  Phase transition from the over-damping to under-damping states is a
ubiquitous phenomenon in physical systems. However, what kind of symmetry is
broken associated with this phase transition remains unclear. Here, we discover
that this phase transition is determined by an anti-parity-time
(anti-$\mathcal{PT}$) symmetry hidden in a single damping linear resonator,
which is significantly different from the conventional
anti-$\mathcal{PT}$-symmetric systems with two or more modes. We show that the
breaking of the anti-$\mathcal{PT}$ symmetry yields the phase transition from
the over-damping to under-damping states, with an exceptional point (EP)
corresponding to the critical-damping state. Moreover, we propose an
optomechanical scheme to show this anti-$\mathcal{PT}$ symmetry breaking by
using the optical spring effect in a quadratic optomechanical system. We also
suggest an optomechanical sensor with the sensitivity enhanced significantly
around the EPs for the anti-$\mathcal{PT}$ symmetry breaking. Our work unveils
the anti-$\mathcal{PT}$ symmetry hidden in damping oscillations and hence opens
up new possibilities for exploiting wide anti-$\mathcal{PT}$ symmetry
applications in single damping linear resonators.
