Dynamic phase transitions in the presence of quenched randomness

We present an extensive study of the effects of quenched disorder on the dynamic phase transitions of kinetic spin models in two dimensions. We undertake a numerical experiment performing Monte Carlo simulations of the square-lattice random-bond Ising and Blume-Capel models under a periodically oscillating magnetic field. For the case of the Blume-Capel model we analyze the universality principles of the dynamic disordered-induced continuous transition at the low-temperature regime of the phase diagram. A detailed finite-size scaling analysis indicates that both nonequilibrium phase transitions belong to the universality class of the corresponding equilibrium random Ising model.

Figure 1

Panel (a) Disorder distributions of the nonequilibrium dynamic order parameter, and panel (b) susceptibility maxima for a lattice size $L=128$ and for both models considered in this paper. The running averages over the samples are shown by the solid lines. Panel (c) shows the signal-to-noise ratio $S/N$ of the dynamic susceptibility, that is, the ratio of the relative variance of the distribution over the square of its mean value as a function of the inverse linear size. The solid lines are second-order polynomial fittings to $1/L$ for the larger system sizes. Clearly, $S/N\left({\chi }_L^Q\right)\to 0$ as $L\to \infty$, indicating that self-averaging is restored in the thermodynamic limit for both kinetic disordered systems [63, 64].

Figure 2

Time series of the magnetization (red solid curves) of the kinetic random-bond Ising model under the presence of a square-wave magnetic field (black dashed lines) for $L=96$ and three values of the half-period of the external field: (a) $t_{1/2}=20$ MCSS, corresponding to a dynamically ordered phase, (b) $t_{1/2}=76$ MCSS, close to the dynamic phase transition, and (c) $t_{1/2}=200$ MCSS, corresponding to a dynamically disordered phase. Note that for the sake of clarity the ratio $h\left(t\right)/h_0$ is displayed.

Figure 3

Period dependencies of the dynamic order parameter of the kinetic random-bond Ising model for $L=96$. Results are shown for the three characteristic cases of the half-period of the external field, following Fig. 2.

Figure 4

Configurations of the local dynamic order parameter $\left\{Q_x\right\}$ of the random-bond kinetic Ising model for $L=96$. The “snapshots” of $\left\{Q_x\right\}$ for each regime are the set of local period-averaged spins during some representative period. Three panels are shown: (a) $t_{1/2}=20\mathrm{MCSS}<t_{1/2}^{\mathrm{c}}$—dynamically ordered phase, (b) $t_{1/2}=76\mathrm{MCSS}\approx t_{1/2}^{\mathrm{c}}$—near the dynamic phase transition, and (c) $t_{1/2}=200\mathrm{MCSS}>t_{1/2}^{\mathrm{c}}$—dynamically disordered phase.

Figure 5

Time series of the magnetization (red solid curves) of the kinetic random-bond $\mathrm{\Delta }=1.975$ Blume-Capel model under the presence of a square-wave magnetic field (black dashed lines) for $L=96$ and three values of the half-period of the external field: (a) $t_{1/2}=20$ MCSS, corresponding to a dynamically ordered phase, (b) $t_{1/2}=85$ MCSS, close to the dynamic phase transition, and (c) $t_{1/2}=200$ MCSS, corresponding to a dynamically disordered phase. Note that for the sake of clarity the ratio $h\left(t\right)/h_0$ is displayed.

Figure 6

Period dependencies of the dynamic order parameter of the kinetic random-bond $\mathrm{\Delta }=1.975$ Blume-Capel model for $L=96$. Results are shown for the three characteristic cases of the half-period of the external field, following Fig. 5.

Figure 7

Configurations of the local dynamic order parameter $\left\{Q_x\right\}$ of the random-bond kinetic $\mathrm{\Delta }=1.975$ Blume-Capel for $L=96$. The snapshots of $\left\{Q_x\right\}$ for each regime are the set of local period-averaged spins during some representative period. Three panels are shown: (a) $t_{1/2}=20\mathrm{MCSS}<t_{1/2}^{\mathrm{c}}$—dynamically ordered phase, (b) $t_{1/2}=85\mathrm{MCSS}\approx t_{1/2}^{\mathrm{c}}$—near the dynamic phase transition, and (c) $t_{1/2}=200\mathrm{MCSS}>t_{1/2}^{\mathrm{c}}$—dynamically disordered phase.

Figure 8

In full analogy to Fig. 7 we show snapshots of the period-averaged dynamic quadrupolar moment conjugate to the crystal-field coupling $\mathrm{\Delta }$. The simulation parameters are exactly the same to those used in Fig. 7 for all three panels (a)–(c).

Figure 9

Half-period dependency of the dynamic order parameter of the kinetic random-bond Ising model. The lower inset illustrates the half-period dependency of the corresponding dynamic susceptibility ${\chi }_L^Q$. The upper inset shows the half-period dependency of the energy and the corresponding heat capacity ${\chi }_L^E$. All results shown refer to a system size of $L=64$ at the critical $t_{1/2}$ region.

Figure 10

Criticality in the kinetic random-bond Ising model ($r=1/7$): (a) Finite-size scaling behavior of the maxima ${\left({\chi }_L^Q\right)}^{*}$ on a log-log scale (main panel) and shift behavior of the corresponding pseudocritical half-periods $t_{1/2}^{*}$ (inset). (b) Double (main panel) and simple (inset) logarithmic scaling behavior of the heat-capacity maxima ${\left({\chi }_L^E\right)}^{*}$. In all cases lines are linear fittings.

Figure 11

Criticality in the kinetic random-bond Blume-Capel model ($r=0.75/1.25;\mathrm{\Delta }=1.975$). The description is analogous to that of Fig. 10.

Figure 12

Half-period dependency of the fourth-order Binder cumulant $U_L$ of the kinetic random-bond Ising (main panel) and Blume-Capel (inset) models for a wide range of system sizes studied.