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A tensor product state approach to spin-1/2 square $J_1$-$J_2$ antiferromagnetic Heisenberg model: evidence for deconfined quantum criticality

The ground state phase of spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model on square lattice around the maximally frustrated regime ($J_2\sim 0.5J_1$) has been debated for decades. Here we study this model using the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point $J_2^{c_1}=0.572(5)J_1$ and critical exponents $ν=0.50(8)$, $η_s=0.28(6)$. In the range of $0.572 < J_2/J_1 \leqslant 0.6 $ we find a paramagnetic ground state with exponentially decaying spin-spin correlation. Up to $24\times 24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent $η_p=0.24(1)$ and an anomalous columnar scaling exponent $η_c=0.28(1)$ at $J_2/J_1=0.6$. These results are consistent with a potential gapless $U(1)$ spin liquid phase. However, since the $U(1)$ spin liquid is unstable due to the instanton effect, a VBS order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point (DQCP) at $J_2^{c_1}$. Remarkably, all the observed critical exponents are consistent with the $J-Q$ model.