The effect of uniform spin-orbit coupling and uniform Zeeman magnetic field on the topological properties of one-dimensional dimerized nano wire

We theoretically demonstrate the interplay of uniform spin-orbit coupling and uniform Zeeman magnetic field on the topological properties of one-dimensional double well nano wire which is known as Su-Schrieffer-Heeger (SSH) model. The system in the absence of Zeeman magnetic field and presence of uniform spin-orbit coupling exhibits topologically trivial/non–trivial insulator depending on the hopping amplitudes and spin-orbit coupling strength. Topological phases of this system can be determined by integers  which are related to the Zak phase of occupied Bloch bands. In the phase diagram, there are three different regions with topologically distinct phases. The system is non-trivial insulator in two of them whereas one of the regions is related to the topologically trivial insulator. We find that the topologically trivial phase in the presence of both uniform spin-orbit coupling and uniform Zeeman magnetic field changes to a topologically non-trivial phase. The number of symmetry protected zero-energy edge states under open boundary conditions are also calculated, which suggest that the topological number  reduces to the  when applying Zeeman field. Furthermore, the symmetries of the Hamiltonian are investigated, implying that the system has time-reversal, particle-hole, chiral and inversion symmetries and belongs to the BDI class either in the presence or absence of uniform Zeeman magnetic field