Nonuniform superconducting states from Majorana flat bands

TL;DR

This paper explores nonuniform superconducting phases, specifically pair density waves and phase crystals, arising from Majorana flat bands in topological superconductors, and analyzes their temperature-dependent stability.

Contribution

It demonstrates the emergence of distinct nonuniform phases in topological superconductors with Majorana flat bands and characterizes their temperature-dependent behavior.

Findings

Pair density wave persists up to 80% of the critical temperature.

Phase crystal appears at lower temperatures with localized phase modulations.

Intermediate region with combined amplitude and phase modulations exists between phases.

Abstract

Zero-energy flat bands within the superconducting gap can give rise to competing ordered phases. We investigate such phases in topological superconductors based on the magnetic adatom platform hosting a flat band of Majorana edge states. Our self-consistent calculations of the superconducting order parameter show the emergence of both a pair density wave with edge-localized amplitude modulations and a phase crystal characterized by edge-localized phase modulations. These two phases lower the free energy of the system by gapping out the Majorana flat band, as dictated by winding numbers, which are primarily tuned by the chemical potential. In fact, at zero temperature the uniform superconducting solution with Majorana flat band never survives and the phase diagram features a pair density wave, while the order parameter transitions into a phase crystal when amplitude modulations are insufficient to hybridize all the Majorana states. A broad intermediate region connects these two phases with comparable modulations in both amplitude and phase. At finite temperatures, the pair density wave survives up to around 80% of the bulk superconducting transition temperature, while the phase crystal only appears at lower temperatures and the intermediate region is strongly suppressed. Our findings establish the ubiquity of emergent nonuniform superconducting phases and their temperature-dependent behavior in topological superconductors.