Sensitive dependence of Poor Man's Majorana modes on the length of the superconductor

TL;DR

This paper models Poor Man's Majorana modes in a finite-length superconductor, revealing their sensitivity to superconductor length and conditions for their existence in realistic systems.

Contribution

It provides a finite-length superconductor model for PMMs, deriving conditions for their existence and analyzing their dependence on system parameters.

Findings

PMMs oscillate between zero and two depending on superconductor length

Four PMMs appear in the long-superconductor limit

Finite-length superconductor prevents separate localization of PMMs at each end

Abstract

In a hybrid system where two quantum dots (QDs) are coupled to a conventional $s$-wave superconductor, Poor Man's Majorana modes (PMMs) have been proposed. Existing theories often idealize the superconductor (SC) as a bulk system or an infinitely long chain, or treat it as another quantum dot with proximity-induced superconductivity, while experiments employ superconducting segments of finite length. Here, we model the SC as a finite-length 1D chain and treat the QDs and SC on equal footing. We obtain the conditions for the existence of PMMs, valid for arbitrary SC length and applicable to arbitrary tunneling strengths and magnetic fields. We find that the number of PMMs is highly sensitive to the SC length: it oscillates between zero and two with a period set by the Fermi wavelength ($\sim1\,\text{\AA}$), while four PMMs appear in the long-SC limit where the effective coupling between the two QDs becomes negligible. We further demonstrate that the PMMs that are separately localized at the two ends of the hybrid system do not exist in the finite-length case. Consequently, only nearly localized PMMs can be identified when the magnetic field is strong enough. In this way, the generalized `sweet spot' of the practical system can be found.