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Quantum triangular ice in the easy-axis ferromagnetic phase

We use spin wave theory to investigate the ground state properties of the $Z_2$-invariant quantum XXZ model on the triangular lattice in the ferromagnetic phase. The Hamiltonian comprises nearest and next-nearest-neighbour Ising couplings, external magnetic fields, and a $Z_2$-invariant ferromagnetic coupling. We show that quantum fluctuations are suppressed in this system, hence linear spin wave theory gives reasonable estimates of the ground state thermodynamic properties. Our results show that, at half-filling (zero magnetic fields), the spontaneous breaking of $Z_2$ symmetry leads to a ferromagnetic phase whose energy spectrum is gapped at all excitations with a maxon dispersion at $\mathbf{k}=0$. This is in sharp contrast to rotational invariant systems with a vanishing phonon dispersion. We show that the $\mathbf{k}=0$ mode enhances the estimated values of the thermodynamic quantities. We obtain the trends of the particle density and the condensate fraction. The density of states and the dynamical structure factors exhibit fascinating peaks at unusual wave vectors, which should be of interest.