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Many-body localization in Ising models with random long-range interactions

We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, $V_{ij}\propto\left|i-j\right|^{-α}$, where the exponent of the interaction range $α$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $α$, the critical exponent experiences a sharp increase at about $α=1$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $α<1$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $α>1$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.