Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-1 2 J 1 -J 2 Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, Z 2 topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.DOI: 10.1103/PhysRevB.94.121111Introduction. Topological phases [1][2][3] are an intriguing form of quantum matter, which have been challenging theorists for the last two decades. Before then, it was believed that Landau symmetry-breaking theory [4] can explain ordering and phase transitions of matter through (spontaneous) breaking of a Hamiltonian symmetry. However, topological phases can preserve all symmetries and still acquire a finite energy gap. Topological phases fall into two broad categories, "intrinsic topological order" [3] on D 2 dimensional lattices, and "symmetry protected topological" (SPT) [5,6] order, which can also exist in one dimension (1D). For the former phase, there is no local unitary transformation to smoothly deform the state into a product state without passing through a phase transition, regardless of the existence of symmetries. The canonical example of an intrinsic topological order is the Z 2 ground state of the toric code [7]. On the other hand, SPTs are undeformable into product states only if protected by a symmetry. The best studied example is surely the Haldane phase of odd-integer spin chains [5,6], including the ground state of the exactly solved Affleck-Kennedy-Lieb-Tasaki (AKLT) [8] model. A key breakthrough was the realization that anyonic statistics associated with intrinsic topological order corresponds to fractionalization of symmetry. Therefore, when intrinsic topological order is coupled with lattice symmetries, the symmetries themselves fractionalize and lead to SPT ordering [9][10][11], which is readily detectable in many numerical methods.In 1973, Anderson [12] conjectured that the spin-
Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on six-leg infinite-length cylinders and scrutinize the properties of the detected phases. We compare these results with those of the corresponding nearest neighbor model. Our results suggest that, for such long-range Hamiltonians, the long-range quantum fluctuations always lead to long-range correlations, where correlators exhibit power-law decays instead of the conventional exponential drops observed for short-range correlated gapped phases. Our results are relevant for comparisons with recent ion-trap quantum simulator experiments that demonstrate highly-controllable long-range spin couplings for several hundred ions.
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