  Hertz-Moriya-Millis theory with dynamical critical exponent $z = 2$ has been
proposed to describe antiferromagnetic quantum criticality in itinerant
electron systems. In this study we show that the dynamical critical exponent
changes from $z = 2$ to $z = 3$ at low temperatures, which results from
effective long-range interactions between spin fluctuations, generated by
Fermi-surface fluctuations beyond the Eliashberg framework. We claim that the
underlying physics for the $z = 3$ antiferromagnetic quantum criticality is the
emergence of fermionized skyrmion excitations at low energies, which form a
Fermi surface, referred as a skyrmion liquid, where the interplay between
itinerant electrons and skyrmions is argued to allow fermionized skyrmions. We
construct a dual field theory in terms of skyrmion excitations, and obtain the
$z = 3$ antiferromagnetic quantum criticality. This demonstration suggests the
expected but nontrivial consistency between weak-coupling and strong-coupling
theories.
