This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems. We start with the linear scheme, which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable. The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable. Moreover, the scheme leads to linear algebraic system to solve at each iteration, and we employ the multigrid solver to solve it efficiently. Numerical results are given to illustrate that the combination of local discontinuous Galerkin (LDG) spatial discretization and the high order temporal scheme is a practical, accurate and efficient simulation tool when solving phase field problems. Namely, we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.