Figure 1

Schematics of domain patterns in a rectangular sample with aspect ratio $\frac{L_y}{L_x}=\kappa$ for the case of uniform Hall conductivity. Thick lines are sample boundaries, with solid ones corresponding to boundaries where the electrochemical potential is fixed, and the dashed ones corresponding to boundaries where periodic boundary conditions are employed. Thin lines are domain walls, and the arrows indicate the direction of the electric field within the domains. The magnitude of the field is constant away from the domain walls. In panel (a), the periodic boundary conditions along both directions are assumed and the shown state of the lowest Lyapunov energy has two horizontal and two vertical domain walls. Due to the periodic boundary conditions, this four-wall state is equal in energy to any state obtained by translating the original one along either $x$ or $y$ direction (or both). The rest of the panels correspond to the case of periodic boundary conditions along $x$, with both the top and the bottom boundaries being grounded. Panel (b) corresponds to the case of $\kappa >0.5$, panel (c) shows one of the degenerate states for $\kappa =0.5$, and panel (d) represents one of a host of different possibilities arising in the case of $\kappa <0.5$.Reuse & Permissions

Figure 2

(Color) Numerically obtained electrochemical potential $V\left(\mathbf{r}\right)$, plotted as a function of $x$ and $y$ on a torus of size $1\times 1$, for four values of the nonuniformity parameter $\alpha$. Other parameters are $n_H=1$ and domain-wall thickness $l_{\text{dw}}=0.11$. The potential is time independent for all cases shown. The transition between four-wall solution at $\alpha =0$ and the one with two walls for $\alpha =0.3$ is continuous; that is, it is not interrupted by the appearance of nonstationary solutions at the intermediate values of $\alpha$.Reuse & Permissions

Figure 3

(Color) Plots of the electrochemical potential $V\left(\mathbf{r},\mathbf{t}\right)$, as a function of $x$ and $y$, at different times $t$ for a case where the potential evolves periodically in time. Times are chosen at eight equally spaced points in the cycle, ordered from left to right in the upper row, followed by left to right in the second row. The system is a $1\times 1$ torus, with $l_{\text{dw}}=0.028$, $n_H=1$, and $\alpha =0.1$. The solution here “tunnels” from one $\alpha =0$ solution to the one related to it by translation by $\widehat{x}/2$.Reuse & Permissions

Figure 4

(Color online) Time trace of $A\equiv \sqrt{{\sum }_i{\left|\frac{\partial {\rho }_i}{\partial t}\right|}^2}$, with the sum being taken over grid points, can be used to extract both the real and imaginary parts of frequencies. The transition from linear regime (where the real part of the frequency ${\omega }_l$ is nonzero) to the saturated one (where the real part of ${\omega }_{\text{sat}}$ vanishes) is apparent.Reuse & Permissions

Figure 5

Plots of absolute value of eigenmodes of the linearized continuity equation for all-periodic boundary conditions, $l_{\text{dw}}=0.028$, and $n_H=1$. The eigenmodes are deviations from the solution with two horizontal walls; thin dashed lines indicate the location of these domain walls. Plus and minus signs indicate whether the deviation is positive or negative, while white color corresponds to zero deviation. For sufficiently big values of the nonuniformity $\alpha$, all of the eigenmode's support is within one domain. On the other hand, for $\alpha =0$, the (unstable) mode is symmetric between two domains.Reuse & Permissions

Figure 6

(Color online) Real and imaginary parts of eigenvalue $a$. The left panel shows the real part of $a$ extracted from linear regime simulations on rougher mesh (filled circles), finer mesh (crosses), as well as the first term in perturbative expansion for the case of $l_{\text{dw}}=0$ (solid line). The dashed lines connecting crosses and filled circles are merely a guide for the eyes. Right panel shows the imaginary part extracted from simulations in the linear regime (with filled circles corresponding to rougher mesh and crosses to finer mesh), the nonlinear regime (open circles), and the first term in the perturbative expansion for the case of $l_{\text{dw}}=0$ (solid line).Reuse & Permissions

Figure 7

Normalized Lyapunov cost of domain wall per unit length, $\delta G\left(\gamma \right)/\delta G\left(\pi /2\right)$, as a function of the angle $\gamma$ between the field in the bulk of the domain and the domain wall. Solid line is the exact result for quartic Lyapunov density. Triangles correspond to the Lyapunov density $g_{\beta }$ of Eq. (A1) with the value $\beta =0.1$, used in our previous work (Refs. 19, 20), while crosses show results for $\beta =1.$Reuse & Permissions