Analysis of the phase transition for the Ising model on the frustrated square lattice

We analyze the phase transition of the frustrated $J_1$-$J_2$ Ising model with antiferromagnetic nearest- and strong next-nearest-neighbor interactions on the square lattice. Using extensive Monte Carlo simulations we show that the nature of the phase transition for $1/2<J_2/J_1\lesssim 1$ is not of the weakly universal type—as commonly believed—but we conclude from the clearly doubly peaked structure of the energy histograms that the transition is of weak first order. Motivated by these results, we analyze the phase transitions via field-theoretic methods; i.e., we calculate the central charge of the underlying field theory via transfer-matrix techniques and present, furthermore, a field-theoretic discussion on the phase-transition behavior of the model. Starting from the conformally invariant fixed point of two decoupled critical Ising models ($J_1=0$), we calculate the effect of the nearest-neighbor coupling term perturbatively using operator product expansions. As an effective action we obtain the Ashkin-Teller model.

Figure 1

Collinear phase of the $J_1-J_2$ square lattice. Red dots stand for up spins, and yellow dots stand for down spins. The two copies $A$ and $B$ of the Ising model with magnetic couplings $J_2$ and lattice spacing $a$ are, respectively, represented with dashed clear and dotted dark blue lines, while the black thin lines correspond to the $J_1$ square lattice. The shaded area represents the unit cell used to derive the continuum limit (Sec. 4b). The coordinates are indicated with respect to the $x$- and $y$ axis of the $A$ sublattice.

Figure 2

Critical temperatures for the phase transition from the paramagnetic into the magnetically ordered phase over the strength of frustration $J_1/J_2$. We adapted the energy scale of the temperature for $J_1/J_2<2$ to $J_2$ and for $J_1/J_2>2$ to $J_1$. Note that the frustration is also given in units of $J_2/J_1$ on the upper $x$ axis.

Figure 3

Energy histograms (solid black lines) for two values of $J_2=0.8J_1$ (left) and $J_2=0.9J_1$ (right) for a $L=1000\text{(left)}\text{and}L=2000$ (right) lattice and the reweighted histograms for slightly lower temperatures (dashed red lines). The two-peaked structure emphasizes the first-order character of the phase transition.

Figure 4

Calculations for the spin-spin correlation functions $\langle S_i{S}_j\rangle$ at $J_1=J_2$ for a $100\times 100$ periodic lattice along one direction. Thus maximal distance is given by $i-j=50$. (Top) Correlations inside the same sublattice – $i-j$ even – for different temperatures around $T_C$. (Middle) Correlations between spins on different sublattice sites – $i-j$ odd – in a larger scale. For all temperatures the correlations decay rapidly and go to zero. Hence, no long-range correlation is observable between the two sublattices. (Bottom) Doubly logarithmic plot of the correlation functions inside the same sublattice for three exemplary temperatures and their related fits.