Generalized CP¹ model from t₁-t₂-J model

A long-wavelength, low-frequency effective theory is obtained from $t_1-t_2-J$ model. The action is written in terms of two-component bose spinor fields (CP^1 fields) and two spinless Fermi fields. The generalized CP^1 model is invariant under U(1) gauge transformations. The bose fields and one of the Fermi fields have charge +1 while the other Fermi field has charge -1 with respect to these transformations. A simple mean-feild theory of a gauge-symmerty breaking, based on a four-fermion interaction, is discussed. An effective theory of frustrated antiferromagnetism is obtained integrating out the Fermi fields around the mean-fields. Another option is used to parametrize the long distance fluctuations in $t_1-t_2-J$ model, with the help of gauge invariant fields. It is argued that the resulting Fermi quasiparticles of the $t_1-t_2-J$ model have both charge and spin. The effective action is rewritten in terms of spin 1/2 Fermi spinor, which has the charge of the holes, and unit vector.