Nematic phase in the $J_1\text{−}{J}_2$ square-lattice Ising model in an external field

The $J_1\text{−}{J}_2$ Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter $\kappa =J_2/\left|J_1\right|$ which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Figure 1

Sketch of the ground state configurations for the $J_1\text{−}{J}_2$ model without external field. Empty circles: $S_i=-1$; filled circles: $S_i=+1$.

Figure 2

Reduced external field versus reduced temperature phase diagram for $\kappa =1$. Full (dotted) lines correspond to continuous (discontinuous) transitions.

Figure 3

Correlation functions versus reduced magnetic field for $\kappa =1$ and $T/|J_1|=0.8$. (a) Positional $\left(M_s\right)$ and orientational order parameters $\left(Q\right)$; (b) local magnetizations; (c) nearest-neighbor correlations; (d) next-nearest-neighbor correlations; (e) three site correlations; (f) square correlations.

Figure 4

Reduced external field versus reduced temperature phase diagram for $\kappa =0.6$. Full (dotted) lines correspond to continuous (discontinuous) transitions.

Figure 5

Correlation functions versus reduced magnetic field for $\kappa =0.6$ and $T/|J_1|=0.65$. (a) Positional $\left(M_s\right)$ and orientational order parameters $\left(Q\right)$; (b) local magnetizations; (c) nearest-neighbor correlations; (d) next-nearest-neighbor correlations; (e) three-site correlations; (f) square correlations.

Figure 6

Results for the $J_1\text{−}{J}_2$ model with $\kappa =0.60$ and $h=0$. Different system sizes (as indicated in the legends) are considered. The results are shown as a function of the reduced temperature. (a) Orientational order parameter (OOP); (b) translational order parameter (TOP); (c) mean energy per site $\langle \mathcal{H}\rangle /L^2$; (d) heat capacity per site at constant field $c_h\equiv {(\partial [\mathcal{H}/L^2]/\partial T)}_h$.

Figure 7

Results for the $J_1\text{−}{J}_2$ model with $\kappa =0.60$ and $T/|J_1|=0.80$. Different system sizes (as indicated in the legends) are considered. The results are shown as a function of the reduced external field, $h/|J_1|$. (a) Orientational order parameter (OOP); (b) translational order parameter (TOP); (c) magnetization per site, $m$; (d) ${\chi }_T\equiv {(\partial m/\partial h)}_T$.

Figure 8

Results for the $J_1\text{−}{J}_2$ model with $\kappa =0.60$ and $T/|J_1|=0.6667$. Different system sizes (as indicated in the legends) are considered. The results are shown as a function of the reduced external field, $h/|J_1|$. (a) Orientational order parameter (OOP); (b) translational order parameter (TOP); (c) magnetization per site, $m$; (d) ${\chi }_T\equiv {(\partial m/\partial h)}_T$-.

Figure 9

Order parameter cumulants, $g_4$ as a function of the reduced external field for two reduced temperatures, different system sizes (see the labels and legends in the plots) and $\kappa =0.6$.

Figure 10

Representative configurations for $\kappa =0.6,h=0,L=256$, close to the order disorder transition. From left to right $T/|J_1|\simeq 0.9756$ (isotropic), $T/|J_1|=0.9718$ (estimated transition temperature), and $T/|J_1|=0.9662$ (ordered phase with orientational and translational order).

Figure 11

Representative configurations for $\kappa =0.6,T/|J_1|=0.50,L=256$, close to the order disorder transitions. From left to right $h/|J_1|=-0.400$ (isotropic), $h/|J_1|=-0.397$ (nematic phase), and $h/|J_1|=-0.390$ (ordered phase with orientational and translational order). The typical length of the rods grows from left to the right. The densities in the lattice gas model are about (from left to right): 0.12, 0.21, and 0.42.