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Recent experiments have produced mounting evidence of Majorana zero modes innanowire-superconductor hybrids. Signatures of an expected topological phasetransition accompanying the onset of these modes nevertheless remain elusive.We investigate a fundamental question concerning this issue: Do well-formedMajorana modes necessarily entail a sharp phase transition in these setups?Assuming reasonable parameters, we argue that finite-size effects candramatically smooth this putative transition into a crossover, even in systemslarge enough to support well-localized Majorana modes. We propose overcomingsuch finite-size effects by examining the behavior of low-lying excited statesthrough tunneling spectroscopy. In particular, the excited-state energiesexhibit characteristic field and density dependence, and scaling with systemsize, that expose an approaching topological phase transition. We suggestseveral experiments for extracting the predicted behavior. As a usefulbyproduct, the protocols also allow one to measure the wire's spin-orbitcoupling directly in its superconducting environment.

Approaching a topological phase transition in Majorana nanowires