Topological order, topological insulators, and the search for the Majorana fermion

Yasaman Bahri, Ashvin Vishwanath

Debanksh Guha, Meghansh Govil, Suraj Kumar Saw +2 more

Rukhsan Ul Haq, Louis Hirsch Kauffman

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

Nandini Trivedi

Chunlei Qu, Ming Gong, Yong Xu +2 more

We show that Majorana fermions (MFs) exist in twoand three-dimensional (2D,3D) fermionic optical lattices with strictly 1D spin-orbit coupling (SOC) which has already been realized in experiments. For a quasi-1D topological BCS superfluid, there are multiple MFs at each end which are topologically protected by a chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero energy flat band. An additional experimentally tunable in-plane Zeeman field drives the system to a topological Fulde-Ferrell (FF) superfluid phase. We find that even though the multiple MFs are robust against the in-plane Zeeman field if the order parameters at the different chains are enforced to be identical, they are destroyed in the self-consistently obtained FF phase where the order parameters are inhomogeneous on the boundaries. Our results are useful to guide the experimentalists on searching for MFs in the context of ultracold fermionic atoms.

Joseph Maciejko, Andreas R uegg, Victor Chua +1 more