  Nambu-Goldstone bosons, or axions, may be ubiquitous. Some of the axions may
have small masses and thus serve as mediators of long-range forces. In this
paper, we study the force mediated by an extremely light axion, $\phi$, between
the visible sector and the dark sector, where dark matter lives. Since nature
does not preserve the CP symmetry, the coupling between dark matter and $\phi$
is generically CP-violating. In this case, the induced force is extremely
long-range and behaves as an effective magnetic field. If the force acts on
electrons or nucleons, the spins of them on Earth precess around a fixed
direction towards the galactic center. This provides an experimental
opportunity for $\phi$ with mass, $m_\phi$, and decay constant, $f_\phi$,
satisfying $m_\phi\lesssim 10^{-25}\,$ eV, $f_\phi\lesssim 10^{14}\,$GeV if the
daily modulation of the effective magnetic field signals in magnetometers is
measured by using the coherent averaging method. The effective magnetic field
induced by an axionic compact object, such as an axion domain wall, is also
discussed.
