Some remarks on pseudogap behavior of nearly antiferromagnetic metals

In the antiferromagnetically ordered phase of a metal, gaps open on parts of the Fermi surface if the Fermi volume is sufficiently large. We discuss simple qualitative and heuristic arguments under what conditions precursor effects, i.e., pseudogaps, are expected in the paramagnetic phase of a metal close to an antiferromagnetic quantum phase transition. At least for weak interactions, we do not expect the formation of pseudogaps in a three-dimensional material. According to our arguments, the upper critical dimension $d_c$ for the formation of pseudogaps is $d_c=2.\mathrm{}$ However, at the present stage we cannot rule out a higher upper critical dimension, $2<~d_c<~3.$ We also discuss briefly the role of statistical interactions in pseudogap phases.