Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model

We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground state properties of the spin-1/2 antiferromagnetic square lattice Heisenberg $J_1$-$J_2$ model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\le J_2\mathrm{/}{J}_1\le 0.62$, separating conventional N'eel and striped antiferromagnetic states for smaller and larger $J_2\mathrm{/}{J}_1$, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and non-zero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a non-zero topological entanglement entropy $\gamma =0.70\pm 0.02$, extremely close to $\gamma =\ln (2)\approx 0.69$ (expected for a $Z_2$ quantum spin liquid) and a non-trivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of {\sl opposite} sign to the topological entanglement entropy.