Exchangeless braiding of Majorana zero modes in weakly coupled Kitaev chains

Exchangeless braiding of Majorana modes is studied in minimal networks of weakly hybridized Kitaev chains of finite length using a rigorous many-body framework. In particular, for two coupled chains it is shown that exchangeless braiding is achieved by $2\ensuremath{\pi}$ rotations of the phase $\ensuremath{\phi}$ of the superconducting order parameter of one of the chains. This braiding protocol is verified by the numerical calculation of the non-Abelian Wilczek-Zee phase of the low-energy many-body subspaces ${\mathcal{H}}_{0}(\ensuremath{\phi})$ based on the Bertsch-Robledo ground-state overlap formula. In the parameter space spanned by the total chain length, the strength of the weak hybridization connecting the chains, and the on-site potential, we identify two regions with different braiding outcomes, i.e., a projective ${\ensuremath{\sigma}}_{x}$ gate and a projective ${\ensuremath{\sigma}}_{z}$-gate phase. The transition between these phases is a continuous crossover, the location of which is reliably given by a simple four-Majorana mode model. This demonstrates the resilience of the anyonic properties of the Majorana modes against finite-size effects and weak links between Kitaev chains.