Universal algebraic growth of entanglement entropy in many-body localized systems with power-law interactions

Power-law interactions play a key role in a large variety of physical systems. In the presence of disorder, these systems may undergo many-body localization for a sufficiently large disorder. Within the many-body localized phase the system presents in time an algebraic growth of entanglement entropy, ${S}_{vN}(t)\ensuremath{\propto}{t}^{\ensuremath{\gamma}}$. Whereas the critical disorder for many-body localization depends on the system parameters, we find by extensive numerical calculations that the exponent $\ensuremath{\gamma}$ acquires a universal value ${\ensuremath{\gamma}}_{c}\ensuremath{\simeq}0.33$ at the many-body localization transition, for different lattice models, decay powers, filling factors, or initial conditions. Moreover, our results suggest an intriguing relation between ${\ensuremath{\gamma}}_{c}$ and the critical minimal decay power of interactions necessary for many-body localization.