Novel antiferromagnetic quantum phase transition in underdoped cuprates

We investigate a zero-temperature itinerant antiferromagnetic transition where the fermions possess a d-wave gap. This problem pertains to both the nodal liquid insulating phase and the d-wave superconducting phase of the underdoped cuprates. We find that a non-trivial quantum phase transition exists, and that the quantum critical point is dominated by a long-ranged interaction ($\mathrm{\mid }x-y{\mathrm{\mid }}^{-2d}$) of the N'{e}el order parameter, which is induced by the Dirac-like fermions near gap nodes. We formulate a Ginzburg-Landau functional and estimate the critical exponents via the large-N expansion method.