Spontaneous Modulation of Superconducting Phase in Kitaev Ladder

We study theoretically two Kitaev chains placed in parallel, i.e., Kitaev ladder, connected through Josephson coupling. The π-junction between the Majorana bound states at the ends of the chains competes with the usual Josephson coupling along the chain, and this frustration leads to modulation of the phase difference of the superconducting order parameter in the two chains. We show that this modulation results in double degeneracy of the ground states, which can be manipulated by external electric and magnetic fields.