We apply continuous similarity transformations (CSTs) to the easy-axis antiferromagnetic XXZ model on the square lattice. The CST flow equations are truncated in momentum space by the scaling dimension d so that all contributions with d≤2 are taken into account. The resulting quartic magnon-conserving effective Hamiltonian is analyzed in the zero-, one-, and two-magnon sector. In this way, a quantitative description of the ground-state energy, the one-magnon dispersion, and its gap as well as of two-magnon bound states is gained for anisotropies ranging from the gapped Ising model to the gapless Heisenberg model. We discuss the critical properties of the gap closing as well as the evolution of the one-magnon roton mininum. The excitation energies of two-magnon bound states are calculated, and their decay into the two-magnon continuum is determined via the inverse participation ratio.
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