalization transition $g(L)$ is independent of system size, $g(L)=g_c\sim1$, and use this criterion to map out the phase diagram of a disordered 1D XXZ spin-1/2 chain. By studying the scaling of $g(L)$ as a function of energy density, we locate the many-body mobility edge. We discuss implications for delocalization phase transition.