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We investigate the effect of a time-reversal breaking impurity term on boththe equilibrium and non-equilibrium critical properties of entanglement entropy(EE) in a three-spin interacting transverse Ising model which can be mapped toa one-dimensional p-wave superconductor with next-nearest-neighbor hopping. Dueto the presence of next-nearest-neighbor hopping, a new topological phase withtwo zero-energy Majorana modes at each end of an open chain appears in thephase diagram. We show that the derivative of EE with respect to one of theparameters of the Hamiltonian can detect the quantum phase transitions byexhibiting cusp like structure at those points; impurity strength ($\la_d$) cansubstantially modify the peak/dip height associated with the cusp. Importantly,we find that the logarithmic scaling of the EE with block size remainsunaffected by the application of the impurity term, although, the coefficient(i.e., central charge) varies logarithmically with the impurity strength for alower range of $\la_d$ and eventually saturates with an exponential dampingfactor ($\sim \exp(-\la_d)$) for the phase boundaries shared with the phasecontaining two Majorana edge modes. On the other hand, it receives a linearcorrection in term of $\la_d$ for an another phase boundary. Finally, we focusto study the effect of the impurity in the time evolution of the EE for thecritical quenching case where impurity term is applied only to the finalHamiltonian. Interestingly, it has been shown that for all the phase boundariesin contrary to the equilibrium case, the saturation value of the EE increaseslogarithmically with the strength of impurity in a certain region of $\la_d$and finally, for higher values of $\la_d$, it increases very slowly which isdictated by an exponential damping factor.

Entanglement entropy of a three-spin interacting spin chain with a time-reversal breaking impurity at one boundary