Exact dynamics of one-qubit system in layered environment

We investigate the exact evolution of the reduced dynamics of a one qubit system as central spin coupled to a femionic layered environment with unlimited number of layers. Also, we study the decoherence induced on central spin by analysis solution is obtained in the limit $N\rightarrow\infty$ of an infinite number of bath spins. Finally, the Nakajima-Zwanzig (NZ) and the time-convolutionless (TCL) projection operator techniques to second order are derived.