  A major challenge in spintronics is to find an efficient means to manipulate
antiferromagnet (AFM) states, which are inert relative to a uniform magnetic
field, due to the vanishingly-small net magnetization. The question is, how
does an AFM response to an inhomogeneous field? Here we address the problem
through a complete classification of the magnetic forces on an AFM domain wall
(DW), revealing the following physical properties: (i) the tiny net
magnetization still responses to the field gradient. (ii) the N\'{e}el order is
sensitive to the field difference between two sublattices. (iii) DW energy has
a quadratic dependence on the magnetic field due to its noncollinear structure.
Remarkably, the first two factors drive DW to the opposite directions in a
nanowire, but the third effect tends to push the DW to the high field region.
Consequently, the competition among these three forces can be applied to
understand the seemingly-contradictory results on AFM motion in literature.
Additionally, our results provide a new route for a speedy manipulating AFM DW;
our numerical simulation indicated that for a synthetic antiferromagnet, the DW
propagating speed can reach tens of kilometers per second, an order of
magnitude higher than that driven by an electric current.
