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Subdiffusion in One-Dimensional Hamiltonian Chains with Sparse Interactions

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Abstract

We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.

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Notes

  1. The first instances of disordered harmonic chains have been introduced in [6, 7]. Because of momentum conservation, the localization length diverges in the bottom of the spectrum for these models, yielding a more complicated phenomenology. We will not deal with such cases here.

  2. More precisely, it rests on the sparsity of loops and the presence of bottlenecks, see [34].

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Acknowledgements

We are most grateful to David A. Huse and J. L. Lebowitz who suggested the study of the disordered models introduced in this paper. The work of F. H. and S. O. was partially supported by the Grant ANR-15-CE40-0020-01 LSD of the French National Research Agency (ANR). F. H. acknowledges also the support of the ANR under grant ANR-14-CE25-0011 EDNHS. W. D. R. acknowledges the support of the Flemish Research Fund FWO under Grants G098919N and G076216N, and the support of KULeuven University under internal Grant C14/16/062.

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Authors and Affiliations

  1. Instituut Theoretische Fysica, KU Leuven, 3001, Leuven, Belgium

    Wojciech De Roeck

  2. Ceremade, UMR-CNRS 7534, Université Paris Dauphine, PSL Research University, 75775, Paris Cedex 16, France

    Francois Huveneers & Stefano Olla

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  1. Wojciech De Roeck
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Correspondence to Francois Huveneers.

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De Roeck, W., Huveneers, F. & Olla, S. Subdiffusion in One-Dimensional Hamiltonian Chains with Sparse Interactions. J Stat Phys 180, 678–698 (2020). https://doi.org/10.1007/s10955-020-02496-1

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