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We study the electronic spectral function of a metal in the vicinity of anantiferromagnetic (AFM) quantum critical point, focusing on a situation wherethe bare bandwidth of the spin fluctuations is significantly smaller than theFermi energy. In this limit, we identify a range of energies where thefermionic quasiparticles near the "hot spots'' on the Fermi surface arestrongly scattered by the quantum critical fluctuations, whereas the damping ofthe AFM fluctuations by the electrons is negligible. Within a one-loopapproximation, there is a parameter range where the $T=0$ spectral function atthe hot spots has a "precursor gap'' feature, with a local maximum at a finitefrequency. However, the ratio of the bare spin wave velocity to the Fermivelocity required to obtain a precursor gap is probably too small to explainexperiments in the electron-doped cuprate superconductors (He et al., Proc.Natl. Acad. Sci 116, 3449 (2019)). At lower frequencies, the Landau damping ofthe AFM fluctuations becomes important, and the electronic spectral functionhas the familiar ${\omega}^{-1/2}$ singularity. Our one-loop perturbativeresults are supported by a numerical Monte Carlo simulation of electronscoupled to an undamped, nearly-critical AFM mode.

Weakly damped bosons and precursor gap in the vicinity of an antiferromagnetic metallic transition