Spin liquids from Majorana zero modes in a Cooper-pair box

We propose a path for constructing diverse interacting spin systems from topological nanowires in Cooper-pair boxes. The wires are grouped into a three-wire building block called a "hexon," consisting of six Majorana zero modes. In the presence of a strong charging energy, the hexon becomes a Cooper-pair box equivalent to two spin-1/2 degrees of freedom. By considering arrays of hexons and controlling the distances between the various wires, one can tune the Hamiltonian governing the low-energy spins, thus providing a route for controllably constructing interacting spin systems in one and two dimensions. We explicitly present realizations of the one-dimensional spin-1/2 XXZ chain, as well as the transverse field Ising model. We propose an experiment capable of revealing the nature of critical points in such effective spin systems by applying a local gate voltage and measuring the induced charge at a distance. To demonstrate the applicability of this approach to two dimensions, we provide a scheme for realizing the topologically ordered Yao-Kivelson spin-liquid model, which has a collective Majorana edge mode, similar to the B phase of Kitaev's honeycomb model.