Self-Propelled Particles with Velocity Reversals and Ferromagnetic Alignment: Active Matter Class with Second-Order Transition to Quasi-Long-Range Polar Order

We introduce and study in two dimensions a new class of dry, aligning active matter that exhibits a direct transition to orientational order, without the phase-separation phenomenology usually observed in this context. Characterized by self-propelled particles with velocity reversals and a ferromagnetic alignment of polarities, systems in this class display quasi-long-range polar order with continuously varying scaling exponents, yet a numerical study of the transition leads to conclude that it does not belong to the Berezinskii-Kosterlitz-Thouless universality class but is best described as a standard critical point with an algebraic divergence of correlations. We rationalize these findings by showing that the interplay between order and density changes the role of defects.

Figure 1

(a) Phase diagram in the (${\rho }_0$, $\eta$) plane. The asymptotic order-disorder transition line is shown in black. The red curve reports the location of the susceptibility peak ${\eta }_{\chi }$ measured for $L=256$ (the error bars are smaller than the symbols). (Inset) The same data in logarithmic scales, with the dashed line marking slope 0.66. (b) Variance $⟨\mathrm{\Delta }{N}^2⟩$ over mean $⟨N⟩$ of the number of particles present in subsystems in the quasiordered (the circles, $\eta =0.2$) and disordered (the squares, $\eta =0.5$) phases for various system sizes $L$ (${\rho }_0=2$). The dashed line corresponds to $\zeta =1.73$. (c),(d) Averaged magnetization as a function of system size for several noise values in the quasiordered and disordered phases (${\rho }_0=1$). In the quasiordered state, $⟨M⟩$ decays algebraically, with an $\eta$-dependent exponent $\kappa (\eta )$. The curves in (c) correspond to, from top to bottom, $\eta =0.05$, 0.1, 0.15, 0.2, 0.24, 0.25, 0.255, and 0.26. In (c) [(d)] the dashed black line marks the slope $-\frac{1}{8}$ ($-1$).

Figure 2

(a) Susceptibility peak maximum ${\chi }_{\max }$ vs system size $L$. The dashed line has the slope 1.75. (Inset) The same data scaled with $L^{-\tilde{\gamma }}$, with $\tilde{\gamma }=1.75$. Here and in (b) and (c), the vertical dashed line delimits system sizes below which the scaling regime is not reached [these points are not used for the fits in (b), (c), and (d)]. (b) Position of ${\chi }_{\max }$ vs $L^{-1}$. The different lines are fits using Eq. (15) (BKT-like scaling) for several values of $\nu$ and Eq. (16) (algebraic scaling). (c) The same as (b) but for divergence of the correlation length with noise. (d) Asymptotic threshold ${\eta }_c$ obtained from fits of $\xi$ and ${\eta }_{\chi }$ with BKT-like scaling (14) and (15) varying the exponent $\nu$. The dashed lines represent the confidence intervals on ${\eta }_c$ given by the fits.

Figure 3

(a),(b) Annihilation of an initially prepared pair of defects deep in the ordered phase. (a) Snapshots of density and polarity orientation fields from our microscopic model (${\rho }_0=4$, $\eta =0.2$, $L^2=1024^2$, and $t=40000$). (b) The same as (a) but from a simulation of hydrodynamic equations (10)–(13) (${\rho }_0=3$, $\eta =0.4$, $a=1$, $L^2=256^2$, and $t=2500$). (c) Time-rescaled radial density profiles around the $+1$ defect in microscopic simulations [the same parameters as in (a)]. The dashed line indicates the transitional density ${\rho }_0^c(\eta )$ corresponding to $\eta =0.2$.