Toy model for anomalous transport and Griffiths effects near the many-body localization transition

Abstract

We introduce and study a toy model for anomalous transport and Griffiths effects in one dimensional quantum disordered isolated systems near the Many-Body Localization (MBL) transitions. The model is constituted by a collection of 1d tight-binding chains with on-site random energies, locally coupled to a weak GOE-like perturbation, which mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in absence of such a coupling the model is localized as expected for the one dimensional Anderson model, increasing the coupling with the GOE perturbation we find a delocalization transition to a conducting one driven by the proliferation of quantum avalanches which does not fit the standard paradigm of Anderson localization. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few, rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponential and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find sub-diffusive transport and power-laws decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region.Concomitantly, the a.c. conductivity vanishes near zero frequency with an anomalous power-law.

Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random field Heisenberg spin chain

“…In such a puzzling context, several progresses have been made to build an analytical theory, able to describe the ergodic-MBL transition [54][55][56][57][58][59][60][61][62]. The most successful description, based on the so-called "avalanche" scenario [63,64], proposes a phenomenological renormalization group (RG) treatment, working in the MBL regime where large insulating blocks compete with small ergodic inclusions.…”

Dirty bosons on the Cayley tree: Bose-Einstein condensation versus ergodicity breaking

“…8, again for weak (µ = 2) and strong (µ = 8) disorder. This quantity is indeed central in studies of non-ergodicity on this type of graphs [21,34,110,152,154]. We have considered the distribution of the correlator C 0i over all sites i for different system sizes [panels (a) and (b) of Fig.…”

Localisation and Sub-Diffusive Transport in Quantum Spin Chains With Dilute Disorder

“…In the following I shall refer to this regime as 'many-body localised', and to the onset of these slow dynamics as the 'manybody localisation transition'. In particular, there has been a great deal of work studying the transport properties close to the many-body localisation transition, with a large body of evidence pointing towards the presence of a Griffths-like subdiffusive regime and other anomalous transport properties [38][39][40][41][42][43][44][45][46][47], and gaining further insights this region could lead to an improved understanding of the many-body localisation transition itself.…”

“…when the density of thermal regions is high), supporting recent arguments that numerical studies of many-body localisation may be vulnerable to significant finite-size and finite-time effects [49] that could give the appearance of a well-defined localised phase which may not be stable in the thermodynamic limit. As well as providing a window onto the breakdown of many-body localisation, I expect this form of disorder to be a useful toy model for Griffiths effects [50] and rare region physics that are believed to play an important role in the many-body localisation transition in systems with random disorder [47,51], as well as in other paradigmatic examples of disordered phases of matter such as the Bose glass [52].…”