The anisotropic Heisenberg model close to the Ising limit: triangular lattice vs. effective models
Stimulated by recent experiments on materials representing the realization of the anisotropic Heisenberg spin-$1/2$ model on the triangular lattice, we explore further properties of such a model in the easy-axis regime $α= J_\perp/J_z < 1$ and the plausibility of finding effective models that capture similar physics. We show that, at finite fields, the magnetization curve as well as the transverse magnetization (superfluid) order parameter $m_\perp$ of the triangular lattice model are indeed qualitatively reproduced by anisotropic Heisenberg models on the honeycomb or the square lattice. At the point of correspondence to the zero-field triangular lattice model, however, the bipartite models are qualitatively different as they remain gapless even at $α\ll 1$ with a small but finite $m_\perp >0 $. Conversely, we present several additional numerical studies of the full model on the triangular lattice which support the appearance of a gap at zero field and $α\ll 1$. In particular, the magnetization curve $m(h)$ as well as the spin stiffness $ρ_s$ indicate a transition/crossover from gappless to gapped regimes at $α\sim α^*$ with $α^* \lesssim 0.5$. We also show that deviations from the linear spin-wave theory and the emergence of the gap can be traced back to the strong effective repulsion between magnon excitations, showcasing similarity to strongly correlated systems.