Two-Dimensional Crystals far from Equilibrium

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Two-Dimensional Crystals far from Equilibrium

Leonardo Galliano, Michael E. Cates, and Ludovic Berthier

Phys. Rev. Lett. 131, 047101 – Published 25 July 2023

See Research News: Two-Dimensional Crystal Found in a Nonequilibrium System

Abstract

When driven by nonequilibrium fluctuations, particle systems may display phase transitions and physical behavior with no equilibrium counterpart. We study a two-dimensional particle model initially proposed to describe driven non-Brownian suspensions undergoing nonequilibrium absorbing phase transitions. We show that when the transition occurs at large density, the dynamics produces long-range crystalline order. In the ordered phase, long-range translational order is observed because equipartition of energy is lacking, phonons are suppressed, and density fluctuations are hyperuniform. Our study offers an explicit microscopic model where nonequilibrium violations of the Mermin-Wagner theorem stabilize crystalline order in two dimensions.

Received 6 March 2023
Accepted 15 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.047101

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Phase transitions

Nonequilibrium systems

Statistical Physics & Thermodynamics

Research News

Two-Dimensional Crystal Found in a Nonequilibrium System

Published 25 July 2023

Crystals cannot form in two-dimensional particle systems at equilibrium. A new study has found a regime where a crystal can form if the system is driven out of equilibrium.

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Authors & Affiliations

Leonardo Galliano ¹, Michael E. Cates², and Ludovic Berthier ^1,3

¹Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, 34095 Montpellier, France
²Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
³Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Article Text

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Issue

Vol. 131, Iss. 4 — 28 July 2023

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Images

Figure 1
Model and absorbing phase transition. (a) Dynamic rules: at each time step $t$ , pairs of overlapping particles (red) are displaced in opposite directions by the same random amount. (b) Time dependence of the average fraction of active particles starting from random initial conditions for $N = 4 \times 10^{4}$ . The activity vanishes for low $ϕ$ , whereas a dynamic steady state with a finite activity is reached at large $ϕ$ . (c)–(d) Absorbing configurations resemble polycrystals (the color codes for orientational order), while (e) active configurations are fully ordered [linear size in (c)–(e) is $L = 50$ ].
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Figure 2
Conserved directed percolation universality class. (a) The time dependence of the activity starting from a fully ordered structure reveals a critical packing fraction $ϕ_{c}$ where $⟨ f (t) ⟩ \sim t^{- α}$ shown with dashed line for $α = 0.42$ , $N = 10^{5}$ . (b) Critical scaling of the average activity for three system sizes. The dashed line is Eq. (1) with $β = 0.64$ , shown in log scale in the inset. (c) Diverging fluctuations of the activity at steady state. The dashed line is Eq. (2) with $γ = 0.49$ . (d) Critical scaling of the relaxation time starting from an ordered configuration in the absorbing (blue) and active (red) phases. The dashed line is Eq. (3) with $ν_{∥} = 1.3$ shown in log scale in the inset. In all panels, $ϕ_{c} = 0.8226$ .
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Figure 3
Transition to long-range order. (a) Discontinuous jump of the global bond-orientational order parameter for different system sizes. The vertical dashed line indicates $ϕ_{c}$ . (b) Coupled decay of the pair correlation function (blue) and bond-orientational correlation function (pink, shifted vertically for clarity) for $ϕ = 0.805$ and $L = 200$ . Dashed line is a fit to exponential decay, $\sim e^{- r / ξ}$ . (c) Diverging correlation length $ξ$ quantifying the typical size of crystalline domains. The blue dashed line is $ξ \sim (ϕ - ϕ_{c})^{- ν_{⊥}}$ with $ν_{⊥} = 0.8$ known from CDP.
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Figure 4
Long-range translational order and hyperuniformity. (a) Mean-squared displacement at several packing fractions for three system sizes above $ϕ_{c}$ . The plateau values do not depend on $L$ , and crystal are thus stable. The plateau values increase slightly with $ϕ$ , since multiple overlaps become more common at larger $ϕ$ . (b) Structure factor of the displacement field for several packing fractions at $L = 300$ . The red dashed line represents the thermal limit where equipartition holds with $S_{u} (q) \sim 1 / q^{2}$ . (c) Static structure factor $S (q)$ for several packing fractions at $L = 200$ . It vanishes as $S (q) \sim q^{2}$ at low $q$ (shown as dashed line) in the active phase: 2D active crystals are hyperuniform.
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