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 $\ensuremath{\alpha}$. 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 $N\ensuremath{\sim}18$ spins, we find indications of a delocalization transition at ${\ensuremath{\alpha}}_c\ensuremath{\approx}1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^*$ the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^*$ EE displays an algebraic growth with the subsystem size $l, {S}_l\ensuremath{\sim}{l}^{\ensuremath{\beta}}$, with $0<\ensuremath{\beta}<1$. We find that ${\ensuremath{\alpha}}^*\ensuremath{\approx}1$ coincides with the delocalization transition ${\ensuremath{\alpha}}_c$ 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 $\ensuremath{\alpha}\ensuremath{\ll}1$. 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 ${\ensuremath{\alpha}}^*<1$.