Excited-eigenstate entanglement properties of XX spin chains with random long-range interactions

Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent alpha. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interactions. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy-level spacings, as obtained by ED for up to N <^>' 18 spins, we find indications of a delocalization transition at alpha c ti 1 in the middle of the energy spectrum. With RSRG-X and ED, we show that for alpha > alpha* the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for alpha < alpha* EE displays an algebraic growth with the subsystem size l, Sl <^>' l beta, with 0 < beta < 1. We find that alpha* ti 1 coincides with the delocalization transition alpha c in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to rainbow proliferation for very long-range interactions alpha << 1. We also investigate the effective temperature dependence of the EE, allowing us to study the half-chain EE of eigenstates at different energy densities, where we find that the crossover in EE occurs at alpha* < 1.