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A path-integral for the t-J model in two dimensions is constructed based on Dirac quantization. Concentrating on the low doping limit, we assume short range antiferromagnetic order of the spin degrees of freedom. Going over to a local spin quantization axis of the dopant fermions, that follows the spin degree of freedom, staggered CP1 fields result and the constraint against double occupancy can be resolved exactly. After a gradient expansion, and after integrating out the fast modes and the dopant fermions, a CP1 field-theory with a massive gauge field is obtained that describes generically incommensurate coplanar magnetic structures, as discussed previously in the context of frustrated quantum antiferromagnets. The analysis of Landau damping shows that in this case, even in the presence of doping, the dynamical critical exponent z=1, as a consequence of coupling of the dopant holes to spin currents. This result agrees with experimental observations in the underdoped region of high temperature superconductors, and departs from the one obtained by Hertz and Millis (z=2) in the case of the Hubbard model. From a renormalization group analysis of the massive CP1 model we determine the quantum critical point at which long-range incommensurate magnetic order ceases to exist as a consequence of doping. |
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