Phase prediction method for pattern formation in time-dependent Ginzburg-Landau dynamics for kinetic Ising model without a priori assumptions of domain patterns

We propose a phase prediction method for pattern formation in a two-dimensional kinetic Ising model with dipole-dipole interactions under the time-dependent Ginzburg-Landau dynamics. Considering the effects of the material thickness by assuming uniformness along the magnetization axis, the model corresponds to thin magnetic materials with long-range repulsive interactions. We formulate a theoretical basis to understand the effects of the material parameters on the formation of the magnetic domain patterns in terms of the equilibrium equations governing the balance between the linear and nonlinear forces in the equilibrium state. Further, we develop a method for predicting the phase in the equilibrium state achieved after the dynamical evolution of a system with given initial parameters. The analytically hard third-order term is approximated using the restricted phase-space approximation [Anzaki et al., Ann. Phys. (NY) 353, 107 (2015)] for the ${\varphi }^4$ models. Although the proposed method does not have perfect concordance with the actual numerical results, it has no arbitrary parameters and functions to tune the prediction. In other words, it is a method with no a priori assumptions of the domain patterns.

Figure 1

Left: The magnetic domain patterns at the end of the simulation for various values of $A$. Right: The time dependences of the external magnetic field (marked B), the average magnetization ${\langle \varphi \rangle }_0$ (marked M), and the correlation ${\langle \delta {\varphi }^2\rangle }_0$ (marked C). The simulation size is $L=512$; the duration of the time evolution is $t_{max}=300$, and the external magnetic field intensity is $B\left(t\right)=R(B_0-v_{\mathrm{B}}t)$, with $B_0=1.5, v_{\mathrm{B}}=0.01$, and $R\left(x\right)=max(x,0)$. The system parameters are $\alpha =1,\beta =1,\rho =1,\gamma =0.2$. The white regions have positive magnetization, whereas the black regions have negative magnetization.

Figure 2

A demonstration of the RPSA for the equilibrium equation [Eq. (21)]. Top: The dashed green line (marked RPSA) indicates the values of ${\langle {\varphi }^{\mathrm{RPSA}}\rangle }_0$ in Eq. (23). The red solid line (marked num) shows the full numerical result ${\langle {\varphi }^{\mathrm{num}}\rangle }_0$ from the time evolutions. Bottom: Moments obtained from the numerical time evolutions. The simulation size is $L=128$, and the duration of the time evolution is $t_{max}=300$. Other parameters are $\alpha =3.5, \beta =2.0$, and $\gamma =2/\pi$. The external field parameters are $B_0=1.5$ and $v_{\mathrm{B}}=0.01$.

Figure 3

Top: Thickness dependences of the moments in the final spin configuration (same as Fig. 2). Bottom: $Q_{max} [=max\left(Q_{\mathit{k}}\right)]$ and $Q_0$, directly obtained from the Green's function without the time evolutions. The numerical setup is similar to that shown in Fig. 2.

Figure 4

Top: The phase diagram of the TDGL dynamics corresponding to Eq. (12) estimated using the phase prediction method proposed in Sec. 8. Orange area (marked *Z), TZ- or Z-breaking phase; blue area (T), T-breaking phase; green area (Symmetric), symmetric phase. The gray solid line shows the contour of the normalized parameters corresponding to the system parameters used in Figs. 2 and 3. Bottom: Patterns corresponding to points a, b, and c on the phase diagram above. The expected patterns corresponding to the phases were obtained. Note that point c shows a pattern with $\varphi \left(\mathit{r}\right)=0$ for all $\mathit{r}$, which is the definition of the symmetric phase.

Figure 5

The numerical simulations for different values of the standard deviation $\sigma$. Left: $\sigma =0.15$. Middle: $\sigma =0.3$ (original). Right: $\sigma =0.6$. System parameters: $(\alpha ,\beta ,\gamma ,B_0,v_{\mathrm{B}})=(1,1,0.3,1.5,0.01)$ with $L=512$.

Figure 6

Size dependences of the numerical simulations. The two-dimensional magnetization pattern in square regions with a size of $128\times 128$ [for $L=256$ and 512, the depicted areas are cropped around the center $(x,y)=(0,0)$ of the entire simulation]. System parameters: $(\alpha ,\beta ,\gamma ,B_0,v_{\mathrm{B}})=(1,1,0.3,1.5,0.01)$ with $L=128$ (left), 256 (middle), and 512 (right).