Equipartition and Entanglement - Relation between ergodicity measures

Similarly to the ergodicity hypothesis in classical chaotic systems, in the quantum setting there is asimilar concept, related to quantum thermalization and equipartition over degrees of freedom anddubbed as the eigenstate thermalization hypothesis. This concept is very useful as it provides a linkbetween classical and quantum chaos. The concept of multifractality of quantum wave-functions is away to break the above ergodicity in terms of chaotization and equipartitioning over degrees offreedom in quantum systems. On the other hand, in quantum information theory it is the entanglemententropy which represents the main measure of ergodicity and thermalization. On the third side, in theeigenstate thermalization hypothesis the fluctuations of local observables and their scaling with thesystem size play the central role. In this talk I will represent an exact relation between the above threemeasures, namely multifractal dimensions, scaling of fluctuations of local observables and the (Renyi)entanglement entropy. I will show that the fractal dimension of the non-ergodic wave function puts anupper bound on its entanglement entropy [1]. I will also provide a couple of explicit examplesdemonstrating that the entanglement entropy may reach its ergodic (Page) value when the wavefunction is still highly non-ergodic and occupies a zero fraction of the total Hilbert space. If time permitsI will briefly discuss some other possible deviations from ergodicity relevant for the chaotic many-body.