Evidence of Kosterlitz-Thouless phase transitions in the Ising model with dipolar interactions

The ferromagnetic bidimensional Ising model with dipolar interactions has been proposed to model ultrathin films with strong out-of-plane anisotropy. The phase diagram presents a rich phenomenology that includes low-temperature phases characterized by stripes of width $n$ ($h_n$) and a high-temperature phase with domains of stripes with mutually perpendicular orientations, named tetragonal liquid (TL). The latter phase can be reached by two possible ways. One of them is the direct transition $h_n$ to TL, and the other one is through an intermediate phase with orientational order but short-range positional disorder, named nematic phase (NM). The regions of the phase diagram where these transitions occur, as well as their character, remain an open question and are the object of the present work. In order to clarify this topic, intensive Monte Carlo simulations were performed by employing short-time dynamics as the main tool for studying the phase transition behavior. The dynamic evolution of the orientational order parameter and its moments are measured for selected values of the ratio between the ferromagnetic exchange and dipolar constants, called $\delta$. The obtained results indicate that the intermediate NM phase is present for $\delta \ge 2$ in narrow ranges of temperatures. Also, the results suggest that both transitions, i.e., $h_n$-NM and NM-TL, have a Kosterlitz-Thouless character. This type of topological transition is observed in continuous bidimensional models and have been proposed for discrete ones, as in the case of the present work.

Figure 1

Log-log plot of the $O_{hv}^2$ time evolution from the paramagnetic initial condition at the indicated temperatures. The system size and interaction parameter used are $L=128$ and $\delta =2$, respectively. Due to superposition of the curves for different temperatures, the data were shifted by a multiplicative factor with the aim of clearly showing their power-law behavior. The inset exhibits the original data (i.e., they are not multiplied by a factor) corresponding to the temperatures at which departures from the power law are observed. For the sake of comparison, $T=0.805$ is also included.

Figure 2

Dynamic exponent of $O_{hv}^2\left(t\right)$ as a function of the temperature, corresponding to the paramagnetic initial condition and $\delta =2$. The solid lines are the linear fits used to estimate the maximum value. More details in the text.

Figure 3

Typical dynamic evolution of the spin configurations during the quenching from high-temperature paramagnetic state to $T=0.79$ [(a)–(c)] and $T=0.83$ [(d)–(f)]. The data correspond to $L=128$ and $\delta =2$. More details in the text.

Figure 4

Typical dynamic evolution of the spin configurations during the annealing from ground state to $T=0.79$ [(a)–(c)] and $T=0.83$ [(d)–(f)]. The data correspond to $L=128$ and $\delta =2$. More details in the text.

Figure 5

Nonequilibrium relaxation of $O_{hv}$,corresponding to $\delta =2$, from the ground-state initial configuration at temperatures within the interval (a) $0.60\le T\le 0.80$ and (b) $0.81\le T\le 0.90$ from top to bottom in both cases. The inset of figure (a) shows the susceptibility $\chi \left(t\right)$ for $0.76\le T\le 0.80$ (from bottom to top). The continuous lines correspond to power-law fits.

Figure 6

Scaling of the nonequilibrium relaxation of $O_{hv}$ from the initial ground state, at the temperatures showed in Fig. 5, by using Eq. (7) with $\lambda =0.0065$. The inset shows the obtained relaxation time as a function of the temperature. The fit with Eq. (8) is also shown with a dashed line.

Figure 7

Phase diagram in the plane $T-\delta$ for $2\le \delta \le 3.6$. The temperatures $T_1$ (triangles) and $T_2$ (squares) were obtained by means of the relaxation and STD analysis from the initial $h_n$ and paramagnetic states, respectively. Dotted lines are for eye guidance and dashed lines represent the phase boundaries between phases $h_n$ and $h_{n+1}$. For the sake of comparison, the data taken from reference [29] are also included: open circles and diamonds refer to the transitions from the high temperature TL to $h_n$ (or NM) and NM to $h_n$, respectively. The inset shows the whole phase diagram, starting from $\delta =0$, where the crosses correspond to the transition $h_n$-TL taken from references [33, 34].

Figure 8

Log-log plot of the $O_{hv}^2$ dynamic evolution from the initial paramagnetic condition at the indicated temperatures for $L=128$ and $\delta =3$. As in the case of $\delta =2$ (see Fig. 1), the data corresponding to different temperatures were shifted by a constant multiplicative factor with the aim of clearly showing their power-law behavior. The inset exhibits the data corresponding to the temperatures at which departures from the power law are observed; for the shake of comparison, $T=1.36$ is also included.

Figure 9

Dynamic exponent of $O_{hv}^2$ as a function of the temperature for the initial paramagnetic condition and $\delta =3$. The solid line corresponds to the linear fits used to determine the maximum value. More details in the text.

Figure 10

Typical dynamic evolution of the spin configurations during quenching from the high-temperature paramagnetic state to $T=1.315$ [(a)–(c)] and $T=1.50$ [(d)–(f)]. The data correspond to $L=128$ and $\delta =3$.

Figure 11

Typical dynamic evolution of the spin configurations during annealing from the ground state to $T=1.315$ [(a)–(c)] and $T=1.50$ [(d)–(f)]. The data correspond to $L=128$ and $\delta =3$.

Figure 12

Nonequilibrium relaxation of $O_{hv}$, corresponding to $\delta =3$,from initial ground state at temperatures within the interval (a) $1.24\le T\le 1.36$ and (b) $1.37\le T\le 1.50$ from top to bottom, in both cases. The inset of figure (a) shows the susceptibility $\chi \left(t\right)$ for $1.24\le T\le 1.35$ (from bottom to top). The continuous lines correspond to power-law fits.

Figure 13

Scaling of the $O_{hv}$ relaxation from the initial ground state at temperatures within the interval $1.37\le T\le 1.50$ by using Eq. (7) with $\lambda =0.045$. The inset shows the relaxation time $\tau \left(T\right)$ together with the fit with Eq. (8) indicated with a dashed line.