Dynamic phase transition in the kinetic spin-$3∕2$ Blume-Capel model under a time-dependent oscillating external field

We present a study, within a mean-field approach, of the stationary states of the kinetic spin-$3∕2$ Blume-Capel model in the presence of a time-dependent oscillating external magnetic field. We use the Glauber-type stochastic dynamics to describe the time evolution of the system. We have found that the behavior of the system strongly depends on the crystal-field interaction. We can identify two types of solutions: a symmetric one where the magnetization $\left(m\right)$ of the system oscillates in time around zero, which corresponds to a paramagnetic phase $\left(P\right)$, and an antisymmetric one where $m$ oscillates in time around a finite value different from zero, namely $\pm 3∕2$ and $\pm 1∕2$ that corresponds to the ferromagnetic-$3∕2$ $\left(F_{3∕2}\right)$ and the ferromagnetic-$1∕2$ $\left(F_{1∕2}\right)$ phases, respectively. There are coexistence regions of the phase space where the $F_{3∕2}$, $F_{1∕2}$ $(F_{3∕2}+F_{1∕2})$, $F_{3∕2}$, $P$ $(F_{3∕2}+P)$, $F_{1∕2}$, $P$ $(F_{1∕2}+P)$, and $F_{3∕2}$, $F_{1∕2}$, $P$ $(F_{3∕2}+F_{1∕2}+P)$ phases coexist, hence the system exhibits seven different phases. We obtain the dynamic phase transition points and find six fundamental phase diagrams which exhibit one or three dynamic tricritical points. We have also calculated the Liapunov exponent to verify the stability of the solutions and the dynamic phase transition points.

Figure 1

Time variations of the magnetization $\left(m\right)$: (a) Exhibiting a paramagnetic phase $\left(P\right)$, $d=-0.75$, $h=1.0$, and $T=0.25$. (b) Exhibiting a ferromagnetic-$1∕2$ $\left(F_{1∕2}\right)$ phase, $d=-0.5$, $h=1.125$, and $T=0.1875$. (c) Exhibiting a ferromagnetic-$3∕2$ $\left(F_{3∕2}\right)$ phase, $d=-0.25$, $h=0.5$, and $T=0.5$. (d) Exhibiting a coexistence region $(F_{1∕2}+P)$, $d=-0.475$, $h=1.375$, and $T=0.025$. (e) Exhibiting a coexistence region $(F_{3∕2}+P)$, $d=0.25$, $h=1.25$, and $T=0.25$. (f) Exhibiting a coexistence region $(F_{3∕2}+F_{1∕2})$, $d=-0.625$, $h=0.125$, and $T=0.05$. (g) Exhibiting a coexistence region $(F_{3∕2}+F_{1∕2}+P)$, $d=-0.5$, $h=0.375$, and $T=0.0625$.

Figure 2

The reduced temperature dependence of the dynamic magnetization $M$, and a symmetric solution which corresponds to the $P$ phase, and nonsymmetric solutions that correspond to the $F_{3∕2}$ and $F_{1∕2}$ phases. $T_{\mathrm{t}}$ is the first-order phase transition temperatures, and $T_{\mathrm{C}}$ and $T_{{\mathrm{C}}^{\prime }}$ are the second-order phase transition temperatures from the $F_{3∕2}$ phase to the $P$ phase and the $F_{1∕2}$ phase to the $P$ phase, respectively. (a) Exhibiting two second-order phase transition, one from the $F_{3∕2}$ phase to the $P$ phase for $d=-0.25$, $h=0.50$; 0.915 is found $T_{\mathrm{C}}$ and the other one from the $F_{1∕2}$ phase to the $P$ phase for $d=-0.5$, $h=1.25$; 0.2125 is found $T_{{\mathrm{C}}^{\prime }}$. (b) Exhibiting two successive phase transitions, the first one is a first-order phase transition from the $F_{3∕2}$ phase to the $F_{1∕2}$ phase and the second one is second-order phase transition the $F_{1∕2}$ phase to the $P$ phase for $d=-0.50$ and $h=0.25$; 0.20 and 0.38 are found $T_{\mathrm{t}}$ and $T_{{\mathrm{C}}^{\prime }}$, respectively. (c) Exhibiting a second-order phase transition from the $F_{1∕2}$ phase to the $P$ phase for $d=-0.5$ and $h=0.25$; 0.38 is found $T_{{\mathrm{C}}^{\prime }}$. (d) Exhibiting a second-order phase transition from the $F_{3∕2}$ phase to the $P$ phase for $d=-0.475$ and $h=0.125$; 0.5575 is found $T_{\mathrm{C}}$. (e) Exhibiting two successive phase transitions, the first one is a first-order phase transition from the $F_{1∕2}$ phase to the $F_{3∕2}$ phase and the second one is second-order phase transition the $F_{3∕2}$ phase to the $P$ phase for $d=-0.475$ and $h=0.125$; 0.20 and 0.5575 found $T_{\mathrm{t}}$ and $T_{\mathrm{C}}$, respectively. (f) Same as Fig. 2d, but $d=0.25$ and $h=1.025$; 0.795 is found $T_{\mathrm{C}}$. (g) Exhibiting two successive phase transitions, the first one is a first-order phase transition from the $P$ phase to the $F_{3∕2}$ phase and the second one is second-order phase transition the $F_{3∕2}$ phase to the $P$ phase for $d=0.25$ and $h=1.025$; 0.48 and 0.795 found $T_{\mathrm{t}}$ and $T_{\mathrm{C}}$, respectively. (h) Same as Fig. 2c, but $d=-0.75$ and $h=0.35$; 0.1475 is found $T_{{\mathrm{C}}^{\prime }}$. (i) Same as Fig. 2g, but the first-order phase transition from the $P$ phase to the $F_{1∕2}$ phase and the second-order phase transition is from the $F_{1∕2}$ phase to the $P$ phase, and $d=-0.75$ and $h=0.35$; 0.08 and 0.1475 found $T_{\mathrm{t}}$ and $T_{{\mathrm{C}}^{\prime }}$, respectively. (j) Same as Fig. 2d, but $d=-0.475$ and $h=0.375$; 0.515 is found $T_{\mathrm{C}}$. (k) Same as Fig. 2e, but $d=-0.475$ and $h=0.375$; 0.04 and 0.515 are found $T_{\mathrm{t}}$ and $T_{\mathrm{C}}$. (l) Same as Fig. 2g, but $d=-0.475$ and $h=0.375$; 0.135 and 0.515 are found $T_{\mathrm{t}}$ and $T_{\mathrm{C}}$.

Figure 3

The values of the Liapunov exponents as a function the reduced temperature. Thick and thin lines represent the ${\lambda }_{\mathrm{s}}$, and ${\lambda }_{\mathrm{n}}$, ${\lambda }_{{\mathrm{n}}^{\prime }}$, respectively, and $T_{\mathrm{t}}$, and $T_{\mathrm{C}}$, $T_{{\mathrm{C}}^{\prime }}$ are the first- and second-order phase transition temperatures, respectively. (a) The system undergoes two successive phase transitions: First, the phase transition is a first-order, because one of the ${\lambda }_{\mathrm{n}}=0$ (${\lambda }_{\mathrm{n}}$ corresponds to the $F_{3∕2}$ phase) and the discontinuity occurs for the other ${\lambda }_{{\mathrm{n}}^{\prime }}$, which corresponds to the $F_{1∕2}$ phase. The first-order phase transition occurs at $T_t=0.20$ and the second one is a second-order phase transition at $T_{{\mathrm{C}}^{\prime }}=0.38$, because ${\lambda }_{\mathrm{n}}={\lambda }_{\mathrm{s}}=0$ for $\mathrm{d}=-0.5$ and $h=0.25$. (b) Same as Fig. 3a, but $d=-0.475$, $h=0.375$, and 0.04 and 0.515 are found $T_{\mathrm{t}}$ and $T_{\mathrm{C}}$, respectively.

Figure 4

Phase diagrams of the spin-3/2 Blume-Capel model in the $(T,h)$ plane. The paramagnetic $\left(P\right)$, ferromagnetic-$3∕2$ $\left(F_{3∕2}\right)$, ferromagnetic-$1∕2$ $\left(F_{1∕2}\right)$, and four different coexistence regions, namely the $F_{3∕2}+P$, $F_{1∕2}+P$, $F_{3∕2}+F_{1∕2}$, and $F_{3∕2}+F_{1∕2}+P$ regions, are found. Dashed and solid lines represent the first- and second-order phase transitions, respectively, and the dynamic tricritical points are indicated with solid circles. (a) $d=0.25$, (b) $d=-0.25$, (c) $d=-0.475$, (d) $d=-0.5$, (e) $d=-0.625$, and (f) $d=-0.75$.