Average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians

To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of $U(1)$ symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions $\nu$ of the spectrum are included in the average. We find indications that, for $\nu\rightarrow0$, the magnitude of the negative $O(1)$ correction is only slightly greater than the one predicted for random pure states. For finite $\nu$, following a phenomenological approach, we derive a simple expression that describes the numerically observed $\nu$ dependence of the $O(1)$ deviation from the prediction for random pure states.