Ising nematic phase in ultrathin magnetic films: A Monte Carlo study

We study the critical properties of a two-dimensional Ising model with competing ferromagnetic exchange and dipolar interactions, which models an ultrathin magnetic film with high out-of-plane anisotropy in the monolayer limit. We present numerical evidence showing that two different scenarios appear in the model for different values of the exchange to dipolar intensities ratio, namely, a single first-order stripe-tetragonal phase transition or two phase transitions at different temperatures with an intermediate Ising nematic phase between the stripe and the tetragonal ones. Our results are very similar to those predicted by Abanov et al. [Phys. Rev. B 51, 1023 (1995)], but suggest a much more complex critical behavior than predicted by those authors for both the stripe-nematic and the nematic-tetragonal phase transitions. We also show that the presence of diverging free energy barriers at the stripe-nematic transition makes possible to obtain by slow cooling a metastable supercooled nematic state down to temperatures well below the transition one.

Figure 1

Energy per spin probability distributions (normalized histograms) for $\delta =2$ and $L=56$.

Figure 2

Orientational order parameter probability distributions (normalized histograms) for $\delta =2$ and $L=56$.

Figure 3

Typical spin configurations in the nematic phases for $\delta =2$ and $L=64$. (a) $T=0.78$, the orientational order parameter for this configuration is $\mid \eta \mid \approx 0.8$; (b) $T=0.807$ and $\mid \eta \mid \approx 0.59$.

Figure 4

Spatial correlation functions along the coordinate directions in a nematic phase for $\delta =2$, $T=0.78$, and $L=92$. The continuous line corresponds to a fitting using a function $f\left(r\right)=A\exp (-r∕\xi )\sin (k_{0}r+\varphi )$, with fitting parameters $k_0=1.47$, $\xi =17.4$, and $\varphi =1.63$. Notice that $k_0\approx \pi ∕2$, the wave vector of the striped structure with $h=2$ (ground state for $\delta =2$). The log-log plot of $C_x$ in the inset shows that it decays at large distances with a power law $r^{-\omega }$, with an exponent $\omega \approx 0.12$.

Figure 5

(Color online) (a) Static structure factors $S(k,0)$ and $S(0,k)$ at $T=0.78$ for different system sizes. The dashed lines correspond to Lorentzian fittings $f\left(k\right)=\lambda ∕({(k-k_0)}^2+{\lambda }^2)$. The upper inset shows the fitted parameter $k_0$ vs $L^{-1}$. The lower inset shows that the maximum of $S\left(\mathbf{k}\right)$ scales as $L^{-1}$. (b) $2L(S(k,0)+S(0,k))$ vs $k∕k_0$ for the same numerical data in (a); the continuous line is a Lorentzian fitting. The data collapse of the different curves shows that the whole curve $S\left(\mathbf{k}\right)$ scales as $L^{-1}$, as predicted by Eq. (8).

Figure 6

(Color online) Static structure factors $S(k,0)$ and $S(0,k)$ at $T=0.95$ for different system sizes. The continuous lines correspond to Lorentzian fittings $f\left(k\right)=\lambda ∕({(k-k_0)}^2+{\lambda }^2)$. The upper inset shows the fitted parameter $k_0$ vs $L^{-1}$. The lower inset shows that the maximum of $S\left(\mathbf{k}\right)$ scales as $L^{-2}$. It can be also shown that the whole curve $S\left(\mathbf{k}\right)$ scales as $L^{-2}$, as predicted by Eq. (10).

Figure 7

Histograms for $\delta =2$ and $L=64$, for temperatures around the nematic-tetragonal transition. (a) Energy per spin; temperatures from left to right: $T=0.804$, 0.805, 0.807, 0.810, and 0.815. (b) Orientational order parameter; same temperatures as in (a), but from right to left.

Figure 8

Histograms for $\delta =1$ and $L=48$. (a) Energy per spin; temperatures from left to right: $T=0.380$, 0.390, 0.395, 0.399, 0.401, 0.405, and 0.410. (b) Orientational order parameter; same temperatures as in (a), but from right to left.

Figure 9

Moments of the energy for $\delta =2$ and different system sizes. (a) Specific heat Eq. (11); (b) fourth-order cumulant Eq. (12).

Figure 10

Moments of the orientational order parameter Eq. (2) for $\delta =2$ and different system sizes. (a) Average of the absolute value; (b) associated susceptibility Eq. (13).

Figure 11

Finite size scaling of the pseudo-critical temperatures for the maxima of the specific heat ($T_1^c$ and $T_2^c$) and the minima of the fourth-order cumulant ($T_1^v$ and $T_2^v$) for $\delta =2$ and system sizes up to $L=64$. The inset shows a zoom of the same curves around $T_2$.

Figure 12

Finite size scaling of the pseudo-critical temperatures for the maxima of the susceptibility for $\delta =2$ and system sizes up to $L=64$.

Figure 13

Finite size scaling of the maxima of the specific heat at $T_1^c$ and $T_2^c$ for $\delta =2$ and system sizes up to $L=64$.

Figure 14

Finite size scaling of the energies per spin at which both maxima in the corresponding probability distribution occur, for the stripe-nematic phase transition for $\delta =2$ and system sizes up to $L=64$; for every value of $L$ the temperature is chosen such that $P\left(e_{\mathit{nem}}\right)=P\left(e_{\mathit{str}}\right)$.

Figure 15

Finite size scaling of the free energy barrier Eq. (14) between phases in the stripe-nematic and the nematic-tetragonal phase transitions for $\delta =2$.

Figure 16

Internal energy per spin $u$ vs $T$ for $\delta =1$ and $N=48\times 48$. Also shown best fits to the low temperature stripe and high temperature tetragonal regions, as described in the text. The best fit parameters for $u_{\mathit{str}}$ are $a_1=1952.54$; $a_2=10.8491$; $a_3=10.8441$; $a_4=5.73016$; $a_5=-58.0638$, and $a_6=7.34061$, and for $u_{\mathit{tetr}}$ are $b_0=-1.47926$; $b_1=0.0817682$; $b_2=0.0559538$ and $b_3=-3.05417$. The ground state energy is $u_g=-0.4677$.

Figure 17

Free energy per spin $f$ vs $T$ for $\delta =1$. The stripe and tetragonal free energies cross each other around $T_m=0.404$. The functions obtained from Eq. (15) are $f_{\mathit{str}}=u_g+\frac{a_1}{1-a_2}{T}^{a_2}+\frac{a_3}{1-a_4}{T}^{a_4}+\frac{a_5}{1-a_6}{T}^{a_6}$ and $f_{\mathit{tetr}}=\frac{b_0}{b_1}T\ln \left(\cosh \frac{b_1}{T}\right)+\frac{b_2}{b_3}T\ln \left(\cosh \frac{b_3}{T}\right)-T\ln \left(2\right)$, where ${\beta }_0=0$ for the tetragonal and ${\beta }_0=\infty$ for the striped phase.

Figure 18

Entropy per spin $s$ vs $T$ for $\delta =1$. The slight discontinuity at the transition temperature is consequence of a weak first-order phase transition.

Figure 19

Cooling-heating curves for $\delta =2$, $L=48$, and $r={10}^{-6}$. The insets show the same curves for a wider range of temperatures. (a) Average energy per spin; (b) average of the absolute value of the orientational order parameter.