Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model

The two-dimensional kinetic Ising model, when exposed to an oscillating applied magnetic field, has been shown to exhibit a nonequilibrium, second-order dynamic phase transition (DPT), whose order parameter $Q$ is the period-averaged magnetization. It has been established that this DPT falls in the same universality class as the equilibrium phase transition in the two-dimensional Ising model in zero applied field. Here we study the scaling of the dynamic order parameter with respect to a nonzero, period-averaged, magnetic “bias” field, $H_b$, for a DPT produced by a square-wave applied field. We find evidence that the scaling exponent, ${\delta }_{\mathrm{d}}$, of $H_b$ at the critical period of the DPT is equal to the exponent for the critical isotherm, ${\delta }_{\mathrm{e}}$, in the equilibrium Ising model. This implies that $H_b$ is a significant component of the field conjugate to $Q$. A finite-size scaling analysis of the dynamic order parameter above the critical period provides further support for this result. We also demonstrate numerically that, for a range of periods and values of $H_b$ in the critical region, a fluctuation-dissipation relation (FDR), with an effective temperature $T_{\mathrm{eff}}(T,P,H_0)$ depending on the period, and possibly the temperature and field amplitude, holds for the variables $Q$ and $H_b$. This FDR justifies the use of the scaled variance of $Q$ as a proxy for the nonequilibrium susceptibility, $\partial ⟨Q⟩∕\partial H_b$, in the critical region.

Figure 1

(Color online) Log-log plot of the dynamic order parameter, $⟨Q⟩$, vs bias field, $H_b$, at $P=P_c$ for $L=90$, 128, 180, and 256. A least-squares fit to power-law scaling of the $L=256$ data, in the range between the labels A and B above, produced a statistically significant fit with scaling exponent $1∕{\delta }_{\mathrm{d}}=0.0673\pm 0.0008$ (corresponding to ${\delta }_{\mathrm{d}}=14.85\pm 0.18$). The dotted line corresponds to the scaling exponent $1∕{\delta }_{\mathrm{d}}=0.0673$. A reference line representing scaling with the equilibrium Ising exponent, ${\delta }_{\mathrm{e}}=15$ $(1∕{\delta }_{\mathrm{e}}=0.0666)$, is shown as the dashed line.

Figure 2

(Color online) Log-log plots of the scaling function ${\mathcal{F}}_{+}(y_1,y_2)$ vs $y_1$ for lattice sizes $L=90$, 128, and 180. The values of $y_1$ plotted are 4.00, 17.2, 30.3, 43.4, 69.7, 149, and [in (b)] 280 and 477. (a) The data are shown for the values of $y_2$ listed on the plot. The best-fit line (dotted) for the last five points of the $L=180$ data at $y_2=3.39$, with slope $-1.76\pm 0.07$, is included along with a reference line (dashed) with the slope $-{\gamma }_{\mathrm{e}}=-1.75$. (b) The data for $y_2=3.39$ with two additional $y_1$ values illustrate the boundary of the regime of power-law scaling.

Figure 3

(Color online) Log-log plot of the scaling function ${\mathcal{F}}_{+}(y_1,y_2)$ vs $y_2=(H_b∕J)L^{\beta \delta ∕\nu }$ for lattice sizes $L=90$, 128, and 180, for the constant values of $y_1$ labeled in the plot. The values of $y_2$ used are $y_2=3.39$, 8.46, 16.9, 33.9, 84.6, and 169. The dotted line represents the best fit to the first five points of the $L=180$ data at $y_1=149$ and has a slope of $1.01\pm 0.01$. The dashed line shows the slope value of 1, expected from Eq. (13).

Figure 4

(Color online) Log-log plot of the scaling function ${\mathcal{F}}_{+}(y_1,y_2)$ vs $y_2=(H_b∕J)L^{\beta \delta ∕\nu }$ for lattice sizes $L=128$, 180, and 256, at the critical period $P_c$, where $y_1=0$. In the $L=256$ data, near the values $y_1=16$, 32, 165, and 335, two closely spaced data points are actually plotted. The best-fit line to the $L=256$ data in the range $8.46<y_2<84.6$, shown as a dotted line in the plot, corresponds to a scaling exponent $1∕{\delta }_{\mathrm{d}}=0.0673\pm 0.0008$. A reference line corresponding to scaling exponent $1∕{\delta }_{\mathrm{e}}=1∕15=0.0666$ is also shown. These results are in complete agreement with those shown in Fig. 1.

Figure 5

(Color online) The scaled fluctuations $X_L^Q$ of the dynamic order parameter plotted vs its susceptibility ${\widehat{\chi }}_L$ to the bias field $H_b$, calculated at $L=180$, for periods $P=140$, 150, 160, 170, and 190 MCSS. The quantity ${\widehat{\chi }}_L$ was calculated using the numerical derivative in Eq. (17). The best-fit lines shown, whose slopes increase monotically with the period $P$ of the data to which they were fit, were calculated as $\left(3.239J\right){\widehat{\chi }}_L+10.42$, $\left(3.557J\right){\widehat{\chi }}_L+8.735$, $\left(3.980J\right){\widehat{\chi }}_L+3.889$, $\left(4.232J\right){\widehat{\chi }}_L+5.868$, and $\left(4.497J\right){\widehat{\chi }}_L+5.371$, respectively.

Figure 6

(Color online) Closeup of Fig. 5, showing the relationship of $X_L^Q$ and ${\widehat{\chi }}_L$ at low values of ${\widehat{\chi }}_L$, which correspond to large values of the bias field $H_b$. For $P=150$, 170, and 190 MCSS, data have been taken (and are shown) down to very low values of ${\widehat{\chi }}_L$, where the breakdown of the linear relationship between $X_L^Q$ and ${\widehat{\chi }}_L$ can be clearly seen. The dashed lines are the same best-fit lines shown in Fig. 5.

Figure 7

(Color online) The scaled fluctuations $X_L^Q$ of the dynamic order parameter plotted vs its susceptibility ${\widehat{\chi }}_L$ to the bias field $H_b$, calculated for lattice size $L=180$, at periods $P=220$ and 250 MCSS. At each period, the data were fit (purely phenomenologically) to two linear relationships. For $P=220$ MCSS, the fits were calculated as $\left(6.265J\right){\widehat{\chi }}_L-7.497$ at low ${\widehat{\chi }}_L$ and $\left(4.161J\right){\widehat{\chi }}_L+19.94$ at high ${\widehat{\chi }}_L$. For $P=250$ MCSS, the fits were $\left(6.485J\right){\widehat{\chi }}_L-9.240$ at low ${\widehat{\chi }}_L$ and $\left(0.2726J\right){\widehat{\chi }}_L+67.92$ at high ${\widehat{\chi }}_L$.

Figure 8

(Color online) The effective temperature $T_{\mathrm{eff}}$, obtained as the slopes of the linear fits to the data in Figs. 5, 7, plotted vs $\theta =(P-P_c)∕P_c$. For the values $\theta =0.606$ and 0.825 ($P=220$ and 250 MCSS), the slopes of both linear regimes fit in Fig. 7 are plotted as $T_{\mathrm{eff}}$ values, using filled squares and diamonds rather than filled circles. The straight line is a weighted least-squares fit to the data below $\theta \approx 0.4$ ($P<190$ MCSS), and has a slope of $2.97J$.

Figure 9

(Color online) Log-log plots of ${\mathcal{G}}_{+}(y_1,y_2)$ and ${\mathcal{G}}_{+}^X(y_1,y_2)$ vs $y_1$, over the range $y_1=30.3–149$, for $y_2=8.46$ at $L=180$. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dash-dash-dotted lines are fits to all four and the last three ${\mathcal{G}}_{+}$ data points, respectively, and correspond to scaling exponents $-1.60\pm 0.03$ and $-1.71\pm 0.05$. The dotted and dash-dotted lines are the result of attempts to fit all four and the last three ${\mathcal{G}}_{+}^X$ data points, respectively. They correspond to nominal scaling exponents $-1.73\pm 0.01$ and $-1.81\pm 0.02$. The dashed line is a reference line corresponding to scaling exponent $-1.75$.

Figure 10

(Color online) Log-log plots of ${\mathcal{G}}_{+}(y_1,y_2)$ and ${\mathcal{G}}_{+}^X(y_1,y_2)$ vs $y_1$, over the range $y_1=30.3–477$, for $y_2=0$ at lattice size $L=180$. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dash-dash-dotted lines show the result of attempts to fit all six and the first five ${\mathcal{G}}_{+}$ data points, and correspond to a nominal scaling exponent $-1.65\pm 0.03$ and a statistically significant scaling exponent $-1.74\pm 0.03$, respectively. The dotted and dash-dotted lines are the result of attempts to fit all six and the first five ${\mathcal{G}}_{+}^X$ data points, and correspond to nominal scaling exponents $-2.01\pm 0.01$ and $-2.10\pm 0.01$, respectively. The dashed line is a reference line corresponding to the scaling exponent $-1.75$.

Figure 11

(Color online) Log-log plot of ${\mathcal{G}}_{+}^{\mid X\mid }(y_1,y_2)$ vs $y_1$, over the range $y_1=30.3–149$, for $y_2=0$ at lattice size $L=180$. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dotted lines are the results of attempts to fit all five and the last four data points with a power-law relationship, and correspond to nominal scaling exponents of $-1.59\pm 0.02$ and $-1.70\pm 0.03$, respectively. The dashed line is a reference line corresponding to a scaling exponent of $-1.75$.

Figure 12

(Color online) Log-log plot of ${\mathcal{G}}_{+}^X(y_1,y_2)$ vs $y_2=(H_b∕J)L^{\beta \delta ∕\nu }$ for lattice sizes $L=128$, 180, and 256, at the critical period $P_c$, where $y_1=0$. The best-fit line to the $L=256$ data in the range $8.46<y_2<84.6$, shown as a dotted line in the plot, corresponds to a scaling exponent $(1-{\delta }_{\mathrm{d}})∕{\delta }_{\mathrm{d}}=-0.914\pm 0.029$. A reference line (dashed), corresponding to a scaling exponent $(1-{\delta }_{\mathrm{e}})∕{\delta }_{\mathrm{e}}=-14∕15\approx -0.933$, is also shown.