Induced Hopf term in the nonlinear σ model

We study the statistics of solitons in the O(3) nonlinear σ model in 2+1 dimensions coupled to fermions with an arbitrary dispersion relation. We show that P and T violation is not a sufficient condition for the appearance of anyons in the spectrum. We give a simple formula for the coefficient of the induced Hopf term, and formulas for the induced Chern-Simons term and fermion number, where appropriate. They depend only on the high- and low-momentum behavior of the fermion dispersion relation, and are stable against perturbations (such as anisotropies) of the dispersion relation which preserve its asymptotic behavior. We apply our results to models of high-T_c materials, and find that, even with P and T violation, anyons in general do not emerge. In a relativistic model, the soliton is always a fermion. We discuss a hypothetical model in which anyons are generated.