We investigate the critical behavior of three-dimensional ferromagnetic CP^{N-1} models, which are characterized by a global U(N) and a local U(1) symmetry. We perform numerical simulations of a lattice model for N=2, 3, and 4. For N=2 we find a critical transition in the Heisenberg O(3) universality class, while for N=3 and 4 the system undergoes a first-order transition. For N=3 the transition is very weak and a clear signature of its discontinuous nature is only observed for sizes L≳50. We also determine the critical behavior for a large class of lattice Hamiltonians in the large-N limit. The results confirm the existence of a stable large-NCP^{N-1} fixed point. However, this evidence contradicts the standard picture obtained in the Landau-Ginzburg-Wilson (LGW) framework using a gauge-invariant order parameter: The presence of a cubic term in the effective LGW field theory for any N≥3 would usually be taken as an indication that these models generically undergo first-order transitions.
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