Desining $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ topological orders using networks of Majorana bound states*

The topologically protected ground states against local perturbations and nontrivial braiding statistics of quasiparticles provide a unique platform for topological quantum computations. However, the experimental realization of topologically ordered states in a controlled way has remained elusive notwithstanding huge attempts in the last few decades. 

In this work, we introduce several systems to simulate class of topologically ordered states with  $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetries. In the first system, we consider a network of Majorana fermions on a hexagonal lattice (square lattice) so that each vertex hosts six pairs of Majorana fermions. The latters are Kondo coupled to a hexagonal lattice of otherwise free magnetic ions, each carrying spin-$1/2$. We show that in the weak coupling limit, a $\mathbb{Z}_2 \times \mathbb{Z}_2$ ($\mathbb{Z}_2$) topological order is induced into the magnetic ions. Next, we introduce a scenario based on the Kramers pairs of Majorana fermions at the end of nanowires, which are experimentally more accessible than the one-dimensional Kitaev chains by a proper combination of heterostructures. We couple them to a hexagonal lattice of quantum dots, allowing electrons to tunnel to the dots.  Using the perturbation theory, we show that the low-energy description of the whole systems is given by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ model of local moments. Majorana surface codes also carry topological order without coupling to free magnetic ions used above. We show that a $\mathbb{Z}_2 \times \mathbb{Z}_2$ topological order is realized in a network of Majorana fermions described above, paving a way to simulate Majorana color codes.