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Multifractality meets entanglement: relation for non-ergodic extended states

In this work we establish a relation between entanglement entropy and fractal dimension $D$ of generic many-body wave functions, by generalizing the result of Don N. Page [Phys. Rev. Lett. 71, 1291] to the case of {\it sparse} random pure states (S-RPS). These S-RPS living in a Hilbert space of size $N$ are defined as normalized vectors with only $N^D$ ($0 \le D \le 1$) random non-zero elements. For $D=1$ these states used by Page represent ergodic states at infinite temperature. However, for $0<D<1$ the S-RPS are non-ergodic and fractal as they are confined in a vanishing ratio $N^D/N$ of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy ${\mathcal{S}_1}(A)$ of a subsystem $A$, with Hilbert space dimension $N_A$, scales as $\overline{\mathcal{S}_1}(A)\sim D\ln N$ for small fractal dimensions $D$, $N^D< N_A$. Remarkably, $\overline{\mathcal{S}_1}(A)$ saturates at its thermal (Page) value at infinite temperature, $\overline{\mathcal{S}_1}(A)\sim \ln N_A$ at larger $D$. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly non-ergodic. Finally, we generalize our results to Renyi entropies $\mathcal{S}_q(A)$ with $q>1$ and to genuine multifractal states and also show that their fluctuations have ergodic behavior in narrower vicinity of the ergodic state, $D=1$.