Long-wavelength effective gauge theory of the doped planar antiferromagnet

The dynamics of holes in a planar, quantum antiferromagnet is modeled using a suitably extended t-J Hamiltonian. A ‘‘slave fermion’’ ansatz is used to express the theory in terms of separate charge and spin variables. A gauge symmetry of the model is identified and an approximate effective Hamiltonian is given in terms of an Abelian gauge potential. The long-wavelength limit is shown to be a relativistic theory of two species of Dirac fermions, corresponding to holes on the two sublattices, coupled with opposite sign to the U(1) gauge field of the ${\mathit{CP}}^1$ nonlinear σ model, which describes the undoped case. The usual constraint of the ${\mathit{CP}}^1$ model is modified to account for the presence of holes. This effective continuum theory has recently been shown to exhibit superconductivity without parity violation. The possible relevance of this model to high-${\mathit{T}}_{\mathit{c}}$ superconductivity is discussed.