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Supplementary movies for “Stability of discrete-symmetry flocks: sandwich state, traveling domains and motility-induced pinning” Swarnajit Chatterjee⋆,1, 2 Mintu Karmakar⋆,3, 4, 5, 6 Matthieu Mangeat,1Heiko Rieger,1and Raja Paul3 1Center for Biophysics & Department for Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany. 2Laboratoire de Physique Théorique et Modélisation, UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise, France. 3School of Mathematical & Computational Sciences, Indian Association for the Cultivation of Science, Kolkata – 700032, India. 4Wenzhou Institute of the University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325011, China. 5School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. 6Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, E08028 Barcelona, Spain. •File movie01_APM_hydrodynamics.mp4 shows the time evolution obtained from the hydrodynamic equations of (a–c) a counter-propagating droplet (σ= 1) and (d–f) a transversely-propagating droplet (σ= 2) inserted in an ordered state (σ= 3) in a 200×200 domain, for varying βand ϵ. (a) Complete reversal of the initial initial liquid phase in a comet-shaped configuration for high temperature (β= 0.75), and (b–c) formation of the sandwich state for low temperature (β= 0.9) with a width decreasing with ϵ, for the counter-propagating droplet. (d) At low velocity (ϵ= 0.6), the transversely-propagating droplet disperses into the stable liquid phase, leaving the initial ordered state intact. (e) At intermediate velocity (ϵ= 1.5), two orthogonally moving clusters, formed by the states of the droplet and the initial liquid, never merge. (f) At larger velocity (ϵ= 2.4), the droplet causes a complete reversal of the liquid phase into the droplet state. Parameters: D= 1,ρ0= 3,rd= 10, and ρd 0= 5ρ0. Colorbar legend: magnetization of the droplet state. •File movie02_APM_droplet_rectangular.mp4 shows the time evolution of (a–c) a counter-propagating droplet and (d–f) a transversely-propagating droplet inserted in an ordered state in a 500 ×50 domain and for several values of ϵ. A counter-propagating droplet with a very small self-propulsion ϵ= 0.3is unable to reverse the initial liquid phase (a), but reverses it completely for an intermediate velocity ϵ= 0.9(b), and creates a sandwich state similar to movie01 for large self-propulsion ϵ= 2.7(c). A transversely propagating droplet at small velocity ϵ= 0.9could not reverse the initial liquid phase (d), although completely reverses it for large self-propulsion ϵ= 2.7(e) or creates a mixed phase containing both states in a transverse direction to the original liquid flow (f). Parameters: D= 1,β= 1,ρ0= 10,rd= 10, and ρd 0= 1.2ρ0. Colorbar legend: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. •File movie03_APM_droplet_counter_beta=1.mp4 shows the time evolution of a counter-propagating droplet inserted in an ordered state at low temperature (β= 1) in a 200 ×200 domain and for several values of ϵ. The droplet induces the formation of a sandwich state whose width decreases with ϵ. Parameters: D= 1,ρ0= 10, rd= 10, and ρd 0= 5ρ0. Colorbar legend: red (σ= 1): right; blue (σ= 3): left. •File movie04_APM_droplet_counter_beta=0.75.mp4 shows the time evolution of a counter-propagating droplet inserted in an ordered state at high temperature (β= 0.75) in a 200 ×200 domain and for several values of ϵ. The droplet causes the formation of a sandwich state when ϵ≤0.9and ϵ≥2.1, while the droplet completely reverses the initial liquid phase within an intermediate range of ϵ(1.2≤ϵ≤1.8). Parameters: D= 1,ρ0= 10, rd= 10, and ρd 0= 5ρ0. Colorbar legend: red (σ= 1): right; blue (σ= 3): left. •File movie05_APM_droplet_transverse.mp4 shows the time evolution of a transversely-propagating droplet inserted in an ordered state in a 200 ×200 domain and for several values of ϵ. For ϵ≤0.6, the droplet disperses into the stable liquid phase, leaving the initial ordered state intact. For 0.9≤ϵ≤1.5, two high-density clusters coexist, each maintaining persistent motion in perpendicular directions, one representing the inserted droplet and the other corresponding to the preexisting liquid phase. For ϵ≥1.8, the droplet causes a complete reversal of the liquid phase into the droplet state. Similarly to movie02, a mixed phase containing both states in a transverse direction to the original liquid flow can be created. Parameters: D= 1,β= 1,ρ0= 10,rd= 10, and ρd 0= 5ρ0. Colorbar legend: green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down.
2 •File movie06_APM_droplet_D=0.3.mp4 shows the time evolution of (a) a counter-propagating droplet and (b) a transversely-propagating droplet inserted in an ordered state in a 200 ×200 domain for D= 0.3. The droplet fully breaks the initial liquid phase, producing a dense lane of its own state. Overtime, this lane becomes unstable due to a spontaneous emergence of transversely moving cluster, forming itself a new lane, which will also break repeatedly. Parameters: β= 1,ϵ= 2.5,ρ0= 10,rd= 10, and ρd 0= 1.2ρ0. Colorbar legend: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. •File movie07_APM_droplet_D=0.2.mp4 shows the time evolution of (a,c) a counter-propagating droplet and (b,d) a transversely-propagating droplet inserted in an ordered state in a 200 ×200 domain for D= 0.2and (a–b) ϵ= 1.2; (c–d) ϵ= 2.5. The droplet forms a jammed cluster which gradually dissolves, as its constituent particles flip into the background liquid state, ultimately restoring the initial ordered phase. Parameters: β= 1, ρ0= 10,rd= 10, and ρd 0= 1.5ρ0. Colorbar legend: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. •File movie08_APM_droplet_D=0.1.mp4 shows the time evolution of (a,c) a counter-propagating droplet and (b,d) a transversely-propagating droplet inserted in an ordered state in a 200 ×200 domain for D= 0.1and (a–b) ϵ= 1.2; (c–d) ϵ= 2.5. The droplet forms a jammed cluster, in addition of spontaneous nucleation of multiple jammed domains with a typical size that decreases with ϵ. Parameters: β= 1,ρ0= 10,rd= 10, and ρd 0= 1.5ρ0. Colorbar legend: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. •File movie09_APM_spontaneous_flock.mp4 shows the fate of the ordered state in a 512×512 domain. (a) Spontaneous nucleation of polar droplets in the liquid phase, leading to a short-range ordered state. Parameters: D= 0.3,β= 1,ϵ= 2.5, and ρ0= 5. (b) Formation of the stripe state, in the liquid-gas coexistence regime, with several liquid domains moving on the gas phase due to the quenching of an ordered state below the liquid spinodal at large velocity ϵ. Parameters: D= 0.7,β= 0.6,ϵ= 2.5, and ρ0= 5. Colorbar legend on the left: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. Colorbar legend on the right: total density ρ. •File movie10_APM_spontaneous_MIP.mp4 shows the formation of the motility-induced (interface) pinning starting from an ordered state in a 256 ×256 domain and for (a) ϵ= 1.2and (b) ϵ= 2.5. Spontaneous nucleation of multiple jammed clusters with a typical size that decreases with ϵ, formed with oppositely moving particles. Parameters: D= 0.3,β= 2, and ρ0= 5. Colorbar legend on the left: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. Colorbar legend on the right: total density ρ. •File movie11_APM_droplet_MIP.mp4 shows a counter-propagating droplet pinning starting from an ordered state in a 256 ×256 domain and for (a) ϵ= 1.2and (b) ϵ= 2.5. The droplet forms a jammed cluster, in addition of spontaneous nucleation of multiple jammed domains with a typical size that decreases with ϵ, similar to movie08 but enhanced by a lower temperature. Parameters: D= 0.3,β= 2,ρ0= 10,rd= 32, and ρd 0= 5ρ0. Colorbar legend on the left: red (σ= 1): right; green (σ= 2): up; blue (σ= 3): left; black (σ= 4): down. Colorbar legend on the right: total density ρ.
