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Abstract

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 Page [Phys. Rev. Lett. 71, 1291 (1993)] to the case of sparse random pure states (SRPS). These SRPS living in a Hilbert space of size N are defined as normalized vectors with only N-D (0 <= D <= 1) random nonzero elements. For D = 1, these states used by Page represent ergodic states at an infinite temperature. However, for 0 < D < 1, the SRPS are nonergodic 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 S-1(A) of a subsystem A, with Hilbert space dimension N-A, scales as (S) over bar (1)(A) similar to D in N for small fractal dimensions D, N-D < N-A. Remarkably, (S) over bar (1)(A) saturates at its thermal (Page) value at an infinite temperature, (S) over bar (1)(A) similar to in 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 nonergodic. Finally, we generalize our results to Renyi entropies S-q(A) with q > 1 and to genuine multifractal states and also show that their fluctuations have ergodic behavior in a narrower vicinity of the ergodic state D = 1.