Tensor network simulation for the frustrated $J_1$-$J_2$ Ising model on the square lattice

By using extensive tensor network calculations, we map out the phase diagram of the frustrated $J_1$-$J_2$ Ising model on the square lattice. In particular, we focus on the cases with controversy in the phase diagram, especially the stripe transition in the regime $g=\mathrm{\mid }{J}_2\mathrm{/}{J}_1\mathrm{\mid }>\frac{1}{2}$, $(J_2>0,J_1<0)$. While recent studies claimed that the phase transition is of first order when $\frac{1}{2}<g<g^{\ast }$ (with the smallest $g^{\ast }$ being $0.67$), our simulations suggest that if there is such a first-order region, it is smaller than those found in earlier studies by other methods. Combining with the analysis of critical properties, we provide evidence that the classical $J_1$-$J_2$ model evolves continuously from two decoupled Ising models ($g\to \mathrm{\infty }$ with central charge $c=1$) to a point belonging to the tricritical Ising universality class (with $c=0.7$) as $g$ decreases to $g^{\ast }\simeq 0.54$.