Designing XY and Dzyaloshinskii-Moriya couplings in Majorana Cooper pair boxes

We theoretically study how to design spin couplings in networks of Majorana Cooper pair boxes (MCBs) connected by multiple normal-metal leads. The inter-box interaction is generated by the conduction-electron-mediated Ruderman--Kittel--Kasuya--Yosida (RKKY) interaction. We show that the connectivity of Majoranas to the leads enables arbitrary types of couplings. As concrete examples, we show the realization of the XY exchange interaction and the Dzyaloshinskii--Moriya (DM) interaction, which are difficult to implement in previously proposed MCB-based schemes. The sign and magnitude of the couplings can be tuned continuously via gate-controlled tunneling amplitudes. These results establish MCBs as a versatile platform for engineered quantum spin systems.