Analytically Solvable Renormalization Group for the Many-Body Localization Transition

Abstract

We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the manybody localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior. arXiv:1807.04285v2 [cond-mat.dis-nn]

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

“…We consider a coarse-grained description of disordered Hamiltonian or Floquet spin chains similar to the simplified RGs analyzed in [1,2], where locally thermalizing and MBL segments of the chain are assumed to be sharply distinct from one another and are characterized by just one and two, respectively, coarse-grained properties. Such segments are referred to as T and I blocks, respectively.…”

“…This "toy" RG has a critical fixed point that does obey oneparameter scaling, but it is physically incorrect in having a spurious symmetry between the MBL and thermal phases. A modification of this toy RG that does not have this incorrect symmetry, but is still somewhat analytically tractable was developed and investigated in [2], and a two-parameter KT-like RG flow was found. Subsequent work [39] argued that a KT-like RG flow follows generally from considering an MBL transition driven by avalanches.…”

“…Two of these recent papers have also suggested that there may be an intermediate "critical" phase that is MBL, where the lengths of locally thermal inclusions within this intermediate MBL phase are distributed according to a power law, and the exponent of this power law varies along the KT-like RG fixed line [2,39]. As we report below, we instead find that the distribution of the lengths of locally thermal regions is a power law in the limit of large lengths only at the critical point, so there is no intermediate phase.…”

“…In this paper we build on the work of [1] and [2] to define a RG description of the MBL transition and the MBL phase which is more detailed, and that captures more of the correct physics. Due to extra features in our RG, it is not fully analytically solvable, however it is somewhat tractable with a combination of analytic and numerical methods.…”

“…

Using a new approximate strong-randomness renormalization group (RG), we study the manybody localized (MBL) phase and phase transition in one-dimensional quantum systems with shortrange interactions and quenched disorder. Our RG is built on those of Zhang et al[1] and Goremykina et al [2], 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 ζ for its effective interactions.

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