Free Majorana Modes in Superconducting Quantum Wires

TL;DR

The paper develops a theoretical framework showing that free Majorana zero modes can arise in one-dimensional superconducting wires with weak attractive interactions when coupled to a spin-1/2 magnetic impurity, via a two-channel Kondo (Emery-Kivelson) mechanism. The low-energy theory $H=H_0+H_m+H_c$ yields a zero-energy edge bound state with a Majorana mode $a$ on the impurity and a second Majorana mode $\gamma_b$ in the wire, with a bound-state wavefunction $\chi_{\epsilon=0}(x)=\sqrt{\Delta/(\hbar v_F)}\, e^{-\Delta|x|/(\hbar v_F)}$ and a bulk gap $\Delta$, analogous to a Kitaev $p$-wave wire. The spin gap stabilizes the symmetric two-channel Kondo fixed point, protecting the zero modes against perturbations and enabling a Kitaev-like correspondence, including a capacitance-edge relation $C=\hbar/(4\Delta)$ and potential experimental probes via local magnetic susceptibility. A concrete realization is provided for two weakly attractive $s$-wave wires, where refermionization leads to a spinless channel that reproduces the Majorana-impurity structure, suggesting routes to engineer non-local Majorana qubits with multiple impurities and to connect topological superconductivity with soluble boundary quantum field theories.

Abstract

An s-wave superconducting wire with attractive interactions can admit a zero-energy bound state equation (solution) at an edge similar to the Jackiw-Rebbi model; this is a specific aspect of low-dimensional quantum systems, Dirac equation and e.g. of the Luther-Emery liquid with a spin gap. In this Letter, K. Le Hur Europhys. Lett., 49 (6), pp. 768-774 (2000), I introduced a magnetic spin-1/2 impurity interacting with such a (spin) bound state in a Luther-Emery wire representing then a Hubbard ladder in a d-wave superconducting state. This method is general and show how Majorana fermions at zero energy, i.e. Majorana zero modes, can take place in a superconducting wire model from the two-channel Kondo effect. Within these two channels (wires), the Luther-Emery form of the superconducting term can be reached within the weak-coupling attractive limit. Due to the interest in Majorana zero modes from magnetic impurities interacting with an s-wave superconductor, I take time to analyze zero-energy edge solutions in my model and present a correspondence with the p-wave superconducting wire in the topological phase through alternatives versions of the quantum field theory. I develop the relation between the edge magnetic susceptibility and the local capacitance measure in a p-wave superconducting wire. I elaborate on the idea that Majorana zero modes, i.e. free Majorana fermions, can be realized with magnetic impurities bridging the gap between two s-wave superconducting wires. The spin gap and the resonance with the impurity can protect the free Majorana solutions when including perturbations.

Figures (1)

• Figure 1: Representation of (possible) Majorana fermions organization for the Hamiltonian $H=H_0+H_m+H_c$. In purple, the Majorana fermions are zero modes. The magnetic impurity is represented on the left through the Majorana fermions $a$ and $b$. Since the zero-energy solution for the wave-function associated to the Majorana fermion $\gamma_b$ produces a typical length scale $\xi=\frac{\hbar v_F}{\Delta}$, we can place (draw) this particle at a certain distance from the Majorana fermion $a$. In Sec. \ref{['wiremodel']}, I show that the two Majorana fermions in purple may be protected through the realization with a magnetic impurity producing a bridge between two s-wave superconducting quantum wires with singlet pairing.