Chiral d density wave ordered pseudo gap phase of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ bilayer. Spin texture and Dzyaloshinskii Moriya interaction

Abstract

We model Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) bilayer system by an inversion symmetry broken and time reversal non invariant Bloch Hamiltonian H. The pseudo-gap phase of the bilayer is assumed to be chiral d density wave ordered. The focal point of the paper is the theoretical study of the spin momentum locking (SML), due to the presence of a strong spin orbit coupling in Bi2212, reported earlier in this phase in a spin and angle resolved photoemission spectroscopic measurement. The non trivial spin texture in k space is found tunable by electric field and also by intercalation.The Dzyaloshinskii Moriya interaction (DMI) coefficient, where DMI being important energy for chiral textures like magnetic skyrmions, are calculated to set the stage for detailed studies in a sequel. We also find that,in the nodal region, in the presence of Zeeman field B , as in a s wave superconductor, for B greater than or equal to B_c (a critical value) a vortex in this system becomes a non-Abelian anyon with a Majorana zero mode required for the fault tolerant quantum computation.