Recent experiments have made great progress in demonstrating the existence of Majorana fermions in engineered topological superconductors. In this talk, I describe exactly solvable models of interacting Majorana fermions in two- and three-dimensional lattices, which exhibit remarkable properties. In particular, despite being translationally invariant, our model in three dimensions supports point-like excitations that are fundamentally immobile, and composite excitations that can only move within submanifolds. These properties define a new kind of topological quantum order in three-dimensional lattice systems, which cannot be described by topological quantum field theory. I will discuss the prospect of large-scale quantum information processing with an array of Majorana fermions, using quantum error correction code based on these solvable models.
1. S. Vijay, T. Hsieh and L. Fu, arXiv:1504.01724 (PRX in press)
2. S. Vijay, J. Haah and L. Fu, 1505.02576