We study the spin-1 2Heisenberg model on the square lattice with first-and second-neighbor antiferromagnetic interactions J1 and J2, which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z2 spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU (2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J2, we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J2/J1 0.44. For 0.5 < J2/J1 < 0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44 < J2/J1 < 0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies. Introduction.-Quantum spin liquid (SL) is an exotic state of matter where a spin system does not form magnetically ordered state or break lattice symmetries even at zero temperature [1]. Understanding spin liquids is important in frustrated magnetic systems and may also hold clues to understanding the non-Fermi liquid of doped Mott materials and high-T c superconductivity [2]. While the exciting properties of SL such as deconfined quasiparticles and fractional statistics have been revealed in many artificially constructed systems [3][4][5][6][7][8][9][10][11][12], the possibility of finding spin liquids in realistic Heisenberg models has attracted much attention over the past 20 years due to its close relation to experimental materials. The prominent example is the kagome antiferromagnet, where recent density matrix renormalization group (DMRG) studies point to a gapped Z 2 SL [10,[13][14][15][16] characterized by a Z 2 topological order and fractionalized spinon and vison excitations [17][18][19][20][21].One of the candidate models for SL is the spin-

The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.

We study the quantum phase diagram of the spin-1/2 Heisenberg model on the kagomé lattice with first-, second-, and third-neighbor interactions J1, J2, and J3 by means of density matrix renormalization group. For small J2 and J3, this model sustains a time-reversal invariant quantum spin liquid phase. With increasing J2 and J3, we find in addition a q = (0, 0) Néel phase, a chiral spin liquid phase, an apparent valence-bond crystal phase, and a complex non-coplanar magnetically ordered state with spins forming the vertices of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and cuboc1 phase break time reversal symmetry in the sense of spontaneous scalar spin chirality. We show that the chiralities in the chiral spin liquid and cuboc1 are distinct, and that these two states are separated by a strong first order phase transition. The transitions from the chiral spin liquid to both the q = (0, 0) phase and to time-reversal symmetric spin liquid, however, are consistent with continuous quantum phase transitions.

We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second (J2) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120• magnetic order phase for J2 0.07J1 and a stripe antiferromagnetic phase for J2 0.15J1. Between these two phases, we identify a spin liquid region characterized by the exponential decaying spin and dimer correlations, as well as the large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin liquid candidates in this region. While the sector with spinon is found to respect the time reversal symmetry, the even sector without a spinon breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin liquid phase and find the optimal parameter region for the gapped Z2 spin liquid. [30,31]. In all these materials, no magnetic order is observed at the temperature much lower than the interaction energy scale. These experimental findings have inspired intensive theoretical studies on the frustrated magnetic systems with strong frustration or competing interactions.Theoretically, the kagome Heisenberg model appears to possess a robust SL. Density matrix renormalization group (DMRG) studies suggest a gapped SL [32][33][34][35], which may be consistent with a Z 2 topological order [34,35]. Variational studies based on the projected fermionic parton wave functions however favor a gapless Dirac SL [36][37][38]. Interestingly, by introducing the second and third neighbor couplings [39][40][41] or the chiral interactions [42], DMRG [40-42] studies recently discovered another topological SL -chiral spin liquid (CSL) [43,44], which breaks time reversal symmetry (TRS) spontaneously and is identified as the ν = 1/2 bosonic fractional quantum Hall state. On the other hand, the nonmagnetic phases in the frustrated honeycomb and square J 1 -J 2 models appear to be conventional valence-bond solid state [45][46][47][48].The spin-1/2 triangular nearest-neighbor antiferromagnetic [69] found the indication of a gapped SL which conserves the TRS in the non-magnetic phase. However, the nature of the quantum phase with the intermediate J 2 remains far from clear.In this Letter, we study the spin-1/2 triangular model with the AF first and second nearest-neighbor J 1 (J 1 )-J 2 couplings based on DMRG calculations. The model Hamiltonian is given aswhere the sums i, j and i, j run over all the first-and second-neighbor bonds, respectively. The first-neighbor couplings J 1 and J 1 are for the vertical and zigzag bonds as shown in Fig. 1(a). We study most systems with J 1 = J 1 unless we specify otherwise. We set J 1 = 1 as the energy scale...

We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-1 2Heisenberg model on a honeycomb lattice with first (J 1 ) and second (J 2 ) neighbor antiferromagnetic interactions, where a Z 2 spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size N = 2 × 6 × 6. With increasing J 2 , we find a Néel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment m 2 s on finite-size cylinders to the thermodynamic limit, we find the Néel order vanishing at J 2 /J 1 0.22. For 0.25 < J 2 /J 1 0.35, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For 0.22 < J 2 /J 1 < 0.25, both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.