Spin liquids are important for a fundamental understanding of quantum magnetism and may find applications in quantum computing, but it is still not clear under which general circumstances these exotic, quantum-disordered phases occur. To gather deeper insights, we analyze a generalization of the spatially anisotropic triangular lattice (SATL), the spatially completely anisotropic triangular lattice (SCATL), using Takahashi's modified spin-wave theory, complemented by exact diagonalizations. For Heisenberg interactions, calculations on the SATL predict quantum disorder for several materials in which experiments have found magnetic long-range order. In the SCATL, the corresponding parameter values yield ordered ground states and this discrepancy disappears. We also study the model with
XY interactions, which can be implemented with current technology in ultracold atoms in optical lattices. This allows to estimate the stability of quantum-disordered phases towards imperfect implementations of the coupling parameters. For both kinds of interactions, we find indications for extended quantum-disordered phases. In part of the parameter space of the
XY model, exact diagonalization hints at a gapped state with chiral long-range order. In both models, our results suggest that two gapped nonmagnetic regions, identified as distinct in the SATL, could actually be continuously connected via the additional anisotropy of the SCATL. Further, we find that different kinds of order are always separated by disordered phases. We propose that this is a quite universal feature of two-dimensional frustrated antiferromagnets with continuous symmetry in the couplings. The SCATL may therefore---besides being relevant for experiments---yield fundamental insight into quantum-disordered phases.
We investigate the extended hard-core Bose-Hubbard model on the triangular lattice as a function of spatial anisotropy with respect to both hopping and nearest-neighbor interaction strength. At half-filling the system can be tuned from decoupled one-dimensional chains to a two-dimensional solid phase with alternating density order by adjusting the anisotropic coupling. At intermediate anisotropy, however, frustration effects dominate and an incommensurate supersolid phase emerges, which is characterized by incommensurate density order as well as an anisotropic superfluid density. We demonstrate that this intermediate phase results from the proliferation of topological defects in the form of quantum bosonic domain walls. Accordingly, the structure factor has peaks at wave vectors, which are linearly related to the number of domain walls in a finite system in agreement with extensive quantum Monte Carlo simulations. We discuss possible connections with the supersolid behavior in the high-temperature superconducting striped phase.