  We study the quasiparticles of the one-band $t$-$J$ and $t$-$t'$-$t"$-$J$
models using a variational approximation that includes spin fluctuations in the
vicinity of the hole. We explain why the spin fluctuations and the longer range
hopping have complementary contributions to the quasiparticle dynamics, and
thus why both are essential to obtain a dispersion in agreement with that
measured experimentally. This is very different from the three-band Emery model
in the strongly-correlated limit, where the same variational approximation
shows that spin fluctuations have a minor effect on the quasiparticle dynamics.
This difference proves that these one-band and three-band models describe
qualitatively different quasiparticles, and therefore they they cannot both be
suitable to describe the physics of underdoped cuprates.
