Renormalization-group study of the many-body localization transition in one dimension

Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length zeta for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length (zeta) over tilde, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, (zeta) over tilde approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.