Ising-like Critical Behavior of Vortex Lattices in an Active Fluid

Turbulent vortex structures emerging in bacterial active fluids can be organized into regular vortex lattices by weak geometrical constraints such as obstacles. Here we show, using a continuum-theoretical approach, that the formation and destruction of these patterns exhibit features of a continuous second-order equilibrium phase transition, including long-range correlations, divergent susceptibility, and critical slowing down. The emerging vorticity field can be mapped onto a two-dimensional (2D) Ising model with antiferromagnetic nearest-neighbor interactions by coarse graining. The resulting effective temperature is found to be proportional to the strength of the nonlinear advection in the continuum model.

Figure 1

Snapshots of the vorticity field $\omega$ in a system of size $N=16\times 16$ obstacles at (a) $\lambda =9.4$ (near the critical point), (b) $\lambda =6$, (c) $\lambda =8$, and at (d) $\lambda =10$. The arrows in (a) denote the velocity field $\mathbf{v}$ and the black circles denote the locations of the obstacles. Gray lines indicate the grid used to calculate spin values. The instantaneous spin lattices shown in (e)–(g) (red for positive, blue for negative spins) are obtained from the vorticity fields in (b)–(d).

Figure 2

(a) Antiferromagnetic order parameter $\mathrm{\Phi }$ as a function of $\lambda$ for different $N$. Solid black line: Onsager’s analytical solution as a function of effective temperature $T_{\mathrm{eff}}(\lambda )$ (shown as the second $x$ axis). Inset: $T_{\mathrm{eff}}(\lambda )$. (b) Binder cumulant $U$ as a function of $\lambda$. The intersection point of the curves for different $N$ marks the critical point ${\lambda }_c\approx 9.4$, here denoted by the dotted line. (c) $\mathrm{\Phi }$ as a function of $|\lambda -{\lambda }_c|$. Power-law behavior with exponent $\beta =1/8$ is shown as a dashed line. The error bars represent the standard error.

Figure 3

Diagonal correlation function $C_{\mathrm{diag}}(r)$ for different $\lambda$. Below and above ${\lambda }_c\approx 9.4$, $C_{\mathrm{diag}}(r)$ decays exponentially to zero or $C_0=⟨m_{\pm }{⟩}^2$, respectively. At ${\lambda }_c$, we observe power-law behavior with an exponent $\eta \approx 1/4$ (slope shown as solid black line). Inset: correlation length $\xi$ as a function of $\lambda$.