| A Quantum spin liquid state is an exotic state of matter where the quantum fluctuations have melted away any form of magnetic order or valence bond order even at zero temperature. Progress in finding realistic spin Hamiltonians with spin-liquid (SL) ground states has accelerated dramatically in the last two years. We carefully study the phase diagrams of various frustrated spin 1/2 systems using the density matrix renormalization group on cylinders with widths up to 12 lattice spacings and total sizes up to 400 sites to search for a spin liquid phase.;For the Heisenberg anti-ferromagnetic model with first neighbor J1 = 1 and second neighbor J2 interactions on the honeycomb lattice, we find an apparently paramagnetic phase at J2 = 0.3, bordered by an antiferromagnetic phase for J2 0.36. The longest correlation length that we find in this intermediate phase is for plaquette valence bond (PVB) order. This correlation length grows strongly with cylinder circumference, indicating either quantum criticality or weak PVB order. For the J1 - J2 XY anti-ferromagnetic interaction model on the honeycomb lattice, we find a surprising antiferromagnetic Ising phase, with ordered moments pointing along the z axis for the intermediate frustration regime 0.22 < J2 < 0.36, despite the absence of any S zSz interactions in the Hamiltonian. Describing the system instead as hard-core bosons with frustrated hopping, this Ising phase is then a Mott insulator with one boson per two-site unit cell, and the Ising order is then CDW order that breaks the Z2 sublattice symmetry of the unit cell. Even with third neighbor J 3 interactions included in these two Hamiltonians, we have yet to find a spin liquid phase.;For the J1 - J2 Heisenberg model on the triangular lattice, we find that there is a possible SL state at J2 = 0.1. On even cylinders, we find an energy splitting between the even and the odd topological sectors, which decays exponentially as the cylinder gets wider. On odd cylinders, we observe the dimerization effect with a dimerization order parameter that decays exponentially with the cylinder's width. These two features are consistent with a Z2 topological spin liquid state. |
