Abstract
We implement a recursive Green's function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random on-site potential. We show that the typical value of the imaginary part of the local FS self-energy, , related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength . In particular, we show that a fractal dimension extracted from jumps discontinuously across the transition, from in the MBL phase to in the thermal phase. Moreover, follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a nonergodic volume in the thermal phase diverges with a Kosterlitz-Thouless–like essential singularity at the critical point and controls the continuous vanishing of as is approached. In contrast, a correlation length () extracted from exhibits a power-law divergence on approaching from the MBL phase.
2 More- Received 23 March 2022
- Revised 31 July 2022
- Accepted 1 August 2022
DOI:https://doi.org/10.1103/PhysRevB.106.054203
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