Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1511.06389 · Physical review. B./Physical review. B

Griffiths effects and slow dynamics in nearly many-body localized systems

Sarang Gopalakrishnan, Kartiek Agarwal, Eugene Demler, David A. Huse, Michael Knap

Published Apr 11, 2016• 150 citations
arXivHTMLPDF

Abstract

The low-frequency response of systems near a many-body localization transition can be dominated by rare regions that are locally critical or ``in the other phase.'' It is known that in one dimension, these rare regions can cause the dc conductivity and diffusion constant to vanish even inside the delocalized thermal phase. Here, we present a general analysis of such Griffiths effects in the thermal phase near the many-body localization transition: we consider both one-dimensional and higher-dimensional systems, subject to quenched randomness, and discuss both linear response (including the frequency- and wave-vector-dependent conductivity) and more general dynamics. In all the regimes we consider, we identify observables that are dominated by rare-region effects. In some cases (one-dimensional systems and Floquet systems with no extensive conserved quantities), essentially all long-time local observables are dominated by rare-region effects; in others, generic observables are instead dominated by hydrodynamic long-time tails throughout the thermal phase, and one must look at specific probes, such as spin echo, to see Griffiths behavior.

DOI
10.1103/physrevb.93.134206

Related papers

Rare‐region effects and dynamics near the many‐body localization transition

Kartiek Agarwal, Ehud Altman, Eugene Demler +3 more

The low‐frequency response of systems near the many‐body localization phase transition, on either side of the transition, is dominated by contributions from rare regions that are locally “in the other phase”, i.e., rare localized regions in a system that is typically thermal, or rare thermal regions in a system that is typically localized. Rare localized regions affect the properties of the thermal phase, especially in one dimension, by acting as bottlenecks for transport and the growth of entanglement, whereas rare thermal regions in the localized phase act as local “baths” and dominate the low‐frequency response of the MBL phase. We review recent progress in understanding these rare‐region effects, and discuss some of the open questions associated with them: in particular, whether and in what circumstances a single rare thermal region can destabilize the many‐body localized phase. image

arXiv:1611.00770•Annalen der Physik•Jan 12, 2017•211 citations

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

Marco Schirò, Marco Tarzia

We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the 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 transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. 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 exponentials 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 MBL 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 subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.

arXiv:1909.07160•Physical review. B./Physical review. B•Jan 17, 2020•15 citations

Anomalous Diffusion and Griffiths Effects Near the Many-Body Localization Transition

Kartiek Agarwal, Sarang Gopalakrishnan, Michael Knap +2 more

We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i) local magnetization fluctuations relax subdiffusively; (ii) the ac conductivity vanishes near zero frequency as a power law; and (iii) the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.

arXiv:1408.3413•Physical Review Letters•Apr 23, 2015•411 citations

Almost conserved operators in nearly many-body localized systems

Nicola Pancotti, Michael Knap, David A. Huse +2 more

We construct almost conserved local operators, that possess a minimal commutator with the Hamiltonian of the system, near the many-body localization transition of a one-dimensional disordered spin chain. We collect statistics of these slow operators for different support sizes and disorder strengths, both using exact diagonalization and tensor networks. Our results show that the scaling of the average of the smallest commutators with the support size is sensitive to Griffiths effects in the thermal phase and the onset of many-body localization. Furthermore, we demonstrate that the probability distributions of the commutators can be analyzed using extreme value theory and that their tails reveal the difference between diffusive and subdiffusive dynamics in the thermal phase.

arXiv:1710.03242•Physical review. B./Physical review. B•Mar 23, 2018•19 citations

Dynamics of entanglement and transport in one-dimensional systems with quenched randomness

Adam Nahum, Jonathan Ruhman, David A. Huse

The thermalization of one-dimensional quantum systems may be severely slowed by weak links, e.g., by the Griffiths regions that dominate the long-time dynamics of a system close to the many-body-localization transition. Here, the authors construct universal coarse-grained descriptions for entanglement production and for the dynamics of quantum operators in systems with weak links. They reveal unexpected differences between these processes (for example, entanglement growth can be slower than linear in time even when operators spread ballistically). The universal physics is explained using simple classical effective models. The authors' analysis of the spreading of quantum information and of conserved quantities shows that thermalization in the presence of weak links has a rich structure, featuring multiple dynamic length scales that diverge with different powers of time.

arXiv:1705.10364•Physical review. B./Physical review. B•Jul 16, 2018•123 citations

Many-body delocalization from embedded thermal inclusion

J. Clayton Peacock, Dries Sels

We numerically study quantum avalanches in one-dimensional disordered spin systems by attaching two XXZ spin chains. One chain has low disorder representing a rare Griffith's region, or thermal inclusion, and the second has larger disorder, i.e., disorder larger than the observed finite-size crossover. Comparing dynamics of this system to identical systems with uniformly large disorder, we find evidence for exponentially slow thermalization (in disorder) within the many-body localized regime when the rare region is present. We observe a decay of the spin imbalance in the bulk of the large disorder region that persists to long times $(\ensuremath{\sim}{10}^{4})$ and find a universal behavior of the spectral function.

arXiv:2303.02748•Physical review. B./Physical review. B•Jul 18, 2023•20 citations
The low-frequency response of systems near a many-body localization transition can be dominated by rare regions that are locally critical or ``in the other phase.'' It is known that in one dimension, these rare regions can cause the dc conductivity and diffusion constant to vanish even inside the delocalized thermal phase. Here, we present a general analysis of such Griffiths effects in the thermal phase near the many-body localization transition: we consider both one-dimensional and higher-dimensional systems, subject to quenched randomness, and discuss both linear response (including the frequency- and wave-vector-dependent conductivity) and more general dynamics. In all the regimes we consider, we identify observables that are dominated by rare-region effects. In some cases (one-dimensional systems and Floquet systems with no extensive conserved quantities), essentially all long-time local observables are dominated by rare-region effects; in others, generic observables are instead dominated by hydrodynamic long-time tails throughout the thermal phase, and one must look at specific probes, such as spin echo, to see Griffiths behavior.
The low‐frequency response of systems near the many‐body localization phase transition, on either side of the transition, is dominated by contributions from rare regions that are locally “in the other phase”, i.e., rare localized regions in a system that is typically thermal, or rare thermal regions in a system that is typically localized. Rare localized regions affect the properties of the thermal phase, especially in one dimension, by acting as bottlenecks for transport and the growth of entanglement, whereas rare thermal regions in the localized phase act as local “baths” and dominate the low‐frequency response of the MBL phase. We review recent progress in understanding these rare‐region effects, and discuss some of the open questions associated with them: in particular, whether and in what circumstances a single rare thermal region can destabilize the many‐body localized phase. image
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the 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 transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. 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 exponentials 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 MBL 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 subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.
We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i) local magnetization fluctuations relax subdiffusively; (ii) the ac conductivity vanishes near zero frequency as a power law; and (iii) the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.
We construct almost conserved local operators, that possess a minimal commutator with the Hamiltonian of the system, near the many-body localization transition of a one-dimensional disordered spin chain. We collect statistics of these slow operators for different support sizes and disorder strengths, both using exact diagonalization and tensor networks. Our results show that the scaling of the average of the smallest commutators with the support size is sensitive to Griffiths effects in the thermal phase and the onset of many-body localization. Furthermore, we demonstrate that the probability distributions of the commutators can be analyzed using extreme value theory and that their tails reveal the difference between diffusive and subdiffusive dynamics in the thermal phase.
The thermalization of one-dimensional quantum systems may be severely slowed by weak links, e.g., by the Griffiths regions that dominate the long-time dynamics of a system close to the many-body-localization transition. Here, the authors construct universal coarse-grained descriptions for entanglement production and for the dynamics of quantum operators in systems with weak links. They reveal unexpected differences between these processes (for example, entanglement growth can be slower than linear in time even when operators spread ballistically). The universal physics is explained using simple classical effective models. The authors' analysis of the spreading of quantum information and of conserved quantities shows that thermalization in the presence of weak links has a rich structure, featuring multiple dynamic length scales that diverge with different powers of time.
We numerically study quantum avalanches in one-dimensional disordered spin systems by attaching two XXZ spin chains. One chain has low disorder representing a rare Griffith's region, or thermal inclusion, and the second has larger disorder, i.e., disorder larger than the observed finite-size crossover. Comparing dynamics of this system to identical systems with uniformly large disorder, we find evidence for exponentially slow thermalization (in disorder) within the many-body localized regime when the rare region is present. We observe a decay of the spin imbalance in the bulk of the large disorder region that persists to long times $(\ensuremath{\sim}{10}^{4})$ and find a universal behavior of the spectral function.