Spontaneous modulation of superconducting phase in Kitaev ladder

We study theoretically the two Kitaev chains put in parallel, i.e., Kitaev ladder, coupled by Josephson junction. The $\pi$-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 the modulation of the phase difference of the superconducting order parameter between the two chains. We show that this modulation gives the double degeneracy of the ground states, which can be manipulated by external electric and magnetic fields.