Topology and zero energy edge states in carbon nanotubes with superconducting pairing

Abstract

We investigate the spectrum of finite-length carbon nanotubes in the presence of onsite and nearest-neighbor superconducting pairing terms. A one-dimensional ladder-type lattice model is developed to explore the low-energy spectrum and the nature of the electronic states. We find that zero energy edge states can emerge in zigzag class carbon nanotubes as a combined effect of curvature-induced Dirac point shift and strong superconducting coupling between nearest-neighbor sites. The chiral symmetry of the system is exploited to define a winding number topological invariant. The associated topological phase diagram shows regions with nontrivial winding number in the plane of chemical potential and superconducting nearest-neighbor pair potential (relative to the onsite pair potential). A one-dimensional continuum model reveals the topological origin of the zero energy edge states: A bulk-edge correspondence is proven, which shows that the condition for nontrivial winding number and that for the emergence of edge states are identical. For armchair class nanotubes, the presence of edge states in the superconducting gap depends on the nanotube's boundary shape. For the minimal boundary condition, the emergence of the subgap states can also be deduced from the winding number.

Majorana quasiparticles in semiconducting carbon nanotubes

“…4(c). The MQPs in our system are also eigenstates of C. In systems with this symmetry, the topological invariant γ − has a clear interpretation as a winding number, γ − = ν/2 [35]. The winding number is an integer, i.e.…”

“…The identity with another definition of the winding number, which uses a flat band Hamiltonian [44], is proven in Appendix C1 in Ref. 35. Let us consider the unitary transformation U c = 1 2…”

Linear and nonlinear transport across a finite Kitaev chain: An exact analytical study

“…In a finite chain, the bulk-edge correspondence [42,43] implies the existence of evanescent state solutions at the system's boundary in the topologically nontrivial phase. These states have a complex wave vector κ, and their wave functions decay away from the edges with a decay length ξ , which for μ = 0 is given by ξ = 2d ln t− t+ .…”

Transport signatures of topological phases in double nanowires probed by spin-polarized STM

“…The promising platforms to engineer topological superconductivity are semiconducting Rashba nanowires (NWs) subjected to a uniform magnetic field [6][7][8][9][10][11][12] or chains of magnetic adatoms [13][14][15][16][17][18][19][20][21]. However, magnetic field and superconductivity have detrimental effects on each other, which has motivated proposals for timereversal invariant topological superconductors to avoid the need of magnetic fields, particular examples being double-NW setups with Karmers pairs of MBSs [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In such setups, two types of proximity induced superconductivity play a crucial role: intrawire (∆) and interwire (∆ c ) superconductivity.…”