Spin-current instability at a magnetic domain wall in a ferromagnetic superfluid: A generation mechanism of eccentric fractional skyrmions

Hiromitsu Takeuchi

Phys. Rev. A 105, 013328 – Published 31 January 2022

Abstract

Spinful superfluids of ultracold atoms are ideal for investigating the intrinsic properties of spin current and texture because they are realized in an isolated, nondissipative system free from impurities, dislocations, and thermal fluctuations. This study theoretically reveals the impact of spin current on a magnetic domain wall in spinful superfluids. An exact wall solution is obtained in the ferromagnetic phase of a spin-1 Bose-Einstein condensate with easy-axis anisotropy at zero temperature. The bosonic-quasiparticle mechanics analytically show that the spin current along the wall becomes unstable if the velocity exceeds the critical spin-current velocities, leading to complicated situations because of the competition between transverse magnons and ripplons. Our direct numerical simulation reveals that this system has a mechanism to generate an eccentric fractional skyrmion, which has a fractional topological charge, but its texture is not similar to that of a meron. This mechanism is in contrast to the generation of conventional skyrmions in easy-axis magnets. The theoretical findings can be examined in the same situation as in a recent experiment on ultracold atoms. In terms of the universality of spontaneous symmetry breaking, unexplored similar phenomena are expected in different physical systems with the same broken symmetry.

Received 16 September 2021
Revised 18 January 2022
Accepted 20 January 2022

DOI:https://doi.org/10.1103/PhysRevA.105.013328

Physics Subject Headings (PhySH)

Domain walls Kelvin-Helmholtz instability Magnons Quantum fluids & solids Skyrmions Spin current Spin texture Vortices in superfluids

Spinor Bose-Einstein condensates

Fluid DynamicsCondensed Matter, Materials & Applied PhysicsNonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Hiromitsu Takeuchi ^*

Department of Physics and Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, Osaka 558-8585, Japan

^*takeuchi@osaka-cu.ac.jp; http://hiromitsu-takeuchi.appspot.com

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Vol. 105, Iss. 1 — January 2022

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Images

Figure 1
Schematic profiles of (a) an AF-core DW and (b) a BA-core DW. Left: The curves show the cross-sectional profile of the density $f_{m}^{2} (m = 0, \pm 1)$ and amplitude of the spin-current velocity $v_{spin}$ parallel to the DWs for $V_{+} = 0$ . The arrows represent the spin texture. Right: Two-dimensional profiles of spin density $s$ (arrow) and density $n$ (gray scale).
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Figure 2
(a) Numerical plots of density profiles $f_{+ 1}^{2}$ (red mark) and $f_{- 1}^{2}$ (blue mark) of an AF-core DW for $c_{s} / c_{n} = - 0.8, - 0.5, - 0.2, - 0.03$ . The dimensionless plots are independent of $\tilde{q} / \tilde{μ}$ . Solid curves show the exact solution ${(f_{\pm 1}^{ex})}^{2}$ for $c_{s} / c_{n} = - 0.5$ given by Eq. (14). (b) Numerical plots of $f_{m}^{2} (m = 0, \pm 1)$ of the lowest-energy DW for $\tilde{q} / \tilde{μ} = - 1, - 0.5, - 0.25, - 0.1$ with $c_{s} / c_{n} = - 0.5$ .
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Figure 3
(a) The maximum transverse spin density $max s_{⊥}$ and (b) the tension $α$ of the lowest-energy DW for $c_{s} / c_{n} = - 0.03, - 0.2, - 0.5, - 0.8$ . (a) The lowest-energy DW is a BA-core DW with $max s_{⊥} > 0$ when $\tilde{q}$ is larger than a critical value ${\tilde{q}}_{C} (c_{s}) < 0$ . The positions of the transition points are indicated by arrows schematically. (b) The analytical results, $α_{ex}$ [Eq. (18)] and $α_{weak}$ [Eq. (19)], for AF-core DWs are compared with the numerical results.
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Figure 4
Bogoliubov excitation spectrum of the first and second lowest-energy excitations at an AF-core DW for $c_{s} / c_{n} = - 1 / 2$ and $V_{+} = 0$ . Circles show the real part (black circle) and the imaginary part (red circle) obtained numerically by solving the BdG equation. Solid curves represent the analytical plots of $Re {\tilde{ω}}_{+}$ (green) and $Im {\tilde{ω}}_{+}$ (orange) in Eq. (35) and $Re ω_{KH}$ (blue) in Eq. (39) with $α = α_{ex}$ .
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Figure 5
Dynamics of the spin-current instability at a BA-core DW for $(c_{s} / c_{n}, \tilde{q} / \tilde{μ}, V_{\pm 1} τ / ξ) = (- 0.5, - 0.05, \pm 0.368)$ . The arrows show the spin density $s$ . The distributions of $s_{z}$ and $n_{0} = {| Ψ_{0} |}^{2}$ are represented by the arrow color and background grayscale, respectively. (a) $t / τ = 0$ : The wall is flat in the initial state. (b) $t / τ = 200$ : The KHI makes the flutter-finger pattern. (c) $t / τ = 625$ : The vortices are released as DW loops, and then spin singularities appear along the DWs at which spin density vanishes locally. (d) $t / τ = 1100$ : A spin singularity survives even on a released loop, forming a fractional skyrmion. The spin texture of skyrmions is schematically illustrated for (e) $N_{s} = 0$ , (f) $N_{s} = 1 / 2$ , and (g) $N_{s} = 1$ , together with the distribution of local ordered states (BA, AF, $F_{\pm}$ ). The phases $\arg Ψ_{- 1}$ and $\arg Ψ_{+ 1}$ , of (d) are plotted in (h) and (i), respectively. The black regions represent $m = \pm 1$ domains, in which $\arg Ψ_{\mp 1}$ is highly fluctuated. The position of the domains with $N_{v} = 1, 2$ is indicated by arrows.
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Figure 6
Snapshots of the phases $arg Ψ_{\pm 1}$ and the transverse spin density $s_{⊥}$ in the time evolution of Fig. 5. The display range of Fig. 5 is shown by red broken lines in the bottom panels. Black areas in the top and middle panels represent the region of $s_{z} > 0$ and $s_{z} < 0$ , respectively. The transverse spin density $s_{⊥}$ has a similar distribution to the background images of $n_{0}$ in Fig. 5. In the later stage of the dynamics, a lot of “notches” appear as white spots along the DWs in the distribution of $s_{⊥}$ , which correspond to the spin singularities in the spin density. The number of branch cuts (jump from $arg Ψ_{\pm 1} = - π$ to $π$ ) terminated by an isolated domain wall is equal to $N_{v} (= N_{s} / 2)$ . A spin singularity exists along a closed DW surrounding an isolated domain, at which a branch cut is terminated. The closed DW with a spin singularity looks like a “C” mark in the distribution plot of $s_{⊥}$ or $n_{0}$ . No singularity happen if the domain contains an even number of terminated branch cuts. Some examples of domains with $N_{s} = 0, 1 / 2, 1$ are denoted by red arrows in the lower right.
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