Majorana Fermions in spin up and down electronic complexes in spin-orbit coupled array of semiconductor quantum dots in proximity to $s$-type superconductor and in magnetic field

Abstract

Semiconductor-s-type superconductor nanowires host spinful fermions and cannot be reduced to a single spinless Kitaev chain hosting single Majorana zero mode. Instead, such systems can be converted into two coupled p-wave Kitaev-like chains associated with different spin sectors. Using the bond Fermion transformation and exact diagonalization, we analyze parity resolved spectra and local spectral functions, demonstrating that zero-energy modes strongly localized at the system boundaries emerge only in one effective chain. Inter-chain coupling lifts parity degeneracy and redistributes the low-energy spectral weight, providing a controlled framework to assess the stability of Majorana-like modes in the finite spinful nanowires.

Figures (7)

• Figure 1: Schematic illustration of an InP nanowire hosting an array of embedded InAsP quantum dots, proximity coupled to an $s$-wave superconductor and subjected to a perpendicular magnetic field, chosen to be orthogonal to the effective Rashba spin orbit coupling and thereby inducing Zeeman splitting.
• Figure 2: Schematic of the two coupled Kitaev chains (upper one is for down spin and lower one is for up spin) in the Majorana and bond Fermion representation, with non-zero bond Fermions $a_{j\downarrow}$ and $a_{j\uparrow}$$(j\neq N)$, and the non local zero mode, living on the two ends of the two chains represent as $a_{N\downarrow}$ and $a_{N\uparrow}$. The coupling between the two chains are shown in red and blue dashed lines.
• Figure 3: Energy spectra of the (a) down and (b) up spin chains in the bond Fermion basis, where we enforce the down spin chain to be topological imposing $\tilde{t} = \Delta_\alpha$ and $V_z = \sqrt{\mu^2 + \Delta^2}$. The parameters are $N=3$, $t=\Delta=1$ and $\alpha=0.8t$, consequently $\tilde{t}=-0.6247$ and $V_z = 1.2806$. Energy is normalized to $|\tilde{t}|$.
• Figure 4: Energy spectra of the total Hamiltonian and the rest of the parameters are same as Fig. \ref{['fig:Energy_spin']}. The black dashed lines correspond to zero at both the plots.
• Figure 5: Site resolved spectral functions $A_j(\omega)$ of the down spin bond Fermions for the chain of $N=3$ quantum dots, evaluated at the chain end and in the bulk. The spectral function is computed with respect to the many-body ground state using exact diagonalization and includes a finite broadening $\eta$. Low-energy spectral weight is strongly localized at the chain ends, while bulk contributions remain suppressed, indicating boundary localized Majorana like excitations whose hybridization.