Phase transitions and multicritical behavior in the Ising model with dipolar interactions

In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width $n$ $\left(h_n\right)$ and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants $\delta$. The obtained results indicate that the $h_1$-TL phase transition line is continuous up to $\delta =1.2585$, while for the $h_2$-TL line a weak first-order character is found in the interval $1.2585\le \delta \le 1.36$ and becomes continuous for $1.37\le \delta \le 1.9$. This result suggests the existence of a tricritical point close to $\delta =1.37$. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on $\delta$ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the $h_1-h_2$ line, i.e., at $\delta =1.2585$, the critical phase corresponding to the $h_1-\mathrm{TL}$ transition coexists with the $h_2$ phase.

Figure 1

Dynamic evolution of the system at $\delta =1.2585$ and $T=0.226$, when it is started from DC. Up (down) spins are indicated with black (white) squares. The evolution times are indicated below each configuration.

Figure 2

Dynamic evolution of $O_{hv}^2,O_{h{v}_1}^2$, and $O_{h{v}_2}^2$ at $\delta =1.2585$ when the system is started from DC at the temperatures (a) $T=0.226$ and (b) $T=0.254$. The power-law fits are indicated with a solid line. More details are given in the text.

Figure 3

Dynamic evolution of (a) $O_{hv}$ and $O_{h{v}_1}$ at $\delta =1.2585$ when the system is started from the $h_1$ OC at the temperature $T=0.254$ and (b) $O_{hv}$ and $O_{h{v}_2}$ when the system is started from the $h_2$ OC at the temperature $T=0.260$. The insets show the evolution of the susceptibilities that exhibit power-law behavior at these temperatures. The solid lines correspond to the fits using the STD equations. More details are given in the text.

Figure 4

STD critical exponents versus $\delta$ obtained for both initial conditions (OC and DC) as indicated. The open symbols correspond to previously reported data in Ref. [16]. More details are given in the text.

Figure 5

Critical exponents versus $\delta$ obtained from initial OC. The open symbols correspond to previously reported data in Ref. [16]. More details are given in the text.

Figure 6

Phase diagram in the $T-\delta$ plane. Open circles are from Ref. [16], crosses are from [6], and pluses were taken from Ref. [15]. Solid symbols correspond to the data obtained in the present work; circles denote the critical temperatures, while the squares and diamonds are the spinodal temperatures $T_{\mathrm{up}}$ and $T_{\mathrm{down}}$, respectively. Dotted lines are to guide the eye, and the dashed line represents the first-order phase transition $h_1-h_2$ line that was taken from Ref. [6]. The inset shows the interval $1.2585\le \delta \le 1.36$ in more detail.

Figure 7

Dynamic evolution of the system at $\delta =1.23$: (a) $O_{hv}$ and $O_{h{v}_1}$ at $T=0.285$ from the ordered initial condition (OC). The inset shows the corresponding susceptibilities. (b) $O_{hv}^2$ and $O_{h{v}_1}^2$ at $T=0.286$ when the system is started from the DC. The solid lines represent the power-law fit by means of STD equations (6) and (7) in (a) and (10) in (b). More details are given in the text.

Figure 8

Estimation of the critical exponents obtained by using STD equations and taking $O_{hv}$ as the order parameter: (a) $\beta$, (b) $z$, (c) $\gamma$, and (d) $\nu$ as a function of $\delta$. The results corresponding to $O_{h{v}_1}$ are also included.