Chiral zero modes in superconducting nanowires with Dresselhaus spin-orbit coupling

Using chiral decomposition, we are able to find analytically the zero modes and the conditions for such modes to exist in the Kitaev ladder model and superconducting nanowires with Dresselhaus spin-orbit coupling. As a result, we are able to calculate the number of zero modes in these systems for arbitrary given parameters in the semi-infinite limit. Moreover, we find that when suitable resonance condition is satisfied exact zero modes exist even in finite systems. Because of this, the energy of the lowest positive energy shows oscillation with respect to the chemical potential.