Many-Body Localization: Transitions in Spin Models

We study the transitions between ergodic and many-body localized phases in spin systems, subject to quenched disorder, including the Heisenberg chain and the central spin model. In both cases systems with common spin lengths $1/2$ and $1$ are investigated via exact numerical diagonalization and random matrix techniques. Particular attention is paid to the sample-to-sample variance $(\Delta_sr)^2$ of the averaged consecutive-gap ratio $\langle r\rangle$ for different disorder realizations. For both types of systems and spin lengths we find a maximum in $\Delta_sr$ as a function of disorder strength, accompanied by an inflection point of $\langle r\rangle$, signaling the transition from ergodicity to many-body localization. The critical disorder strength is found to be somewhat smaller than the values reported in the recent literature. Further information about the transitions can be gained from the probability distribution of expectation values within a given disorder realization.