Nonuniversal surface behavior of dynamic phase transitions

We have studied the dynamic phase transition (DPT) of the kinetic Ising model in systems with surfaces within the mean-field approximation. Varying the surface exchange coupling strength $J_s$, the amplitude of the externally applied oscillating field $h_0$, and its period $P$, we explore the dynamic behavior of the layer-dependent magnetization and the associated DPTs. The surface phase diagram shows several features that resemble those of the equilibrium case, with an extraordinary bulk transition and a surface transition for high $J_s$ values, independent from the value of $h_0$. For low $J_s$, however, $h_0$ is found to be a crucial parameter that leads to nonuniversal surface behavior at the ordinary bulk transition point. Specifically, we observed here a bulk-supported surface DPT for high field amplitudes $h_0$ and correspondingly short critical periods $P_c$, whereas this surface transition simultaneous to the bulk one is suppressed for slow critical dynamics occurring for low values of $h_0$. The suppression of the DPT for low $h_0$ not only occurs for the topmost surface layer, but also affects a significant number of subsurface layers. We find that the key physical quantity that explains this nonuniversal behavior is the time correlation between the dynamic surface and bulk magnetizations at the bulk critical point. This time correlation has to pass a threshold value to trigger a bulk-induced DPT in the surface layers. Otherwise, dynamic phase transitions are absent at the surface in stark contrast to the equilibrium behavior of the corresponding thermodynamic Ising model. Also, we have analyzed the penetration depth of the dynamically ordered phase for the surface DPT that occurs for large $J_s$ values. Here we find that the penetration depth depends strongly on $J_s$ and behaves identically to the corresponding equilibrium Ising model.

Figure 1

Schematic representation of the simple cubic lattice with the (100) surface orientation utilized in this study. Each atom in layer $L=1$ (surface) has four in-plane nearest neighbors and interacts with them via an exchange coupling strength $J_s$, which is represented by dashed lines. Additionally, it has one other nearest neighbor in $L=2$, to which it is coupled via exchange coupling strength $J_b$, represented with solid lines. For $L>1$, each atom interacts with its six nearest neighbors through $J_b$, of which four are in-plane neighbors in $L$, one is in the layer above $(L-1)$, and one is in the layer below $(L+1)$.

Figure 2

(a) Magnetization of the surface and bulk layers as a function of time (normalized by $\tau$) for an oscillating field amplitude $h_0=0.080$ and for oscillating periods of $P=2P_c$, $1.2P_c$, and $0.8P_c(P_c=93.37585\tau )$. The magnetic field normalized to double its amplitude is represented by dashed (black) lines. The dynamic order parameter $Q$ of the surface and bulk layers is shown as a function of the period $P$ of the oscillating field normalized to $P_c$ for (b) $h_0=0.080$ and (d) $h_0=0.375$. (c) and (e) Susceptibilities of the surface ${\chi }_s$ and bulk ${\chi }_b$ normalized to ${\chi }_0$, as a function of $P/P_c$, corresponding to the order parameters in (b) and (d), respectively. All surface quantities are represented with thin (blue) solid lines and the bulk quantities with thick (red) solid lines. The surface coupling constant is $J_s=J_b$ in all cases.

Figure 3

Magnetization $m$ of the surface [thin (blue) solid line] and the bulk [thick (red) solid line] as a function of time $t$ normalized to $P_c$, for different values of $h_0$. The period of the oscillating field in each case is the $P_c$ value that corresponds to each respective amplitude $h_0$. The magnetic field normalized to double its amplitude is depicted as a dashed (black) line in each figure. A decreasing phase shift between surface and bulk magnetizations can be observed upon changing from (a) to (d), which corresponds to successively increasing $h_0$ values as indicated in each figure. The surface coupling constant is $J_s=J_b$ in all cases.

Figure 4

(a) Surface susceptibility ${\chi }_s$ at $P_c$ normalized by the bulk susceptibility at the same point, as a function of $h_0$ [47]. (b) Time correlation $C_{sb}$ between the surface and the bulk magnetizations at $P_c$ as defined in Eq. (11), as a function of $h_0$. (c) Relationship between the surface susceptibility peak at $P_c$ and $C_{sb}$. In all figures, closed circles represent numerical calculations with the solid line being a guide to the eye. The surface coupling constant is $J_s=J_b$ in all cases.

Figure 5

(a) Critical exponent ${\beta }_s$ as a function of $h_0$, obtained by fitting the computed data of $Q_s$ with Eq. (12). The dashed line represents the equilibrium surface critical exponent in the MFA ${\beta }_s^{\mathrm{eq}}=1$. (b) Ratio $A_s/A_b$ between the prefactors of Eq. (12) as a function of $h_0$. In both cases, closed circles represent the numerical calculations and the solid line is a guide to the eye. The surface coupling constant is $J_s=J_b$ in all cases.

Figure 6

Normalized susceptibility $\chi /{\chi }_0$ maps as a function of the normalized period $P/P_c$ and the layer index $L$, using a logarithmic (base 10) scale for $\chi /{\chi }_0$. The darkest regions represent high values of the susceptibility, i.e., peaks, and the light color corresponds to low values of the susceptibility. (a) The amplitude of the oscillating field is $h_0=0.080$ and the peak that comes from the bulk DPT does not extend to the topmost layers. (b) The amplitude of the oscillating field is $h_0=0.375$ and the peak extends all the way to the surface layer $L=1$. The surface coupling constant is $J_s=J_b$ in both cases.

Figure 7

Normalized surface susceptibility ${\chi }_s/{\chi }_0$ as a function of the normalized period $P/P_c$ for different $J_s$ values above the special point. The $J_s$ values, to which each curve corresponds, are given in the figure. They are, from left to right, $J_s=1.35J_b$, $1.45J_b$, and $1.55J_b$. (a) The $h_0=0.080$ case and (b) the $h_0=0.375$ case.

Figure 8

(a) and (b) Normalized susceptibility $\chi /{\chi }_0$ maps as a function of the normalized period $P/P_c$ and the layer index $L$, using a logarithmic (base 10) scale for $\chi /{\chi }_0$, for $h_0=0.375$ and (a) $J_s=1.3J_b$ and (b) $J_s=1.5J_b$. Two peaks are observed: one occurring at $P_c^s$, with $P/P_c>1$, corresponds to the surface transition and the other one at $P/P_c=1$ is the bulk transition. In (a) the surface transition penetrates more deeply into the bulk than in (b). (c) Penetration depth $\lambda$ of the surface transition in units of $L$, as a function of $J_s$. Here $\lambda$ has been obtained by fitting Eq. (13) to our numerical susceptibility $\chi$ vs $L$ data. For the DPT case (shown as open squares) all fits were done at the surface critical period $P_c^s$. In the equilibrium case (shown as closed triangles), the fits of $\chi$ vs $L$ were performed at the surface critical temperature $T_c^s$.

Figure 9

Normalized susceptibility $\chi /{\chi }_0$ maps as a function of the normalized period $P/P_c$ and the layer index $L$, using a logarithmic (base 10) scale for $\chi /{\chi }_0$, for (a)–(d) $h_0=0.080$ and (e)–(h) $h_0=0.375$. The respective values of $J_s$ are shown in each panel.

Figure 10

The $J_s-P$ surface phase diagrams for different amplitudes of the oscillating field $h_0$. The lines represent the boundaries between the phases and are determined from the respective positions of the susceptibility peaks. The surface phase boundaries are depicted as solid (blue) lines and the bulk phase boundaries as dashed (red) lines. The vertical axis corresponds to the period normalized to the bulk critical period in each case and the horizontal axis to the ratio between surface and bulk coupling strengths.

Figure 11

Normalized surface susceptibility at $P/P_c=1$ as a function of the time correlation $C_{sb}$ between surface and bulk magnetization. Values have been computed for different values of $h_0$ and $J_s(<1.25J_b)$. Circles (black) represent values calculated for $J_s=J_b$, squares (red) correspond to $J_s=0.85J_b$, and triangles (green) to $J_s=1.15J_b$.