Magnetic walls in the anisotropic $XY$-spin system in an oscillating magnetic field

Wall structures associated with dynamic phase transitions in the anisotropic $XY$-spin system in a temporally oscillating magnetic field $h\cos \left(\Omega t\right)$ in a one-dimensional system are analyzed by using the time-dependent Ginzburg-Landau model. It is numerically confirmed that there exist two types of magnetic walls, i.e., the Néel and Bloch walls, and is found that the transition between the two walls can occur for changing $h$ or $\Omega$. The phase diagram for the stable regions of each wall is obtained by both numerical and analytical methods. Furthermore, the critical behavior of the modulus of the Bloch wall around the Néel-Bloch transition point is studied, and it is found that the transition can be either continuous or discontinuous with respect to $h$, depending on $\Omega$.

Figure 1

The limit cycle attractor(s) in the four dynamic phases: (a) Ising-SRO, (b) Ising-SBO, (c) $XY$-SRO, and (d) $XY$-SBO. The horizontal and vertical axes represent $X$ and $\dot{X}$ components in (a) and (b) (Ising phases), and $X$ and $Y$ components in (c) and (d) ($XY$ phases), respectively. The other phases except the Ising-SRO phase have plural coexisting stable limit cycles.

Figure 2

Phase diagram of stable uniform solutions and domain wall solutions $(\gamma =0.55)$. The phase diagram for $\gamma >1∕2$ is qualitatively the same as this figure. For spatially uniform solutions, there appear two oscillations, the Ising-SRO and Ising-SBO phases, but the $XY$-type oscillation is not observed. In the Ising-SBO phase, the domain wall solution is present, and only the Néel wall is observed. For domain wall solutions, see Sec. 3.

Figure 3

Typical phase diagrams of stable uniform solutions and domain wall solutions for $0<\gamma <1∕2$: (a) $\gamma =0.2$, (b) $\gamma =0.3$, (c) $\gamma =0.33$, and (d) $\gamma =0.34$. Phase diagrams represent the regions of stable uniform solutions and stable domain wall solutions for each $\gamma$. For domain wall solutions, see Sec. 3. Here “XN” and “XB” denote the $XY$-SRO Néel and $XY$-SRO Bloch wall regions, respectively. Symbols denote the numerically obtained Néel-Bloch transition line, solid lines represent the transition lines for the uniform solutions obtained by the Floquet analysis, and the dotted lines stand for the Néel-Bloch transition lines obtained by the Fourier expansion approximation (see Sec. 4C). $h_1$ and $h_2$ stand for the $XY$-SRO Néel-Bloch transition line [Eq. (4.22)] and the Ising-SBO Néel-Bloch transition line [Eq. (4.24)], respectively. The Néel-Bloch transition in the Ising-SBO phase is observed in the region $\tilde{\gamma }<\gamma <1∕2$, where $\tilde{\gamma }$ is a numerical value in $0.2<\tilde{\gamma }<0.3$. The region of the Ising-SBO Néel wall rapidly expands as $\gamma$ approaches $\gamma =1∕3$. In the $h=0$ case, the Bloch and Néel walls are stable, respectively, for $\mid \gamma \mid <1∕3$ and $1∕3<\mid \gamma \mid$. The $XY$-SBO phase exists in a narrow region between the $XY$-SRO and Ising-SBO phases.

Figure 4

Typical phase diagrams of stable uniform solutions and domain wall solutions for $\gamma <0$: (a) $\gamma =-0.4$ and (b) $\gamma =-0.05$. Phase diagrams for $\gamma <-1∕3$ and $-1∕3<\gamma <0$ are qualitatively the same as (a) and (b), respectively. Diagrams represent the regions of stable uniform solutions and stable domain wall solutions for each $\gamma$. For domain wall solutions, see Sec. 3. Symbols denote the numerically obtained Néel-Bloch transition line, solid lines represent the transition lines for the uniform solutions obtained by the Floquet analysis, and the dotted line stands for the Néel-Bloch transition line $h_1$ obtained by the Fourier expansion approximation (see Sec. 4C).

Figure 5

Periodic oscillation of an Ising-SBO Néel wall for $\gamma =0.34$, $h=0.8$, and $\Omega =1.5$. The solid lines indicate $X(z,t)$, and the broken lines $Y(z,t)$. In the Néel wall case, $Y(z,t)$ vanishes.

Figure 6

Periodic oscillation of an Ising-SBO Bloch wall for $\gamma =0.34$, $h=1.2$, and $\Omega =1.5$. The setting of the graph is the same as Fig. 5. A Bloch wall has a nonvanishing $Y$ component.

Figure 7

Periodic oscillation of an $XY$-SRO Néel wall for $\gamma =-0.05$, $h=1.5$, and $\Omega =1.5$. The solid and broken lines indicate $X(z,t)$ and $Y(z,t)$. $\overline{X(z,t)}$ vanishes even though $X(z,t)\ne 0$ at any time.

Figure 8

Periodic oscillation of an $XY$-SRO Bloch wall for $\gamma =-0.05$, $h=0.5$, and $\Omega =1.5$. The setting of the graph is the same as Fig. 7. $\overline{X(z,t)}$ does not vanish in the wall region.

Figure 9

A schematic figure of the modulus of the Bloch wall. The solid line indicates $\overline{X(z,t)}$, and the broken line indicates $\overline{Y(z,t)}$. The above figure corresponds to the Ising-SBO case, where $Y_B$ is the modulus of the Bloch wall. In the $XY$-SRO case, the modulus $X_B$ is the peak value of $\overline{X(z,t)}$.

Figure 10

The relation between $h$ and $Y_B\left(h\right)$ around the Ising-SBO Néel-Bloch transition point $(h=h_2)$. The parameter values are $\gamma =0.3$ and $\Omega =0.5$. The symbols and solid line represent the numerical result and the function form $Y_B=0.379\sqrt{0.675-h}$.

Figure 11

The relation between $h$ and $X_B\left(h\right)$ around the $XY$-SRO Néel-Bloch transition point $(h=h_1)$. Parameter values are $\gamma =-0.05$ and $\Omega =1.0$. The symbols and broken line represent the numerical results and the function form $X_B=2.90\sqrt{0.6689-h}$. The Néel-Bloch transition is continuous, and the relation $X_B\propto \sqrt{h_1-h}$ holds below the transition point.

Figure 12

The relation between $h$ and $X_B\left(h\right)$ around the $XY$-SRO Néel-Bloch transition point, when $\gamma =-0.05$ and $\Omega =0.5$. The symbols + (×) indicates the numerical results for $h$ being varied downward (upward). The result shows a hysteresis, and therefore the Néel-Bloch transition is the first order transition.

Figure 13

Néel (left) and Bloch (right) walls in the absence of an applied field. The horizontal axis is the spatial coordinate $z$, and the solid and dotted lines represent $X$ and $Y$ components of the wall solutions, respectively. For the Néel wall, the $Y$ component vanishes but for the Bloch wall it is present. The Bloch wall structure corresponds to the case for $p=1$ and $q=1$ [see Eq. (A8)] with the chirality $q∕p=1$.