Weak first-order phase transitions in the frustrated square lattice J1-J2 classical Ising model

The classical $J_1$-$J_2$ Ising model on the square lattice is a minimal model of frustrated magnetism whose phase boundaries have remained under scrutiny for decades. Signs of first-order phase transitions have appeared in some studies, but strong evidence remains lacking. The current consensus, based upon the numerical data and theoretical arguments in [S. Jin et al., Phys. Rev. Lett. \textbf{108}, 045702 (2012)], is that first-order phase transitions are ruled out in the region $g = J_2/|J_1|\gtrsim 0.67$. We point out a loophole in the basis for this consensus, and we find strong evidence that the phase boundary is instead weak first-order at $0.67\lesssim g<\infty$ such that it asymptotically becomes second-order when $g\rightarrow\infty$. We also find strong evidence that the phase boundary is first-order in the region $0.5<g\lesssim0.67$. We establish these results with adiabatic evolution of matrix product states directly in the thermodynamic limit, and with the theory of finite entanglement scaling. We also find suggestive evidence that when $g\rightarrow0.5^+$, the phase boundary becomes of an anomalous first-order type wherein the correlation length is very large in one of the coexisting phases but very small in the other.