Probing localization properties of the MBL transition via an imaginary vector potential

Identifying and measuring the ``localization length'' in many-body systems in the vicinity of a many-body localization transition is difficult. Following Hatano and Nelson, a recent work (Heußen, White, Refael, Phys. Rev. B 103, 064201) introduced an ``imaginary vector potential'' to a disordered ring of interacting fermions, in order to measure a localization length corresponding to the end-to-end Green's function of the corresponding open chain. We employ this non-hermitian localization technique to extract the distribution of the localization length at the transition point. We find that the localization-length distributions at the critical points are well-described by extreme value distributions. We explain the appearance of these distributions, as well as connect our study to the localization length identified in the avalanche model of the many-body localization transition.