Non-trivial fusion of Majorana zero modes in interacting quantum dots arrays

Motivated by the recent experimental reports of Majorana zero modes (MZMs) in quantum-dot systems at the sweet spot (t_h = ∆), where the electronic hopping t_h is equal to the superconducting coupling parameter ∆, we study the time-dependent spectroscopy corresponding to the non-trivial fusion of MZMs. Here the expression non-trivial refers to the fusion of Majoranas from different pairs of MZMs with pre-defined parities. We employ a time-dependent real-space local density-of-states method to investigate the fusion outcomes of MZMs in canonical chains, as well as in a Y-shape array of interacting quantum dots. In the case of two pairs of MZMs, after the fusion we find equal-height peaks in the electron and hole components of the local density-of-states, reflecting fusion into the vacuum I and fermion Ψ channels. For π-junction quantum dots, where the superconducting phase has opposite signs on the left and right portions of the chain, after the fusion we show the formation of an exotic two-site MZM [1] near the center of the wire coexisting with another single-site MZM. Finally, we study the fusion of three MZMs in the Y-shape geometry. In this case, after the fusion we observed the formation of another exotic multi-site MZM (spread now over three near central sites). Our results could be realized in quantum dot systems in the context of tunneling conductance experiments.