Location of the Potts-critical end point in the frustrated Ising model on the square lattice

We report on Monte Carlo simulations for the two-dimensional frustrated $J_1$-$J_2$ Ising model on the square lattice. Recent analysis has shown that for the phase transition from the paramagnetic state to the antiferromagnetic collinear state different phase-transition scenarios apply depending on the value of the frustration $J_2/J_1$. In particular a region with critical Ashkin-Teller-like behavior, i.e., a second-order phase transition with varying critical exponents, and a noncritical region with first-order indications were verified. However, the exact transition point ${[J_2/J_1]}_C$ between both scenarios was under debate. In this paper we present Monte Carlo data which strengthens the conclusion of Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)] that the transition point is at a value of $J_2/J_1\approx 0.67$ and that double-peak structures in the energy histograms for larger values of $J_2/J_1$ are unstable in a scaling analysis.

Figure 1

Phase diagram for the frustrated Ising model for $J_2>J_1/2$. Two different regions are identified where the phase transitions show different behavior. In particular the shaded area in the middle is discussed in this work by means of histograms and critical exponents ($\nu$, red triangles; $\gamma$, blue diamonds). Data for the critical temperatures (green circles) are from Ref. 7.

Figure 2

Histograms for different lattice sizes recorded for critical (size-dependent) temperatures at $J_2=0.8J_1$. After the emergence of a double peak at intermediate system sizes $L=1000,1200$ the vanishing of this first-order signature is observed for even larger systems ($L=1500,2000$).

Figure 3

Computation of critical exponents $\nu$ and $\gamma$ from the Binder cumulants and susceptibility of the magnetic order parameter at $J_2=0.8J_1$. (a) and (c) show size-dependent values of the derivative ${[dU/dT]}_{T_C}$ and susceptibility ${[\chi /L^2]}_{T_C}$ and algebraic fits that yield $\nu =0.78\left(1\right)$ and $\gamma =1.36\left(1\right)$. In (b), additionally, a scaling collapse of the Binder cumulant is shown for given $T_C$ and $\nu$; three original Binder cumulants are shown in the inset.

Figure 4

Flowgrams for the susceptibility $\chi$ at the transition temperature $T_C$ for several ratios $J_2/J_1$. Only for $J_2\gtrsim 0.7J_1$ is the scaling nearly constant in $L^{-7/4}$ and Ashkin-Teller-like behavior obtained.