Ising Transition Driven by Frustration in a 2D Classical Model with Continuous Symmetry

We study the thermal properties of the classical antiferromagnetic Heisenberg model with both nearest ($J_1$) and next-nearest ($J_2$) exchange couplings on the square lattice by extensive Monte Carlo simulations. We show that, for $J_2/J_1>1/2$, thermal fluctuations give rise to an effective $Z_2$ symmetry leading to a finite-temperature phase transition. We provide strong numerical evidence that this transition is in the 2D Ising universality class, and that $T_c\to 0$ with an infinite slope when $J_2/J_1\to 1/2$.

Figure 1

(a) Finite-size susceptibility $\chi (L)/L$ and (b) Binder cumulant $U_4$ as a function of the temperature for different sizes; (c) Binder cumulant $U_4$ as a function of $1/L$ for different temperatures ($0.1965,0.1966,\dots ,0.1973$, from top to bottom). The lines are guides to the eye and the horizontal line marks the value for $U_4$ at the critical point for the Ising universality class [15]. All data are for $J_2/J_1=0.55$.

Figure 2

Inverse of the correlation length ${\xi }^{-1}$ as a function of the temperature for different sizes of the lattice and $J_2/J_1=0.55$. The critical exponent $\nu$ and $T_c$ can be extracted from the behavior of ${\xi }^{-1}$ as a function of the temperature. The arrow marks the resulting $T_c$.

Figure 3

(a) Specific heat per site $C$ as a function of the temperature for different sizes of the lattice and $J_2/J_1=0.55$. The lines are guides to the eye. (b) Maximum of the specific heat per site $C_{\max }(L)$ as a function of $L$. The line is a three-parameter fit (see text).

Figure 4

Monte Carlo results for the critical temperature $T_c$ as a function of the frustrating ratio $J_2/J_1$. The line is an extrapolation of the large $J_2$ data down to $J_2=0$ (see text).