Phase transitions in the frustrated Ising model on the square lattice

Abstract

We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions ${J}_{1}<0$ (nearest neighbor, ferromagnetic) and ${J}_{2}>0$ (second neighbor, antiferromagnetic). The striped phase breaks a ${Z}_{4}$ symmetry and is stabilized at low temperatures for $g={J}_{2}/|{J}_{1}|>1/2$. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The ${J}_{1}$-${J}_{2}$ model has a line of very weak first-order phase transitions in the whole region $1/2<g<{g}^{*}$, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from $g={g}^{*}$ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point ${g}^{*}$ in the ${J}_{1}$-${J}_{2}$ model. The continuous transitions near ${g}^{*}$ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of $g>1/2$. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.