  Efficient and energy stable high order time marching schemes are very
important but not easy to construct for the study of nonlinear phase dynamics.
In this paper, we propose and study two linearly stabilized second order
semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses
backward differentiation formula and the other uses Crank-Nicolson method to
discretize linear terms. In both schemes, the nonlinear bulk forces are treated
explicitly with two second-order stabilization terms. This treatment leads to
linear elliptic systems with constant coefficients, for which lots of robust
and efficient solvers are available. The discrete energy dissipation properties
are proved for both schemes. Rigorous error analysis is carried out to show
that, when the time step-size is small enough, second order accuracy in time is
obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where
$\varepsilon$ is the characteristic interface thickness. Numerical results are
presented to verify the accuracy and efficiency of proposed schemes.
