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Identifying and measuring the "localization length'' in many-body systems inthe vicinity of a many-body localization transition is difficult. FollowingHatano and Nelson, a recent work (Heuben, White, Refael, PRB 103, 064201(2021)) introduced an "imaginary vector potential'' to a disordered ring ofinteracting fermions, in order to define a many-body localization length(corresponding, in the non-interacting case, to the end-to-end Green's functionof the hermitian system). We extend these results, by connecting thislocalization length to the length scale appearing in the avalanche model ofdelocalization. We use this connection to derive the distribution of thelocalization length at the MBL transition, finding good agreement with ournumerical observations. Our results demonstrate how a localization lengthdefined as such probes the localization of the underlying ring, without theneed to explicitly construct the l-bits.

Probing localization properties of many-body Hamiltonians via an imaginary vector potential