# Many-Body Localization & Griffiths Effects
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**Anchor paper:** K. Agarwal, S. Gopalakrishnan, M. Knap, M. Müller, E. Demler, "Anomalous diffusion and Griffiths effects near the many-body localization transition," *Phys. Rev. Lett.* 114, 160401 (2015). [arXiv:1408.3413]
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## Background
Many-body localization (MBL) is a phenomenon where strong quenched disorder in an interacting quantum system prevents thermalization, producing a localized phase with extensively many local conserved quantities and only logarithmic entanglement growth. By 2014 the MBL phase itself was reasonably well characterized, but far less was understood about the ergodic (thermal) phase in the immediate vicinity of the MBL transition — specifically whether transport there remained conventionally diffusive all the way up to the transition, or whether proximity to localization already left an imprint on transport.
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## New results
Studying the disordered 1D XXZ spin chain near the MBL transition, the authors found the delocalized phase is anomalous on three fronts: local magnetization fluctuations relax subdiffusively (C_zz(t) ~ 1/t^β), the AC conductivity vanishes at zero frequency as a power law σ(ω) ~ ω^α, and the distribution of local resistivities broadens toward low frequency. They argued a single mechanism explains all three: rare, atypically disordered regions that are locally insulating (a many-body Griffiths phase) act as bottlenecks dominating transport even though the bulk is thermal. A phenomenological random resistor-capacitor network model gives an exact scaling relation between the exponents (α + 2β = 1), unifying the picture.
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![Subdiffusive transport signatures near the MBL transition](https://arxiv.org/html/1408.3413v2/allthreems.png)
*Fig. 2 — optical conductivity σ(ω), return probability C_zz(t), and resistivity-distribution width vs. frequency/time for several disorder strengths, with power-law fits extracting the exponents.*
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## Related work in this direction
- K. Agarwal, S. Gopalakrishnan, E.A. Demler, D.A. Huse, M. Knap, "Griffiths effects and slow dynamics in nearly many-body localized systems," PRB 93, 134206 (2016) — extends the rare-region analysis to higher dimensions and more general disordered models.
- K. Agarwal, E. Altman, E. Demler, S. Gopalakrishnan, D.A. Huse, M. Knap, "Rare-region effects and dynamics near the many-body localization transition," Annalen der Physik 529, 1600326 (2017) — broadens the framework to show rare regions govern slow dynamics approaching the transition from *either* side.
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**Driven/Floquet Localization & Symmetry Engineering** is nested here: it asks the natural next question — what happens to this disorder/localization physics once periodic driving is added.
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# Parents
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* Disorder & Ergodicity Breaking⏎