We use the density matrix renormalization group method to calculate several energy eigenvalues of the frustrated S=1/2 square-lattice J_{1}-J_{2} Heisenberg model on 2L×L cylinders with L≤10. We identify excited-level crossings versus the coupling ratio g=J_{2}/J_{1} and study their drifts with the system size L. The lowest singlet-triplet and singlet-quintuplet crossings converge rapidly (with corrections ∝L^{-2}) to different g values, and we argue that these correspond to ground-state transitions between the Néel antiferromagnet and a gapless spin liquid, at g_{c1}≈0.46, and between the spin liquid and a valence-bond solid at g_{c2}≈0.52. Previous studies of order parameters were not able to positively discriminate between an extended spin liquid phase and a critical point. We expect level-crossing analysis to be a generically powerful tool in density matrix renormalization group studies of quantum phase transitions.

We construct a class of projected entangled pair states (PEPS) which is exactly the resonating valence bond (RVB) wavefunctions endowed with both short range and long range valence bonds. With an energetically preferred RVB pattern, the wavefunction is simplified to live in a one parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin 1/2 J1 − J2 antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with even or odd number of fluxes penetrating the cylinder and even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin liquid phase at the optimum parameter. Introduction -Resonant valence bond (RVB) states, which were first proposed by Anderson [1] to describe a possible ground state for the S = 1/2 antiferromagnetic Heisenberg model on a triangular lattice, and later to explain the possible mechanism of high-T c cuprates [2, 3], provide us a rich tool box to construct the so called spin liquid states. The equal weight superposition of the nearest neighbor (NN) RVB state on square lattice was shown to be critical [4,5]. Several numerical work [6][7][8][9] have demonstrated that the equal weight NN RVB states on the kagome and triangular lattices are Z 2 spin liquid states. Recently numerical breakthroughs claimed a spin liquid ground state for the Kagome Heisenberg model [10,11] and the frustrated spin 1/2 J 1 − J 2 antiferromagnetic (AF) Heisenberg model on the square lattice [12,13]. However, these work did not give direct access to the topological nature of the spin liquid states, therefore, a simple variational wavefunction approach is highly desirable. Although the variational energy of the NN RVB state on the kagome lattice [7,9] is still higher than the energy obtained via the density matrix renormalization group (DMRG) method [10], the topological nature is well understood within the formalism of the projected entangled pair states (PEPSs) [7]. On the other hand, from a projective wavefunction [14] approach supplemented by a projective symmetry group (PSG) analysis all possible spin-liquid states on the triangular [15], and Honeycomb [19] lattices have been obtained and classified but, for all lattices, the energetically favorable states are believed to involve longer range RVB. As a result, it is natural to think that a general