The spin-$1/2$ $J_1-J_2$ Heisenberg antiferromagnet on a square lattice:a plaquette renormalized tensor network study

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E, \textbf{83}, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$-${J}_{2}$ model on an $L\times L$ square lattice with $L$=8,16 and 32. By treating tensor elements as variational parameters, we obtain the ground states for different $J_2/J_1$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations and plaquette order parameters. In addition to the well-known N\'{e}el-order and collinear-order at low and high ${J}_2/{J}_1$, we observe a plaquette-like order at ${J}_2/J_1\approx 0.5$. A continuous transition between the N\'{e}el order and the plaquette-like order near $J_2^{c_1}\approx 0.40 J_1$ is observed. The collinear order emerges at ${J}_2^{c_2} \approx 0.62J_1$ through a first-order phase transition.