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Emergent order in hydrodynamic spin lattices

Abstract

Macroscale analogues1,2,3 of microscopic spin systems offer direct insights into fundamental physical principles, thereby advancing our understanding of synchronization phenomena4 and informing the design of novel classes of chiral metamaterials5,6,7. Here we introduce hydrodynamic spin lattices (HSLs) of ‘walking’ droplets as a class of active spin systems with particle–wave coupling. HSLs reveal various non-equilibrium symmetry-breaking phenomena, including transitions from antiferromagnetic to ferromagnetic order that can be controlled by varying the lattice geometry and system rotation8. Theoretical predictions based on a generalized Kuramoto model4 derived from first principles rationalize our experimental observations, establishing HSLs as a versatile platform for exploring active phase oscillator dynamics. The tunability of HSLs suggests exciting directions for future research, from active spin–wave dynamics to hydrodynamic analogue computation and droplet-based topological insulators.

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Fig. 1: Spontaneous antiferromagnetic order in a 1D HSL.
Fig. 2: Tuning collective order through the lattice parameters in experiments and theory.
Fig. 3: Inducing global polarization through applied rotation.
Fig. 4: Rotation-induced polarization transition from antiferromagnetic to ferromagnetic order in 2D square lattices.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

We gratefully acknowledge financial support from the NSF through CMMI-1727565 (J.W.M.B. and P.J.S.) and DMS-1719637 (R.R.R.), MIT Solomon Buchsbaum Research Fund (J.D.), and CNRS Momentum programme (G.P.).

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Contributions

P.J.S. conceived the study, led the experimental developments and the writing of the paper, and contributed to the theoretical modelling. S.E.T. and R.R.R. contributed to the theoretical modelling. G.P. contributed to the conception and execution of the preliminary experiments. A.G. contributed to the preliminary experiments. J.D. contributed to the theoretical modelling and the writing of the paper. J.W.M.B. contributed to the conception of the experiments and theory, and to writing the paper.

Corresponding authors

Correspondence to Pedro J. Sáenz or John W. M. Bush.

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Peer review information Nature thanks Michael Shats and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Schematic of the experimental set-up.

The test section was mounted on an optical table and vibrated vertically by an electromagnetic shaker. The shaker was connected to the bath by a thin steel rod coupled with a linear air bearing. The forcing acceleration was monitored by two piezoelectric accelerometers. The bath was enclosed within a transparent acrylic chamber to ensure that ambient air currents did not affect the experiments. A d.c. motor housed inside the hollow air bearing enabled system rotation.

Extended Data Fig. 2 Experimental spin flips.

ae, Snapshots at different times illustrate a typical spin-flip event arising in a 1D spin lattice (Supplementary Video 6). At each time, the panels at left are colour-coded according to the instantaneous spin Si (same colour map as in Fig. 1d), while those on the right depict the recent walker trajectories, colour-coded according to the local speed. Perturbed by the wave fields emitted by its neighbours, the middle walker initially follows an elliptical path. The length of the minor axis decreases until the walker trajectory essentially becomes a straight line across the well centre. Subsequently, the process is reversed, resulting in the walker rotating in the opposite sense. The three walkers shown are part of the 1D antiferromagnetic lattice described in Fig. 1h with γ/γF = 86.0%.

Extended Data Fig. 3 Wave coupling.

a, Experimental visualization of the wave field generated by a single walker in a 1D lattice with the same geometry as in Fig. 1h and γ/γF = 92.0%. The submerged wells can be identified as the regions with a different shade of grey. b, Superposition of the wave field shown in a and the zeros of the drop-centred Bessel function \({J}_{0}({k}_{\text{F}}|{\bf{x}} \mbox{-} {{\bf{x}}}_{{\rm{i}}}|)\). c, Wave field of a bouncer computed with the theoretical model developed previously68 for walkers over variable topography. The bouncer is located at (xy) = (3D/8, 0) in a 2D square lattice with the same well diameter D and centre-to-centre separation L as in a and γ/γF = 88.0%. Solid blue lines denote the submerged wells and dashed lines the zeros of a Bessel function J0 centred at the drop position.

Extended Data Fig. 4 Emergent order for varying lattice spacing.

a, b, The dependence of the average spin correlation, χ (a) and the mean phase difference, Δϕ (b) on the lattice spacing, L, as predicted by the Bessel model (equation (16)) with τ = 0.4 s, ω0 = 3.3 s−1, and \( {\mathcal F} \) = 70 s−2. The average droplet-induced wave field, which is approximated by a Bessel function  J0(kFL), centred in the neighbouring well at L = 0, is provided for comparison in a. c, d, Dominant (c) and subdominant (d) synchronization modes for the corresponding sections (A–D) in ab.

Extended Data Fig. 5 Emergent order for varying bath acceleration.

Comparison of the experimentally observed average spin correlation, χ, with the predictions of the Bessel model (equation (16)) and the generalized Kuramoto model (equation (23)). Solid lines are fits resulting from smoothing the piecewise linear plot given by connecting the points. Bessel model parameters: L = 17.7 mm, r = 1.8 mm, λF = 2π/kF = 5.1 mm. The interaction parameter \( {\mathcal F} \) is varied across the range 70 < \( {\mathcal F} \) < 130 s−2 and transformed back to γ/γF using the relation from Extended Data Table 1. To simplify the simulations, we fix the relaxation time to be τ = 0.1 s and the natural angular frequency to be ω0 = 3.3 s−1, values consistent with experimental observation (Fig. 1c). The effective walker mass is set to mw = 1.65m, in line with prior work61. GK model parameters: α is varied over the range 8.5 < α < 15 s−2, while maintaining β < 0 and a constant ratio |β/α| = 0.3. By dividing the expressions for α and β in equations (25), (26) by a factor of two, in accordance with the mismatch between the GK and Bessel models discussed in Supplementary Fig. 3, the minimum χ predicted by the GK model emerged in the vicinity of γc.

Extended Data Fig. 6 Emergent order in large 2D square lattices.

Simulations of the Bessel model (equation (16)) and the generalized Kuramoto model (equation (23)) for a 50 × 50 square lattice demonstrate the emergence of antiferromagnetic and ferromagnetic order in 2D for various lattice spacings and bath accelerations. a, The lattice spacing determines the emergent antiferromagnetic (ADM+) or ferromagnetic (FM+) order in a manner predicted by our reduced theory (equation (24)). b, c, Specifically, preferred in-phase rotation between neighbouring pairs can be clearly observed in the antiferromagnetic AFM+ (b), and ferromagnetic FM+ (c) regimes. We note that b and c correspond to simulations of the Bessel model with the spacings indicated on a. d, The emergent in-phase antiferromagnetic order (AFM+) as a function of bath acceleration. Bessel model parameters in a: τ = 0.1 s, = 72 s−2, 16.8 ≤ L ≤ 19 mm, ω0 = 3.3 s−1, and λF = 4.95 mm. Bessel model parameters in d are the same as in a, but with L = 17.1 mm and the interaction parameter varies across the range 65 ≤ ≤ 85 s−2, which is transformed back to γ/γF using the relation from Extended Data Table 1. GK model parameters in d: α is varied over the range 9.5 < α < 13 s−2 while maintaining β < 0 and a constant ratio |β/α| = 0.07, τ = 0.2 s and ω0 = 3.3 s−1. In all cases, each data point results from averaging 50 simulations of 600 s each, to ensure statistical significance.

Extended Data Fig. 7 Emergent order for different vertical bouncing synchronizations and lattice geometries.

a, Oblique view of a 2D spin lattice where the walker in the centre is bouncing vertically in-phase and out-of-phase with its left and right neighbours, respectively. b, Average spin correlation for lattices with the same geometry as those described in Fig. 1 when the walkers all bounce in phase (blue, result presented in text), out of phase (green), or have randomly distributed bouncing phases (red). Vertically in-phase pairs promote in-phase orbital antiferromagnetic order (AFM+), whereas vertically out-of-phase pairs promote out-of-phase orbital antiferromagnetic order (AFM). A random distribution of vertical phases thus leads to competing orbital synchronization modes, which has an effect on the emergent spin correlation. Solid lines are fits resulting from smoothing the piecewise linear plot given by connecting the points. c, Triangular HSLs with a lattice spacing tuned to promote antiferromagnetic order can be used to investigate frustration effects.

Extended Data Fig. 8 Emergent collective order in simulations.

Simulations of square HSLs with the theoretical model developed previously68 is used to explore collective order in 2D. a, b, For appropriate lattice spacings, in-phase ferromagnetic order FM+ (a) and in-phase antiferromagnetic order AFM+ (b) are simulated. Left, schematic; middle, wave field and walker’s trajectories; right, time evolution of the orbital phases.

Extended Data Fig. 9 Model tunability.

Proof-of-concept simulations performed with the model developed previously68 illustrate the tunability and potential for future research of HSLs. Left, schematic; middle, wave field and drop trajectories; and right, orbital phase evolution. a, An HSL tuned to promote FM+ along the horizontal direction, but AFM+ across vertical pairs. b, FM+ lattice geometry with a random vertical and horizontal shift ±ε in the position of each well. c, FM+ lattice geometry with two drop sizes (and so, two walker speeds). d, FM+ lattice geometry with coupling strength controlled locally through the thickness of the liquid layer.

Extended Data Table 1 Model parameters and physical variables for HSLs

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Sáenz, P.J., Pucci, G., Turton, S.E. et al. Emergent order in hydrodynamic spin lattices. Nature 596, 58–62 (2021). https://doi.org/10.1038/s41586-021-03682-1

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