Dark-soliton-like magnetic domain walls in a two-dimensional ferromagnetic superfluid

We report a stable magnetic domain wall in a uniform ferromagnetic spin-1 condensate, characterized by the magnetization having a dark soliton profile with nonvanishing superfluid density. We find exact stationary solutions for a particular ratio of interaction parameters with and without magnetic fields, and develop an accurate analytic solution applicable to the whole ferromagnetic phase. In the absence of magnetic fields, this domain wall relates various distinct solitary excitations in binary condensates through $\text{SO}\left(3\right)$ spin rotations, which otherwise are unconnected. Remarkably, studying the dynamics of a quasi-two-dimensional (quasi-2D) system we show that standing wave excitations of the domain wall oscillate without decay, being stable against the snake instability. The domain wall is dynamically unstable to modes that cause the magnetization to grow perpendicularly while leaving the domain wall unchanged. Real-time dynamics in the presence of white noise reveals that this “spin twist” instability does not destroy the topological structure of the magnetic domain wall.

Figure 1

(a) Schematic of a transverse magnetic domain wall (along $y$ axis) in a system of background density $n_b$. The arrows represent the transverse magnetization vector $(F_x,F_y)$, and the background color shows the superfluid density. A comparison between analytical predictions (lines) and numerical results (symbols) for the (b) density and (c) $F_x$ spin density for various values of $g_s/g_n$ and $q$. The inset shows two complete profiles of $F_x$ at the exactly solvable point with $q=0$ and $q\ne 0$, respectively. Here, ${\xi }_n=\hslash /\sqrt{M{g}_n{n}_b}$ is the density healing length.

Figure 2

One period of evolution for standing wave deformations of a MDW confined by hard-wall potentials in a square domain $x,y\in [-L,L]$ with $L=40{\xi }_n$. The equilibrium configurations are shown in the second and fourth columns. (a1)–(a5) A standing wave on an open MDW with free end-points attached on the hard-wall boundaries; the initial configuration (MDW core location) is $x=Acos(\pi y/L)$, $y\in [-L,L]$ with $L=40{\xi }_n$ and $A=2.5{\xi }_n$. (b1)–(b5), (c1)–(c5) Standing waves on a closed MDW. The initial configurations are determined by $R_0-\sqrt{x^2+y^2}+Asin[sarctan(y/x)]=0$ where $R_0=L/2$, $s=2$ (dipole mode) for (b1) and $s=3$ (triple mode) for (c1). Oscillation periods $T$ are: (a) $T\simeq 620t_0$; (b) $T\simeq 620t_0$; (c) $T\simeq 300t_0$, where $t_0=\hslash /g_n{n}_b$ and $g_s/g_n=-0.1$. (d1)–(d5) Results with white noise added to the initial condition of (c1), causing a $\sim 1$% increase in particle number. Such stability also stands for the open domain walls. Note that for (a), (b), and (c), $|F_x|=|\mathbf{F}|$.

Figure 3

The transverse magnetization $F_x$ during the time evolution. $\left({\text{c}}^{\prime }1\right)\text{–}\left({\text{c}}^{\prime }5\right)$ and $\left({\text{d}}^{\prime }1\right)\text{–}\left({\text{d}}^{\prime }5\right)$ correspond to Fig. 2 and (d1)–(d5), respectively.

Figure 4

Unstable spin-twist modes. (a) Spectrum of the unstable modes for $q=0$ and two values of $g_s/g_n$. For $g_s/g_n=-1/2$, the bifurcation point ($\omega =0$) occurs exactly at $k_y{\xi }_n=1/\sqrt{2}$. Inset shows the magnitude of the long wavelength instability as $g_s/g_n$ varies. (b) The spin-texture created by the unstable mode [50] at $k_y{\xi }_n\simeq 0.445$ where the maximum imaginary frequency is reached for $q=0$ and $g_s/g_n=-1/2$. White circles with $+$ and $-$ indicating positive and negative circulation spin-vortices, respectively. The red arrows and the background color represent transverse magnetization field $(F_x,F_y)$ and longitudinal magnetization $F_z$, respectively.

Figure 5

(a) Shows the spin-density cross section $F_x(x,y=0)$ of the open MDW shown in Fig. 2 at different times. The soliton-like profile of the magnetization is preserved during the motion. (b) Shows the periodic behavior of the overlap function for the open domain-wall configuration [Fig. 2], demonstrating that the standing wave on the MDW persists without decay over long time periods.

Figure 6

Spin-exchange dynamics during the MDW oscillation shown in Fig. 2. 2–2, 2–2 show dynamics of component densities $2n_{\pm 1}$ and $n_0$, respectively.