Interacting Majorana fermions

Majorana fermions are the real (in a mathematical sense) counterparts of complex fermions like ordinary electrons. The promise of topological quantum computing has lead to substantial experimental progress in realizing these particles in various synthetic platforms. The realization of Majorana fermions motivates a fundamental question: what phases of matter can emerge if many Majorana fermions are allowed to interact? Here we review recent progress in this direction on the proposed experimental setups, analytical and numerical results on low-dimensional lattice models, and the exactly solvable Sachdev-Ye-Kitaev model. The early progress thus far suggests that strongly correlated phases of matter with Majorana building blocks can exhibit many novel phenomena, such as emergent spacetime supersymmetry, topological order and the physics of black-holes, in condensed matter systems. They may also provide alternative avenues for universal topological quantum computing through the realization of the Fibonacci phase and measurement-based only surface codes.