Designing $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ topological orders in networks of Majorana bound states

Abstract: Topological orders have been intrinsically identified in a class of systems such as fractional quantum Hall states and spin liquids. Accessing such states often requires extreme conditions such as low temperatures, high magnetic fields, pure samples, etc. Another approach would be to engineer the topological orders in systems with more accessible ingredients. In this work, we present networks of Majorana bound states, which are currently accessible in semiconductor nanowires proximitized to conventional superconductors, and show that the effective low-energy theory is topologically ordered. We first demonstrate the main principles in a lattice made of Kitaev superconducting chains comprising both spin species. The lattice is coupled to free magnetic moments through the Kondo interaction. We then show that at the weak coupling limit, effective ring spin interactions are induced between magnetic moments with a topological order enjoying a local $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetry. We then show that the same topological order and also the $\mathbb{Z}_2$ one can be engineered in architecture patterns of semiconductor nanowires hosting Majorana bound states. The basic blocks of patterns are the time-reversal Majorana Cooper boxes coupled to each other by metallic leads, and the Majorana states are allowed to tunnel to quantum dots sitting on the vertices of the lattices. In the limit of strong onsite Coulomb interactions, where the charge fluctuations are suppressed, the magnetic moments of dots on the square and honeycomb lattices are described by topologically ordered spin models with underlying $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetries, respectively. Finally, we show that the latter topological order can also be realized in a network of purely Majorana zero modes in the absence of coupling to quantum dots.