Many-body localization and enhanced nonergodic subdiffusive regime in the presence of random long-range interactions

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions ${V}_ij$ decaying as power-law ${V}_ij/{({r}_ij)}^{\ensuremath{\alpha}}$ with distance ${r}_ij$. We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder ${h}_c^r$, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance $I(t)$, which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder ${h}_c^I$, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than ${h}_c^r$. In between ${h}_c^r$ and ${h}_c^I$ there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side ($h<{h}_c^r$) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For $h<{h}₀<{h}_c^r$, the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.