Relation Between Entanglement and Multifractality

From the fundamental point of view, multifractality provides a way of ergodicity breaking in terms of chaotization and equipartitioning. However, in quantum information theory the entanglement is the main measure of ergodicity and thermalization. Thus, it is of particular interest to consider the role of entanglement and multifractality in quantum dynamics, ergodicity and thermalization and their mutual relations in many-body disordered systems in many-body localized and thermalizing parameter range.

Recently we have found an exact relation between the entanglement entropy and fractal dimensions giving the upper bound for the entanglement entropy for any eigenstate with a given fractal dimension. In addition, we provide an explicit example demonstrating that the entanglement entropy may reach its ergodic (Page) value when the wave function is still highly non-ergodic and occupies a zero fraction of the total Hilbert space.

The explanation of this phenomenon can be easily given by the structure of the reduced density matrix. Indeed, for fractal support sets ND smaller than the subsystem size there are only few non-zero elements in the density matrix dominated by the diagonal contribution. As soon as fractal dimension is larger D > 0.5 the diagonal of this density matrix is filled leading to the saturation of the entanglement entropy at the Page value.