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# Quasiparticles, flat bands and the melting of hydrodynamic matter

## Abstract

The concept of quasiparticles—long-lived low-energy particle-like excitations—has become a cornerstone of condensed quantum matter, where it explains a variety of emergent many-body phenomena such as superfluidity and superconductivity. Here we use quasiparticles to explain the collective behaviour of a classical system of hydrodynamically interacting particles in two dimensions. In the disordered phase of this matter, measurements reveal a subpopulation of long-lived particle pairs. Modelling and simulation of the ordered crystalline phase identify the pairs as quasiparticles, emerging at the Dirac cones of the spectrum. The quasiparticles stimulate supersonic pairing avalanches, bringing about the melting of the crystal. In hexagonal crystals, where the intrinsic three-fold symmetry of the hydrodynamic interaction matches that of the crystal, the spectrum forms a flat band dense with ultra-slow, low-frequency phonons whose collective interactions induce a much sharper melting transition. Altogether, these findings demonstrate the usefulness of concepts from quantum matter theory in understanding many-body physics in classical dissipative settings.

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## Data availability

Data supporting the figures within this paper are deposited at https://doi.org/10.5281/zenodo.7295031.

## Code availability

The code used for the analysis of the experiment, analytic modelling and simulations in this study is available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was supported by the Institute for Basic Science (project code IBS-R020). We thank I. Michael and Y.-K. Cho for their essential help in constructing the microfluidic channels. T.T. thanks S.A. Safran for crucial comments on quasiparticle spectra.

## Author information

Authors

### Contributions

I.S. performed the experiment, analysed the measurements and ran simulations. H.K.P. and T.T. designed and supervised the research. T.T. conceived the study of quasiparticles and flat bands in hydrodynamic systems and developed the physical theory. H.K.P., I.S. and T.T. conducted the research and wrote and revised the manuscript.

### Corresponding authors

Correspondence to Imran Saeed, Hyuk Kyu Pak or Tsvi Tlusty.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Nature Physics thanks Philip Pincus and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Comparison of experiment and simulations.

A. The pair velocity up in simulations, showing the direction $${\hat{{{{\bf{u}}}}}}_{{{{\rm{p}}}}} \sim (\cos 2\theta ,\sin 2\theta )$$ (Left) and the magnitude $$\left\vert {{{{\bf{u}}}}}_{{{{\rm{p}}}}}\right\vert \sim {r}^{-2}$$ (Right), with a geometric factor α = 0.34 estimated from the experiment. Solid lines are the theoretical predictions (as in the experiment, Fig. 1b). B. Distribution of velocity w.r.t. center of mass (in units of u) of all particles (gold) and in the pairs (blue) in simulations, for areal densities ρ = 1.8% (left), and 5.0% (right) (as in the experiment, Fig. 1c). C. Lifetime of pairs (in R/u units) as a function of pair size r/R for for areal densities ρ = 0.9;1.3;1.8 % in simulations (as in the experiment, Fig. 1d). Data are presented as means ± SEM, shown as error bars with whiskers, for a sample size n = 2 × 105 − 7.3 × 105 of pairs measured in each simulation (as in Fig. 1d). Solid lines are fits to a sum of two exponentials.

### Extended Data Fig. 2 The spectrum computed from simulations.

The spectrum $${\omega }_{{{{\bf{k}}}}}^{+}=-{\omega }_{{{{\bf{k}}}}}^{-}$$ computed in simulations of square (left), and hexagonal (right) lattices (Methods). Compare to theoretical spectra (compare to Figs. 1e,f and 4a). Both lattices include 51 × 51 particles.

### Extended Data Fig. 3 Pair correlation function.

The pair correlation function g(r) computed in simulations of square (left) and hexagonal (right) lattices (Methods). Times are measured in units of τ = a3/(u2). Both lattices include 51 × 51 particles.

### Extended Data Fig. 4 Scaling in the melting transition.

Progression of the mean squared deviation (MSD) in square and hexagonal crystals for a/R = 5, 6, 8, where time is normalized by (R/u)(a/R)7/2 = τ(a/R)1/2.

## Supplementary information

### Supplementary Video 1

Measurement of the disordered system. Motion of particles in the experimental system described in Fig. 1a–d.

### Supplementary Video 2

Melting of a square lattice. Progression of the configuration and the angle-averaged structure factor S(k) for the simulation described in Fig. 2.

### Supplementary Video 3

Supersonic quasiparticle avalanche. Progression of a simulation starting from an isolated pair, as described in Fig. 3.

### Supplementary Video 4

Melting of a hexagonal lattice. Progression of the configuration and the angle-averaged structure factor S(k) for the simulation described in Fig. 5.

### Supplementary Video 5

Melting by flat-band modes. Progression of a simulation starting from an isolated pair in a hexagonal lattice, as described in Fig. 6a.

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Saeed, I., Pak, H.K. & Tlusty, T. Quasiparticles, flat bands and the melting of hydrodynamic matter. Nat. Phys. (2023). https://doi.org/10.1038/s41567-022-01893-5

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• DOI: https://doi.org/10.1038/s41567-022-01893-5