Topological states in normal and superconducting p -wave chains

Abstract

We study a two-band model of fermions in a 1d chain with an antisymmetric hybridization that breaks inversion symmetry. We find that for certain values of its parameters, the sp-chain maps formally into a p-wave superconducting chain, the archetypical 1d system exhibiting Majorana fermions. The eigenspectra, including the existence of zero energy modes in the topological phase, agree for both models. The end states too share several similarities in both models, such as the behavior of the localization length, the non-trivial topological index and robustness to disorder. However, we show by mapping the s-and p-fermions to two copies of Majoranas, that the excitations in the ends of a finite sp chain are indeed conventional fermions though endowed with protected topological properties. Our results are obtained by a scattering approach in a semi-infinite chain with an edge defect treated within the T -matrix approximation. We augment the analytical results with exact numerical diagonalization that allow us to extend our results to arbitrary parameters and also to disordered systems.

Fermi points and topological quantum phase transitions in a multi-band superconductor

“…The problem can be viewed as a generalization of Kitaev's model to two orbitals and only interband pairing. We also have the anti-symmetric hybridisation term that, under some conditions, was shown to be responsible for topological phases [13][14][15]. The simplest Hamiltonian in the momentum space that describes those types of superconductivity and hybridization can be written as…”

“…In addition, it was recently shown [13][14][15] an intimate relation between a two band insulator with antisymmetric hybridization and the Kitaev model, as regards to the topological properties and their end states. By tuning the parameters of the 1D sp chain, the system can be driven, through a topological quantum phase transition, from a trivial to a topological insulator.…”

Induced p-wave superconductivity without spin–orbit interactions

“…These zero energy modes are associated with topological transitions in the system [13,31] as we will see further on in the text. Finally, the excitation energies are given by,…”

Detection of topological phases by quasilocal operators

“…This may be obtained using the scaling behavior of the energy gap E g ∼ (g − g c ) νz , where z is the dynamical critical exponent. Since the energy spectrum is linear, we have z = 1 and the gap scales linearly which leads to ν = 1 (as shown for instance in 22 ). Generalizing the Kitaev model to a multi-band model with an anti-symmetric coupling between the two bands leads to a rich phase diagram that displays a topological transition between a Weyl-like phase and a conventional superconductor that turns out to be in a different universality class of the Kitaev model 23 .…”