Excited-Eigenstate Entanglement Properties of XX Spin Chains with Random Long-Range Interactions

Abstract

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 $α$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interaction. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy level spacings, as obtained by ED for up to $N\sim 18$ spins, we find indications of a delocalization transition at $α_c \approx 1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $α>α^*$ the entanglement entropy (EE) of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $α<α^*$ EE displays an algebraic growth with the subsystem size $l$, $S_l\sim l^β$, with $0<β<1$. We find that $α^* \approx 1$ coincides with the delocalization transition $α_c$ 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 {\it rainbow} proliferation for very long-range interactions $α\ll 1$. We also investigate the effective temperature dependence of the EE allowing us to study the half-chain entanglement entropy of eigenstates at different energy densities, where we find that the crossover in EE occurs at $α^* < 1$.