Magnetic properties of the t–J model at low doping

A path-integral for the t–J model in two dimensions is constructed based on Dirac quantization, with an action found originally by Wiegmann [Phys. Rev. Lett. 60 (1988) 821 ; Nucl. Phys. B 323 (1989) 311]. 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. The staggered CP1 fields are split into slow and fast modes, such that 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 a spin current, as opposed to a spin density. 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 spin density waves in the Hubbard model.