I will discuss our work on the topological phases of a p-wave superconducting wire and Kitaev model on honeycomb lattice. I will elaborate on some of the topological invariants which characterize such phases. In the first part of the talk, I will consider time-independent p-wave superconducting wire. The topological phase diagrams and zero energy Majorana end modes will be discussed. par The effects of time-reversal symmetry breaking will be examined. In the second part of the talk, I will show that in p-wave superconducting wire chemical potentials which vary periodically in time can produce end modes even when the corresponding time-independent system has no such end modes. I will discuss various properties of the dynamically generated edge states that we have found numerically and analytically using Floquet theory. Finally, I will present a new topological invariant which can separately predict the numbers of end modes which have Floquet eigenvalues equal to +1 and -1. At the end, I will discuss Kitaev model on honeycomb lattice for time-independent and time-dependent cases. The phase diagrams for zigzag and armchair zero energy modes will be discussed. I will show that the driving frequencies at which Majorana edge modes appear or disappear in this two-dimensional system can be completely understood by mapping it to a one-dimensional system in which the edge momentum appears as one of the parameters of the model. References: 1) W. DeGottardi, M. Thakurathi, S. Vishveshwara and D. Sen, Phys. Rev. B 88, 165111 (2013) 2) M. Thakurathi, A. A. Patel, D. Sen and A. Dutta, Phys. Rev. B 88, 155133 (2013) 3) M. Thakurathi, K. Sengupta and D. Sen, Phys. Rev. B 89, 235434 (2014)

Centre For High Energy Physics, IISc Bangalore