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Breakdown of hydrodynamics below four dimensions in a fracton fluid


Hydrodynamics is a universal effective theory that describes the thermalization of chaotic many-body systems, and depends only on the symmetries of the underlying theory. Although the Navier–Stokes equations can describe classical liquids and gases, quantum fluids of ultracold atoms or quark–gluon plasma, they cannot yet describe the phases of matter where particle motion is kinematically constrained. Here we present the nonlinear fluctuating hydrodynamics of models with simultaneous charge/mass, dipole/centre of mass and momentum conservation. This hydrodynamic effective theory is unstable below four spatial dimensions: dipole-conserving fluids at rest are unstable to fluctuations, which drive the system to a dynamical universality class with qualitatively distinct features from conventional fluids. In one spatial dimension, our construction is reminiscent of the well-established renormalization group flow of the stochastic Navier–Stokes equations; however, the fixed point we find possesses subdiffusive scaling rather than the superdiffusive scaling of the Kardar–Parisi–Zhang universality class. We numerically simulate many-body classical dynamics in one- and two-dimensional models with dipole and momentum conservation, and find evidence for the predicted breakdown of hydrodynamics. Our theory provides a controlled example of how kinematic constraints lead to a rich landscape of dynamical universality classes in high-dimensional models.

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Fig. 1: Momentum correlation function.
Fig. 2: Temporal dependence of 1/k* showing anomalous scaling.
Fig. 3: Magnon-like dispersion relation at the DMC fixed point.
Fig. 4: Anomalous scaling in model A in two dimensions.

Data availability

The data used to generate Figs. 1–4 and Supplementary Figs. 1 and 2 are provided in Supplementary Data 2.

Code availability

The code used to run simulations is provided in Supplementary Data 1.


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We acknowledge useful discussions with L. Delacrétaz, B. Halperin, X. Huang, K. Jensen, R. Nandkishore, L. Radzihovsky and D. T. Son. This work was supported by the Department of Energy through award DE-SC0019380 (P.G.), the Simons Foundation through award no. 620869 (P.G.), the National Science Foundation under CAREER Award DMR-2145544 (A.L.), the Gordon and Betty Moore Foundation’s EPiQS Initiative via grants GBMF4302 and GBMF8686 (J.F.R.-N.) and GBMF10279 (J.G. and A.L.), and by Research Fellowships from the Alfred P. Sloan Foundation under grant FG-2020-13615 (P.G.) and FG-2020-13795 (A.L.).

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Authors and Affiliations



P.G. and A.L. developed the theoretical framework and classical models. J.F.R.-N. and J.G. performed the numerical simulations and statistical analysis of the data, and prepared the figures. P.G. and A.L. wrote the manuscript with input from all the authors.

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Correspondence to Paolo Glorioso or Andrew Lucas.

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Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Figs. 1 and 2 and Sections 1–7.

Supplementary Data 1

MATLAB codes used to run the simulations and generate the data for models A and B.

Supplementary Data 2

Raw data used to generate Figs. 1–4 and Supplementary Figs. 1 and 2.

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Glorioso, P., Guo, J., Rodriguez-Nieva, J.F. et al. Breakdown of hydrodynamics below four dimensions in a fracton fluid. Nat. Phys. (2022).

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