$U(1)$-symmetric infinite projected entangled-pair state study of the spin-$1/2$ square $J_{1}-J_{2}$ Heisenberg model

Abstract: We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair state (iPEPS) ansatz to investigate the ground state phase diagram of the spin-$1/2$ square $J_{1}-J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound, improving previous tensor-network-based results. By studying the finite-$D$ scaling of the magnetically order parameter, we find a N\'{e}el phase for $J_2/J_1<0.53$. For $0.53<J_2/J_1<0.61$, a non-magnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\xi \sim D^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'{e}el-to-VBS transition at $J^{c_1}_{2}/J_1=0.530(5)$, where estimated critical anomalous exponents are $\eta_{s} \sim 0.6$ and $\eta_{d} \sim 1.9$ for spin and dimer correlations respectively. We show that the associated VBS order parameter monotonically increases with $J_2/J_1$ and finally a first-order quantum phase transition takes place at $J^{c_2}_{2}/J_1=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.