Dynamic phase transition of the Blume-Capel model in an oscillating magnetic field

We employ numerical simulations and finite-size scaling techniques to investigate the properties of the dynamic phase transition that is encountered in the Blume-Capel model subjected to a periodically oscillating magnetic field. We mainly focus on the study of the two-dimensional system for various values of the crystal-field coupling in the second-order transition regime. Our results indicate that the present nonequilibrium phase transition belongs to the universality class of the equilibrium Ising model and allow us to construct a dynamic phase diagram, in analogy with the equilibrium case, at least for the range of parameters considered. Finally, we present some complementary results for the three-dimensional model, where again the obtained estimates for the critical exponents fall into the universality class of the corresponding three-dimensional equilibrium Ising ferromagnet.

Figure 1

Time series of the magnetization (solid red curves) of the $\mathrm{\Delta }=1$ kinetic spin-1 Blume-Capel model under the presence of a square-wave magnetic field (dashed black lines) for $L=128$ and three values of the half-period of the external field: (a) $t_{1/2}=200$ MCSS, corresponding to a dynamically disordered phase; (b) $t_{1/2}=113$ MCSS, close to the dynamic phase transition; and (c) $t_{1/2}=50$ MCSS, corresponding to a dynamically ordered phase. Note that for the sake of clarity the ratio $h\left(t\right)/h_0$ is displayed.

Figure 2

Period dependencies of the dynamic order parameter of the $\mathrm{\Delta }=1$ kinetic spin-1 Blume-Capel model for $L=128$. Results are shown for three characteristic cases of the half-period of the external field: $t_{1/2}=200$ MCSS (blue line), $t_{1/2}=113$ MCSS (red line), and $t_{1/2}=50$ MCSS (black line). The strongly fluctuating trace (red line) corresponds to the vicinity of the dynamic phase transition, given that $t_{1/2}\approx t_{1/2}^{\mathrm{c}}=112.3\pm 1.3$, as shown below.

Figure 3

Configurations of the local dynamic order parameter $\left\{Q_x\right\}$ of the $\mathrm{\Delta }=1.75$ kinetic spin-1 Blume-Capel model for $L=128$. The “snapshots” of $\left\{Q_x\right\}$ for each regime are the set of local period-averaged spins during some representative period: (a) $t_{1/2}=100$ MCSS $>t_{1/2}^{\mathrm{c}}$, dynamically disordered phase; (b) $t_{1/2}=43$ MCSS $\approx t_{1/2}^{\mathrm{c}}$, near the dynamic phase transition; and (c) $t_{1/2}=20$ MCSS $<t_{1/2}^{\mathrm{c}}$, dynamically ordered phase.

Figure 4

In full analogy with Fig. 3 we show snapshots of the period-averaged quadrupole moment conjugate to the crystal-field coupling $\mathrm{\Delta }$. Simulation parameters are exactly the same as those used in Figs. 3–3.

Figure 5

Half-period dependency of the dynamic order parameter of the kinetic spin-1 $\mathrm{\Delta }=1.5$ Blume-Capel model for a wide range of system sizes studied. Inset: Half-period dependency of the corresponding dynamic susceptibility ${\chi }_L^Q$.

Figure 6

Estimation of the critical half-period $t_{1/2}^{\mathrm{c}}$ and the correlation length's exponent $\nu$ of the kinetic spin-1 Blume-Capel model for all values of $\mathrm{\Delta }$ considered. Solid lines are fits of the form of (8).

Figure 7

Finite-size scaling analysis of the maxima ${\left({\chi }_L^Q\right)}^{*}$ of the kinetic spin-1 Blume-Capel model for all values of $\mathrm{\Delta }$ considered. Solid lines are fits of the form of (9).

Figure 8

Finite-size scaling analysis of the dynamic order parameter estimated at the critical half-period, ${(\langle |Q|\rangle }_L{)}^{*}$, of the 2D kinetic spin-1 Blume-Capel model at $\mathrm{\Delta }=1$. The solid line is a power-law fit of the form of (10).

Figure 9

Half-period dependency of the energy of the kinetic spin-1 $\mathrm{\Delta }=1.5$ Blume-Capel model for a wide range of system sizes studied. Inset: Half-period dependency of the corresponding heat capacity ${\chi }_L^E$.

Figure 10

Illustration of the logarithmic scaling behavior of the maxima of the heat capacity, ${\left({\chi }_L^E\right)}^{*}$, for all values of $\mathrm{\Delta }$ considered in this work. Solid lines are fits of the form of (11).

Figure 11

Half-period dependency of the fourth-order Binder cumulant $U_L$ of the kinetic spin-1 $\mathrm{\Delta }=1.5$ Blume-Capel model for a wide range of system sizes studied. Inset: An enhancement of the intersection area.

Figure 12

Dynamic $\mathrm{\Delta }-t_{1/2}^{\mathrm{c}}$ phase boundary of the kinetic spin-1 square-lattice Blume-Capel model. Inset: For the sake of completeness we provide a part of the phase diagram of the equilibrium counterpart on the $(\mathrm{\Delta }-T_{\mathrm{c}})$ plane using the results from Ref. [47] that were used in the present work. The dotted line is a simple guide for the eye.

Figure 13

Finite-size scaling analysis of the maxima ${\left({\chi }_L^Q\right)}^{*}$. Inset: Shift behavior of the corresponding pseudocritical half-periods $t_{1/2}^{*}$ of the 3D kinetic spin-1 Blume-Capel model.

Figure 14

Finite-size scaling analysis of the dynamic order parameter estimated at the critical half-period, ${(\langle |Q|\rangle }_L{)}^{*}$, of the 3D kinetic spin-1 Blume-Capel model. The solid line is a power-law fit of the form of (10).

Figure 15

Finite-size scaling analysis of the maxima ${\left({\chi }_L^E\right)}^{*}$ of the 3D kinetic spin-1 Blume-Capel model. The solid line is a power-law fit of the form of (12).