We present a new scenario for the breakdown of ferromagnetic order in two-dimensional quantum magnets with competing ferromagnetic and antiferromagnetic interactions. Dynamical effects lead to the formation of magnon bound states, which undergo Bose-Einstein condensation, giving rise to magnetic multipole order. This scenario is explored in some detail for extended Heisenberg models on a square lattice and a triangular lattice. On a square lattice, two-magnon bound states are most stable, giving rise to bond-centered spin nematic (quadratic) order. In particular, we present numerical evidence confirming the existence of a state with d-wave nematic correlations but no long range spin order, lying between the saturated ferromagnetic and antiferromagnetic phases. On the other hand, in a multiple spin exchange model on a triangular lattice, three-magnon bound states are most stably formed, leading to the appearance of magnetic octupole order.
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