Probing localization properties of many-body Hamiltonians 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 paper [S. Heussen, C. D. White, and G. Refael, Phys. Rev. B 103, 064201 (2021)] introduced an "imaginary vector potential" to a disordered ring of interacting fermions, in order to define a many-body localization length (corresponding, in the noninteracting case, to the end-to-end Green's function of the Hermitian system). We extend these results, by connecting this localization length to the length scale appearing in the avalanche model of delocalization. We use this connection to derive the distribution of the localization length at the MBL transition, finding good agreement with our numerical observations. Our results demonstrate how a localization length defined as such probes the localization of the underlying ring, without the need to explicitly construct the l-bits.