# 2015 · Anomalous Diffusion and Griffiths Effects Near the MBL Transition
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**Paper focus:** Agarwal, Gopalakrishnan, Knap, Müller, and Demler, *Anomalous Diffusion and Griffiths Effects Near the Many-Body Localization Transition* (2015). The paper's enduring contribution is a quantitative link between rare bottlenecks, slow density relaxation, and low-frequency conductivity. This page explains how to read its evidence and scaling argument. The broader [Griffiths page](https://knowen.org/nodes/33539) discusses the subsequent landscape, including avalanche mechanisms and the unsettled extrapolation from finite systems to asymptotic phases. [1](https://doi.org/10.1103/PhysRevLett.114.160401)
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### Figure to read
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Disorder-dependent dynamical exponents in the original XXZ-chain study. [Figure 1 in the source paper](https://arxiv.org/pdf/1408.3413#page=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). Figure linked rather than reproduced; the archived manuscript does not provide an explicit open reproduction license.
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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.
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### The model and the measured quantities
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The study considers a spin-one-half XXZ chain with random longitudinal fields. Exchange terms move spin excitations, interactions couple neighboring spins, and randomness creates unequal local environments. Total longitudinal magnetization is conserved, making spin transport a natural diagnostic. The authors use exact-diagonalization calculations in finite chains and compare their behavior with a phenomenological model built from broadly distributed resistances. This combination tests both microscopic dynamics and a proposed coarse-grained mechanism. [1](https://doi.org/10.1103/PhysRevLett.114.160401)
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Three observables provide complementary views. A local spin autocorrelation measures how long memory persists at its starting point. Optical conductivity measures response as a function of frequency. The distribution of resistivities tests whether the sample-to-sample fluctuations become unusually broad. An average transport coefficient alone would discard much of the rare-region information. That is why the distribution is central rather than an optional statistical supplement.
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### Reconstructing the scaling argument
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Suppose a density packet has a characteristic width proportional to $t^\beta$. If one scaling length controls the spreading, the return probability has a related decay, and the effective diffusivity scales as $t^{2\beta-1}$. Converting time to inverse frequency and assuming a nonsingular static susceptibility gives $\sigma(\omega)\sim\omega^{1-2\beta}$. Writing the conductivity exponent as $\alpha$ yields $\alpha+2\beta=1$. These are connected assumptions and consequences, not independent exact statements about every disordered chain. [1](https://doi.org/10.1103/PhysRevLett.114.160401)
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For ordinary diffusion, $\beta=1/2$ gives a finite low-frequency conductivity in this scaling picture. Smaller positive $\beta$ gives a conductivity that decreases with frequency and a density packet that spreads more slowly. A broad distribution of resistors supplies an intuitive explanation: increasingly long observation times encounter increasingly severe bottlenecks. The low-frequency limit therefore cannot always be represented by one typical local resistance.
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### Reading the original figures critically
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Figure 1 summarizes fitted dynamical behavior as disorder is varied and compares it with a spectral diagnostic. Figure 2 displays the frequency and time-domain measurements behind the scaling discussion. When inspecting them, identify the actual fitting windows and finite-size restrictions. A fitted exponent is meaningful evidence about those windows, but its uncertainty includes the possibility of a crossover outside them. Agreement between different observables is stronger evidence than any one straight line on logarithmic axes.
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The later Griffiths analysis by Gopalakrishnan and collaborators generalizes the role of dimension, conservation laws, and probe choice. It shows why local slow dynamics need not imply the same transport law in every setting. This places the original paper's one-dimensional scaling in a wider family of rare-region effects rather than elevating its exponent relation into a universal formula. [2](https://doi.org/10.1103/PhysRevB.93.134206)
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### What the paper does not settle
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Avalanche theories subsequently emphasize thermal inclusions that can grow into a localized background. That mechanism addresses the stability of localization from the opposite side of the rare-region problem. It does not erase the relevance of insulating bottlenecks, but changes what is required to infer an asymptotic phase boundary. [3](https://doi.org/10.1103/PhysRevLett.121.140601) The current numerical review documents finite-size drifts and the difficulty of resolving very slow thermalization. [4](https://doi.org/10.1088/1361-6633/ad9756)
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The appropriate takeaway is a testable connection between a microscopic disordered model, broad transport statistics, and linked dynamical observables. For a new experiment, reproduce that logic rather than only its phase labels: compare independent exponents, inspect disorder distributions, and vary the available time or length scale. The [spin-noise page](https://knowen.org/nodes/33540) illustrates how this reasoning extends to a measured fluctuation spectrum, where the probe's coupling introduces another layer of physical filtering.
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### References
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1. Agarwal, Kartiek; Gopalakrishnan, Sarang; Knap, Michael; Mueller, Markus; Demler, Eugene. [Anomalous diffusion and Griffiths effects near the many-body localization transition](https://doi.org/10.1103/PhysRevLett.114.160401). Phys. Rev. Lett. 114, 160401 (2015). [Open manuscript](https://arxiv.org/abs/1408.3413).
2. Gopalakrishnan, Sarang; Agarwal, Kartiek; Demler, Eugene; Huse, David A.; Knap, Michael. [Griffiths effects and slow dynamics in nearly many-body localized systems](https://doi.org/10.1103/PhysRevB.93.134206). Phys. Rev. B 93, 134206 (2016). [Open manuscript](https://arxiv.org/abs/1511.06389).
3. Thiery, Thimothée; Huveneers, François; Müller, Markus; De Roeck, Wojciech. [Many-body delocalization as a quantum avalanche](https://doi.org/10.1103/PhysRevLett.121.140601). Phys. Rev. Lett. 121, 140601 (2018). [Open manuscript](https://arxiv.org/abs/1706.09338).
4. Sierant, Piotr; Lewenstein, Maciej; Scardicchio, Antonello; Vidmar, Lev; Zakrzewski, Jakub. [Many-body localization in the age of classical computing
                    *](https://doi.org/10.1088/1361-6633/ad9756). Reports on Progress in Physics 88, 026502 (2025). [Open manuscript](https://arxiv.org/abs/2403.07111).
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*Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.*
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# Parents
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* Griffiths Physics & the MBL Transition⏎