Renormalization group crossover in the critical dynamics of field theories with mode coupling terms

Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossover between an unstable fixed point, characterized by a dynamic critical exponent $z=d/2$, and a stable fixed point with $z=2$. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent $\kappa =4/d$. The crossover is regulated by a conservation length scale ${\mathcal{R}}_0$, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond ${\mathcal{R}}_0$, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents $z=3/2$ and $z=2$ in the critical slowdown of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.

Figure 1

Renormalization group flow and crossover. Top: Flow diagram on the $(X_l,f_l)$ plane for $d=3$. When the initial friction ${\eta }_0$ is small, $X_0\sim 1$, the flow converges toward the unstable fixed point, $z=d/2$, and remains in its proximity for many iterations, before crossing over to the stable $z=2$ fixed point. Bottom: running parameters and critical exponent $z$ as a function of the iteration step along a flow line at small ${\eta }_0$. The initial values of the parameters are $f_0=2, X_0=0.9999$, and $w_0=3$.

Figure 2

Different critical regions. Different values of $k, \xi$, and ${\mathcal{R}}_0$ correspond to different critical behaviors. The dark red region corresponds to conservative critical dynamics with $z=d/2$, while the light green region corresponds to dissipative critical dynamics with $z=2$. We set $\mathrm{\Lambda }=1$ so that physical values for lengths are $k^{-1}>1, \xi >1$, and ${\mathcal{R}}_0>1$. On the critical manifold relaxation is studied in the $(k^{-1},{\mathcal{R}}_0)$ plane: The two different regimes are separated by the curve ${\mathcal{R}}_0=k^{-d/4}$. Off the critical manifold, relaxation is studied in the $(\xi ,{\mathcal{R}}_0)$ plane: The two different regimes are separated by the curve ${\mathcal{R}}_0={\xi }^{d/4}$. The black dashed line represents, respectively, ${\mathcal{R}}_0=k^{-1}$ or ${\mathcal{R}}_0=\xi$. The figure refers to the $d=3$ case.

Figure 3

Static critical behavior. (a) Average scalar polarization for temperatures $0.5\le T\le 2.5$ and for different sizes ($N=512$, 1000, 2197, 4096, 8000). An ordering transition occurs at approximately $T_c\simeq 1.5$. (b) Susceptibility as a function of temperature, same sizes as in panel (a); the maximum of each curve is located at a temperature that decreases with increasing the size of the system and approaches the critical temperature $T_c$ in the thermodynamic limit. (c) Finite-size scaling of the susceptibility. Curves at $N=2197,4096,8000$ satisfy finite-size scaling with exponents $\nu =0.707$ and $\gamma /\nu =1.973$, as predicted by the theory of the Heisenberg model [41].

Figure 4

Numerical protocol in the $(\xi ,{\mathcal{R}}_0)$ plane. Depending on the values of $\xi$ and ${\mathcal{R}}_0$, the system explores two different critical regimes (see also Fig. 2). Simulations performed at fixed $\widehat{\eta }$ and different $T$ correspond to exploring the $(\xi ,{\mathcal{R}}_0)$ plane along horizontal segments. Since the size of the system is finite ($L\le 20$), only a limited window of $\xi$ can be accessed and the length of such segments is finite ($1={\mathrm{\Lambda }}^{-1}<\xi <{\xi }_{\max }=10$). According to the RG prediction, for all values of $\widehat{\eta }$ corresponding to ${\mathcal{R}}_0>{10}^{3/4}$ the segments belong entirely to the conservative region (segment a). For larger values of $\widehat{\eta }$, such that $1<{\mathcal{R}}_0<{10}^{3/4}$, the segments cross from the conservative region to the dissipative one (segment b): In this case, there is no sufficient span in each region to extract the exponent $z$ from the $\tau$ vs $\xi$ plot. Since the minimum physical value of ${\mathcal{R}}_0$ is ${\mathrm{\Lambda }}^{-1}=1$, larger values of $\widehat{\eta }$ are all equivalent to the ${\mathcal{R}}_0=1$ case (segment c).

Figure 5

Dynamic critical exponents. Relaxation time versus correlation length in $d=3$, for $L=20, N=8000$, and $T\in [1.48:2.00]$, at various values of the friction coefficient, $\widehat{\eta }=1.0,2.0,4.0$. Each point is an average over 10 samples, apart from the lowest $T$ (largest $\xi$ and $\tau$) at $\widehat{\eta }=4$, for which we have four samples (one such sample takes 7 days to run on a i7-8700, 3.20 GHz CPU). Lines are the best fit to $z=1.5$ (low friction, $\widehat{\eta }=1.0,2.0$) and $z=2$ (large friction, $\widehat{\eta }=4.0$).

Figure 6

Dynamic scaling for correlations. Test of the dynamic scaling hypothesis on the dynamic correlation functions at $k=0$. Upper panels: spatiotemporal correlation functions at various values of the temperature for $\widehat{\eta }=1$ (a) and $\widehat{\eta }=4$ (b). Lower panels: same curves plotted as a function of $t/{\xi }^z$ with, respectively, $z=1.5$ (c) and $z=2$ (d): In both cases, the functions verify the dynamic scaling hypothesis.

Figure 7

Inertial behavior: experiments vs models. (a) Normalized dynamical correlation functions $C(k,t)/C(k,0)$ at nonzero values of $k$; in all three cases, $k$ has been chosen in such a way to have $k\xi =1$, to reproduce the scaling situation found in experiments on natural swarms [22] (Vicsek swarm $k=0.717$, natural swarm $k=0.798$, and ISM $k=0.673$). (b) The relaxation form factor $h(t/\tau )\equiv \dot{C}(t/\tau )/C(t/\tau )$ goes to 1 for overdamped exponential relaxation, while it goes to 0 for inertial relaxation [22]. The fixed-network ISM reproduces the correlation form of real swarms in a rather compelling way.