The question of when and how isolated many-body quantum systems thermalize, and why ergodicity-breaking transitions occur, is one of the central topics of non-equilibrium quantum physics. Such transitions are most conveniently studied in random matrix models. One of the simplest is the Rosenzweig-Porter model, which, between the ergodic and the localized phase, also hosts an intermediate fractal phase with fractal states, which are delocalized but confined to a part of the Hilbert space of dimension $N^{D_q} < N$, where $N$ is the dimension of the full Hilbert space and $D_q \in \left[0, 1\right]$ is the fractal dimension. In its most basic form the model is given by a single parameter $\gamma$ and has the ergodic phase for $\gamma \le 1$, fractal phase for $\gamma \in (1, 2)$, and localized phase for $\gamma \ge 2$. Embedding the model in the many-body Hilbert space of spins $1/2$ allows us to divide the system into two compact bipartitions and compute the entanglement entropy between them. Although the random matrices of the Rosenzweig-Porter model are full, and thus describe a system with unphysical many-body interactions in which every spin configuration is coupled to every other, similarities with physical models have been observed. One of them is the fading ergodicity regime, in which the fluctuations of matrix elements of local observables begin to depart from their ergodic values already before the ergodicity-breaking transition, even though the averages of these observables remain unchanged. A similarity has also been demonstrated between the average entanglement entropy of a single-site bipartition and one local observable of a system in which fading ergodicity regime was detected. In this work we relate the averages and the fluctuations of the eigenstate entanglement entropy of the Rosenzweig-Porter model to fading ergodicity and to the fractality of the eigenstates for different bipartition fractions $f$. We derive fading ergodicity ansatz for the entanglement entropy as a function of $f$ and demonstrate the existence of a characteristic point $\gamma_{c,2} = 2 - f$, while the fluctuations of the eigenstate entanglement yield the characteristic point $\gamma_{c,1} = 1 + \frac{f}{2}$. We also study the growth of entanglement from an initial weakly entangled state and show that it evolves on two timescales. The onset of growth is governed by the square root of the Thouless time, with a derived factor dependent on $f$, and the saturation by the Thouless time. Finally, we demonstrate the agreement between the average eigenstate entanglement and the saturation values of the time evolution, as well as between the eigenstate-to-eigenstate fluctuations and the long-time temporal fluctuations.
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