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

We investigate the effect of a time-reversal breaking impurity term on both the equilibrium and non-equilibrium critical properties of entanglement entropy (EE) in a three-spin interacting transverse Ising model which can be mapped to a one-dimensional p-wave superconductor with next-nearest-neighbor hopping. Due to the presence of next-nearest-neighbor hopping, a new topological phase with two zero-energy Majorana modes at each end of an open chain appears in the phase diagram. We show that the derivative of EE with respect to one of the parameters of the Hamiltonian can detect the quantum phase transitions by exhibiting cusp like structure at those points; impurity strength (\la_d) can substantially modify the peak/dip height associated with the cusp. Importantly, we find that the logarithmic scaling of the EE with block size remains unaffected by the application of the impurity term, although, the coefficient (i.e., central charge) varies logarithmically with the impurity strength for a lower range of \la_d and eventually saturates with an exponential damping factor (\sim \exp(-\la_d)) for the phase boundaries shared with the phase containing two Majorana edge modes. On the other hand, it receives a linear correction in term of \la_d for an another phase boundary. Finally, we focus to study the effect of the impurity in the time evolution of the EE for the critical quenching case where impurity term is applied only to the final Hamiltonian. Interestingly, it has been shown that for all the phase boundaries in contrary to the equilibrium case, the saturation value of the EE increases logarithmically 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 is dictated by an exponential damping factor.Comment: 10 pages, 10 figure