Extended order parameter and conjugate field for the dynamic phase transition in a Ginzburg-Landau mean-field model in an oscillating field

We present numerical evidence for an extended order parameter and conjugate field for the dynamic phase transition in a Ginzburg-Landau mean-field model driven by an oscillating field. The order parameter, previously taken to be the time-averaged magnetization, comprises the deviations of the Fourier components of the magnetization from their values at the critical period. The conjugate field, previously taken to be the time-averaged magnetic field, comprises the even Fourier components of the field. The scaling exponents $\beta$ and $\delta$ associated with the extended order parameter and conjugate field are shown numerically to be consistent with their values in the equilibrium mean-field model.

Figure 1

Dynamic phase diagram illustrating the bifurcation of Fourier coefficients $m_0$, $m_2$, and $m_4$ below $P_C$. Here $a=-\frac{3\sqrt{3}}{4}$, $b=\frac{3\sqrt{3}}{8}$, and $H_1=1.5$, for which it is found that $P_C=5.319357661995$. Note that the values $+|m_{2j}|$ and $-|m_{2j}|$ are displayed below $P_c$ for simplicity; the two stable asymmetric loops actually have opposite complex Fourier components $m_{2j}$ and $-m_{2j}$.

Figure 2

Critical scaling of order parameter scaling variables $z_k$ with respect to the scaled period $\epsilon$. The plots for $z_k$, $k=0,\cdots ,7$ are shown individually in the figure, along with a reference line representing scaling with exponent 1/2.

Figure 3

Plot of $h\left(t\right)$ and $m\left(t\right)$ for GL model with applied field $h\left(t\right)=H_{1}sin\left(\omega t\right)+H_{2}sin\left(2\omega t\right)$, with $H_1=1.5$ and $\omega =\frac{2\pi }{P_c}$. The thin curves (black online) represent $h\left(t\right)$; the thick curves (red online) represent $m\left(t\right)$. The cases $H_2=0.0,0.5$, and 1.0 are represented by solid, dashed, and dotted curves, respectively.

Figure 4

Critical scaling of the variables $z_k$ ($k=0,\cdots ,7$) with respect to $h_0$, $h_2$ and $h_4$. The scaling with respect to $h_0$, $h_2$, and $h_4$ are represented by solid lines (red), dashed lines (green), and dotted lines (blue), respectively. The black reference line shows exact scaling with exponent 1/3.

Figure 5

Crossover of critical scaling of $z_0=\left|m_0\right|$ with respect to $h_0$ at the period value $P=5.3193577>P_c$. The data for $z_0$ are represented by the thin line (red). The thick black dashed curve represents scaling with exponent 1/3, while the thick black dash-dotted curve represents scaling with exponent 1.

Figure 6

Illustration of the three bracketed terms ($T_1$, $T_2$, $T_3$) from Eq. (8) with respect to $h_0$ at the period $P=5.3193577$. The signs of $T_2$ and $T_3$ remain positive throughout. The sign of $T_1$ switches from positive to negative just above $h_0={10}^{-12}$. Due to the logarithmic scale used, the absolute value $|T_1|$ is therefore plotted. The sum of the three terms, denoted $\Sigma =T_1+T_2+T_3$, is also shown. The thick black dashed curve shows represents scaling with exponent 1, while the thick black dotted curve represents scaling with exponent 3.