  Many body localization (MBL) represents a unique physical phenomenon,
providing a testing ground for exploring thermalization, or more precisely its
failure. Here we characterize the MBL phase geometrically by the many-body
quantum metric (MBQM), defined in the parameter space of twist boundary. We
find that MBQM scales linearly as a function of the inverse system length in
the MBL phase, and grows faster in the ergodic phase. We validate our theory
using the disordered hardcore Bose-Hubbard model, and characterize the ergodic
to MBL phase transition via the localization length scale defined from the
MBQM. MBQM provides an intuitive and experimentally accessible method to
characterize MBL phases.
