Geometric Characterization of Many-Body Localization

Many-body localization (MBL) is the breaking of ergodicity in disordered interacting systems wherein all eigenstates localize. MBL is interesting both for understanding the fundamental physics of thermalization and for potential applications as quantum memory. MBL was originally characterized by its entanglement entropy evolution and system size scaling. As a localized phase, it is natural to characterize states by their localization length. Indeed, phenomenological models of MBL propose a set of local integrals that lead to a hierarchy of length scales. To present, there have been a variety of localization lengths defined typically based in part on Fock space localization. Here we attempt to extract a real space localization length through the many-body quantum metric (MBQM), which has been defined in the development of the modern theory of insulators and shown to be related to the positional variance in single particle systems. We study the one-dimensional disorder Fermi-Hubbard model. We find that the MBQM can indeed be used to characterize the phase transition and construct a phase diagram as a function of disorder strength and interaction strength. We also find that the MBQM gives rise to a natural localization length in the thermodynamic limit.