Superconducting Spiral Phase in the two-dimensional t-J model

We analyse the t-t'-t''-J model, relevant to the superconducting cuprates. By using chiral perturbation theory we have determined the ground state to be a spiral for small doping \delta << 1 near half filling. In this limit the solution does not contain any uncontrolled approximations. We evaluate the spin-wave Green's functions and address the issue of stability of the spiral state, leading to the phase diagram of the model. At t'=t''=0 the spiral state is unstable towards a local enhancement of the spiral pitch, and the nature of the true ground state remains unclear. However, for values of t' and t'' corresponding to real cuprates the (1,0) spiral state is stabilized by quantum fluctuations (``order from disorder'' effect). We show that at \delta = 0.119 the spiral is commensurate with the lattice with a period of 8 lattice spacings. It is also demonstrated that spin-wave mediated superconductivity develops in the spiral state and a lower limit for the superconducting gap is derived. Even though one cannot classify the gap symmetry according to the lattice representations (s,p,d,...) since the symmetry of the lattice is spontaneously broken by the spiral, the gap always has lines of nodes along the (1,\pm 1) directions.