  Long-wavelength spin fluctuations prohibit antiferromagnetic long-range order
at finite temperature in two dimensions. Nevertheless, the correlation length
starts to grow rapidly at a crossover temperature, leading to critical slowing
down and to a renormalized-classical regime over a wide range of temperature,
between a fraction of the mean-field transition temperature and the
zero-temperature ordered state. This leads to a single-particle pseudogap of
the kind observed in electron-doped cuprates. Very few theoretical methods can
claim an accurate description of this regime. The challenge is that in this
regime Fermi-liquid quasiparticles are already destroyed and new quasiparticles
of the ordered state are not fully formed yet. Here, we study this problem for
the two-dimensional Hubbard model by first generalizing the two-particle
self-consistent approach. Using a special algorithm, spin fluctuations are
treated in the thermodynamic limit even for large correlation lengths. The
effects of Kanamori-Br\"uckner screening, of classical and of quantum
fluctuations are taken into account. Results are presented at half-filling for
the one-band Hubbard model with nearest-neighbor hopping. They agree well with
available benchmark diagrammatic quantum Monte Carlo at high temperature where
the pseudogap opens up. In addition to temperature-dependent spectral
properties, we find quantum corrections to the zero-temperature renormalized
mean-field antiferromagnetic gap. Finally, analytic continuation of the
Matsubara results for spectral functions show that the pseudogap opens up at
significantly higher temperature than was previously identified based on the
Matsubara data only.
