Approaching a topological phase transition in Majorana nanowires

TL;DR

This paper explores how finite-size effects can smooth the topological phase transition in Majorana nanowires, proposing spectroscopic methods to detect the transition and measure spin-orbit coupling.

Contribution

It demonstrates that finite-size effects can obscure the phase transition, and introduces spectroscopic techniques to reveal the transition and measure spin-orbit coupling.

Findings

Finite-size effects can smooth the topological transition into a crossover.

Low-lying excited states show characteristic dependence on field, density, and system size.

Proposed experiments can detect the transition and measure spin-orbit coupling.

Abstract

Recent experiments have produced mounting evidence of Majorana zero modes in nanowire-superconductor hybrids. Signatures of an expected topological phase transition accompanying the onset of these modes nevertheless remain elusive. We investigate a fundamental question concerning this issue: Do well-formed Majorana modes necessarily entail a sharp phase transition in these setups? Assuming reasonable parameters, we argue that finite-size effects can dramatically smooth this putative transition into a crossover, even in systems large enough to support well-localized Majorana modes. We propose overcoming such finite-size effects by examining the behavior of low-lying excited states through tunneling spectroscopy. In particular, the excited-state energies exhibit characteristic field and density dependence, and scaling with system size, that expose an approaching topological phase transition. We suggest several experiments for extracting the predicted behavior. As a useful byproduct, the protocols also allow one to measure the wire's spin-orbit coupling directly in its superconducting environment.