We have performed de Haas-van Alphen (dHvA) measurements of the heavy-fermion superconductor CeCoIn$_5$ down to 2 mK above the upper critical field. We find that the dHvA amplitudes show an anomalous suppression, concomitantly with a shift of the dHvA frequency, below the transition temperature $T_{rm n}=20$ mK. We suggest that the change is owing to magnetic breakdown caused by a field-induced antiferromagnetic (AFM) state emerging below $T_{rm n}$, revealing the origin of the field-induced quantum critical point (QCP) in CeCoIn$_5$. The field dependence of $T_{rm n}$ is found to be very weak for 7--10 T, implying that an enhancement of AFM order by suppressing the critical spin fluctuations near the AFM QCP competes with the field suppression effect on the AFM phase. We suggest that the appearance of a field-induced AFM phase is a generic feature of unconventional superconductors, which emerge near an AFM QCP, including CeCoIn$_5$, CeRhIn$_5$, and high-$T_{rm c}$ cuprates.
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