Griffiths Physics & the MBL Transition

Rare regions can control the dynamics of a disordered system even when they occupy a tiny fraction of its volume. Near many-body localization, the relevant question is whether unusually insulating intervals act as bottlenecks in a thermal background, and how their statistics appear in transport. Agarwal and collaborators helped make that connection quantitative. The resulting Griffiths picture remains a useful explanation of slow regimes, but its scope must be separated from the stronger claim of a stable asymptotic phase.

Figure to read

Disorder-dependent dynamical exponents in the original XXZ-chain study. Figure 1 in the source paper. Agarwal, Kartiek; Gopalakrishnan, Sarang; Knap, Michael; Mueller, Markus; Demler, Eugene. Anomalous diffusion and Griffiths effects near the many-body localization transition, Phys. Rev. Lett. 114, 160401 (2015). Figure linked rather than reproduced; the archived manuscript does not provide an explicit open reproduction license.

Compare the transport and relaxation exponents with the spectral diagnostic. Treat the plotted boundary as the interpretation of the finite-size 2015 study, not a settled modern phase boundary.

Why rare events produce common power laws

Consider a one-dimensional chain containing an insulating interval of length $\ell$. A simple rare-region model assigns it probability proportional to $e^{-a\ell}$ and a crossing time proportional to $e^{b\ell}$, with positive model-dependent constants. Eliminating length gives a broad waiting-time tail. The key point is the competition between rarity and slowness: exponentially uncommon intervals can still dominate transport because they are exponentially hard to cross. This is a phenomenological construction, not an exact identity for every disordered Hamiltonian. 1

In one dimension, current cannot easily go around such a barrier. In higher dimensions it may bypass a compact insulating inclusion, changing the transport consequences. Local autocorrelations can nevertheless remain sensitive to rare regions. This distinction explains why the same rare-region mechanism need not generate the same conductivity law in every dimension. Gopalakrishnan, Agarwal, and collaborators develop this observable-dependent perspective, including the competition with hydrodynamic tails. 2

The scaling connection

The 2015 anomalous-diffusion paper studies disordered XXZ chains and a resistor-capacitor phenomenology. If a characteristic spreading length grows as $t^\beta$, ordinary diffusion corresponds to $\beta=1/2$, while subdiffusion has smaller positive $\beta$. Under the paper's scaling assumptions, the low-frequency conductivity exponent obeys $\alpha+2\beta=1$ when $\sigma(\omega)\sim\omega^\alpha$. Thus two experimentally different quantities test one proposed dynamical description. 1

That relation is useful precisely because it can fail. A finite frequency window, mixed mechanisms, an evolving susceptibility, or multiple relevant scales can spoil a simple collapse. The practical question is whether the independently extracted exponents agree over a defensible range, not whether a fitted curve looks straight on logarithmic axes. The landmark-paper page explains how to read the original evidence without converting finite-size estimates into universal constants.

The reverse rare region: a thermal seed

The avalanche perspective asks what happens to an unusually thermal inclusion inside a localized background. If coupling to adjacent degrees of freedom allows the thermal region to grow, the inclusion can become an expanding internal bath. Thiery and collaborators formulate this mechanism as a route to delocalization. It addresses a different spatial arrangement from an insulating bottleneck in a thermal conductor, although both theories depend on rare regions. 3

Ha, Morningstar, and Huse further connect avalanche propagation to strong many-body resonances, using a bath-coupled finite system to expose the relevant structures. Such studies clarify a mechanism, but interpretation still depends on how the diagnostic relates to the fully isolated thermodynamic limit. Their value is to sharpen the microscopic question: which resonances actually connect the putatively localized degrees of freedom? 4

What remains open

The modern numerical literature emphasizes substantial finite-size drift and the difficulty of separating extremely slow relaxation from absent relaxation. A convincing assessment combines dynamics, spectra, distributions, and scaling rather than assigning an MBL transition from one observable. The current review by Sierant and collaborators provides a guide to those unresolved issues. 5

For this map, the defensible conclusion is that Agarwal's work supplies a mechanism and quantitative diagnostics for anomalously slow disordered dynamics. Whether a particular measured regime persists indefinitely requires additional evidence. Connections to spin noise and magnetometry are especially productive: they ask how the statistics of bottlenecks survive the spatial and frequency filtering imposed by an actual measurement. A next-generation test would compare several such filters on the same system and determine whether one rare-region model explains them consistently.

References

1. Agarwal, Kartiek; Gopalakrishnan, Sarang; Knap, Michael; Mueller, Markus; Demler, Eugene. Anomalous diffusion and Griffiths effects near the many-body localization transition. Phys. Rev. Lett. 114, 160401 (2015). Open manuscript.
2. Gopalakrishnan, Sarang; Agarwal, Kartiek; Demler, Eugene; Huse, David A.; Knap, Michael. Griffiths effects and slow dynamics in nearly many-body localized systems. Phys. Rev. B 93, 134206 (2016). Open manuscript.
3. Thiery, Thimothée; Huveneers, François; Müller, Markus; De Roeck, Wojciech. Many-body delocalization as a quantum avalanche. Phys. Rev. Lett. 121, 140601 (2018). Open manuscript.
4. Ha, Hyunsoo; Morningstar, Alan; Huse, David A.. Many-body resonances in the avalanche instability of many-body localization. Phys. Rev. Lett. 130, 250405 (2023). Open manuscript.
5. Sierant, Piotr; Lewenstein, Maciej; Scardicchio, Antonello; Vidmar, Lev; Zakrzewski, Jakub. Many-body localization in the age of classical computing
   *. Reports on Progress in Physics 88, 026502 (2025). Open manuscript.

Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.