Solvable model of driven matter with pinning

We present a simple model of driven matter in a 1D medium with pinning impurities, applicable to magnetic domains walls, confined colloids, and other systems. We find rich dynamics, including hysteresis, reentrance, quasiperiodicity, and two distinct routes to chaos. In contrast to other minimal models of driven matter, the model is solvable: we derive the full phase diagram for small $N$, and for large $N$, we derive expressions for order parameters and several bifurcation curves. The model is also realistic. Its collective states match those seen in the experiments of magnetic domain walls.

Figure 1

States and bifurcations for $J=-K$. (a) Pinned ($E=0.4,K=0.5$), (b) half-pinned ($E=0.4,K=1.1$), (c) sync ($E=0.4,K=2.1$), (d) periodic in $x$, fixed in $\theta$ ($E=0.7,K=1.0$), (e) periodic in both $x$ and $\theta$ ($E=0.7,K=1.8$), (f) chaotic ($E=1.1,K=1.0$). Lyapunov exponents highlight the bifurcations for two different values of $E$. (g) $E=0.7$, (h) $E=1.1$. The system is integrated with $(dt,T)=(0.01,100)$ using an RK4 method.

Figure 2

Bifurcation diagram for $N=2$ particles for $J=-K$. Black lines denote theoretical predictions, and colors signify the largest Lyapunov exponent. In the periodic and chaotic parameter regions, bistability was sometimes observed.

Figure 3

Bifurcation diagram, power spectral density and time series. $K=-J=-1.5$. Simulation is performed with $N=2$ swarmalators. For plotting the bifurcation diagram we have simulated our model for $T=2000$ time units with step-size $dt=0.01$ by RK4 method. Then last $5\%$ data were considered and the local minimum was plotted. The same numerics were used for the PSD. The time series of $x_1$ are shown for four different values of $E$ over $T=250$ time units starting from the initial time $t=0$.

Figure 4

Bifurcation diagrams in $(K,E)$ space for $J=K$. Colors indicate the largest Lyapunov exponent (${\lambda }_{\text{max}}$) of the system. Black curves denote analytically calculated stability boundaries. SN stands for saddle node bifurcation and SNIPER stands for saddle node infinite period bifurcation.

Figure 5

Model Robustness. (a), (b) Bifurcation diagrams for noneven coupling $J=-cK$ where $c=\sqrt{3},1/\sqrt{3}$ are qualitatively identical to the even coupling $J=-K$ case study presented in the text. (c) Bifurcation diagram for static states when the pinning is relaxed from being perfected symmetric $({\alpha }_1,{\alpha }_2)=(0,\pi )$ to asymmetric $({\alpha }_1,{\alpha }_2)=(0,a\pi )$ for $0\le a\le 1$. Notice the half-pinned and sync states become reentrant; for $a\approx 0.8$ and increasing $K$, the system transitions as pinned $\to$ half-pinned $\to$ sync $\to$ half-pinned $\to$ sync. We have set $E=0.1$ here.

Figure 6

Match to micromagnetic simulations of two domain walls. Top row (a)–(f), high resolution micromagnetic simulations from Ref. [31]. Middle row (g)–(l), $J=-K$ and bottom row (m)–(r), $J=K$ model Eqs. (1) and (2). Numerical parameters: $(dt,T)=(0.25,200)$. Initial $(x_i,{\theta }_i)$ were drawn uniformly at random from $[0,2\pi ]$. Multistability was sometimes observed. Reprinted Supplementary Figs. 7 and 8 with permission from Hrabec et al. [ ()]. Copyright (2023) by the American Physical Society.

Figure 7

Large $N$ limit for $J=-K$. $K-E$ space for (a) $N=3$, (b) $N=4$, and (c) $N\gg 1$. Notice the $N=3,4$ plots are much different to the $N=2$ plot shown in the main text. Instead, they resemble the $N\to \infty$ plot shown in (c) where we use $N=100$.

Figure 8

Large $N$ limit for $J=K$. (a) Pinned state (blue dots, $K=1$) and antipinned state (red stars, $K=-2$) plotted on same graph to save space for $E=0$ and $N=200$ particles. (b) Sync state for $(K,E)=(3,0)$. (c) Order parameters for $E=0$. (d) Bifurcation diagram in $(K,E)$ space. Black curves show theoretical predictions. Colors denote Lyapunov exponents, computed using $N=10$ particles, which well approximated the $N\gg 1$ limit (simulations for larger $N$ were prohibitive).

Figure 9

Emerging states of the $2\mathrm{D}$ model. We have fixed $E=0$. $b=1, J_x=J_y=K$ and ${\alpha }_{x_i}={\alpha }_{y_i}={\beta }_i=2\pi i/N$. (a) Order parameters $S_{x_{+}}$ (blue diamonds) and $S_{x_{-}}$ (red dots). (b) Order parameters $S_{y_{+}}$ (blue diamonds) and $S_{y_{-}}$ (red dots). $(dt,T,N)=(0.5,500,1000)$. Last 50% data were taken to calculate the order parameters. Initially, $x_i,y_i$, and ${\theta }_i$ are chosen randomly from $[0,2\pi ]$.

Figure 10

Comparison of $N=2,3,4,5$ for $J=K$. As $N$ increases, the shape starts to change. At $N=5$ the shape is almost identical to that of the $N=\infty$ results, plotted in Fig. 11 below for convenience.

Figure 11

Bifurcation diagram: $N\gg 1$ for $J=K$. We take $N=100$ here. We refer the reader to Ref. [34] for details.