Topological quantum phase transitions in the spin–singlet superconductor with Rashba and Dresselhaus (110) spin–orbit couplings

• Majorana fermion can emerge in the spin–orbit coupled singlet superconductor.
• Pfaffian invariant and 1D winding number can be used to identify the nontrivial topological phase where Majorana flat band exists.
• All the possible phase diagrams in the spin–orbit coupled singlet superconductor are demonstrated.
• Majorana flat band only exists in the yy direction in our model.
• Majorana flat band has a significant experimental signature in the tunneling conductance measurement.

We examine the topological properties of a spin–singlet superconductor with Rashba and Dresselhaus (110) spin–orbit couplings. We demonstrate that there are several topological invariants in the Bogoliubov–de Gennes (BdG) Hamiltonian by symmetry analysis. In particular, the Pfaffian invariant PP for the particle–hole symmetry can be used to demonstrate all the possible phase diagrams of the BdG Hamiltonian. We find that the edge spectrum is either Dirac cone or flat band which supports the emergence of the Majorana fermion in this system. For the Majorana flat bands, an edge index, namely the Pfaffian invariant P(ky)P(ky) or the winding number W(ky)W(ky), is needed to make them topologically stable. These edge indices can also be used in determining the location of the Majorana flat bands.