In this paper, we propose first- and second-order exponential scalar auxiliary variable (ESAV) schemes for solving the phase field crystal equation with the periodic boundary condition. Specifically, the scalar auxiliary variable (SAV) in this work is constructed based on an exponential function, which differs from the square root form commonly used in the traditional SAV method. This feature allows the proposed schemes that only need to solve one linear system with constant coefficients at each time step. To construct the first-order ESAV scheme, we utilize the backward Euler method. Combining the ESAV method with the leapfrog approach, the second-order ESAV scheme is developed. Theoretically, the proposed schemes are proven to be uniquely solvable and unconditionally energy stable. Furthermore, the error estimates of the second-order ESAV scheme is rigorously proved. The numerical results are provided to demonstrate the effectiveness of the proposed schemes.