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 $m_philesssim 10^{-25},$ eV, $f_philesssim 10^{14},$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.
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