Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators

TL;DR

This paper demonstrates that Majorana vortex end modes can exist in time-reversal invariant insulators, including trivial ones, with fully gapped surfaces, expanding potential material platforms for Majorana physics.

Contribution

It shows that Majorana vortex end modes are robust in higher-order topological insulators and trivial insulators with gapped surfaces, unaffected by surface mass terms.

Findings

Majorana vortex end modes appear when chemical potential is between critical values.

MVEMs persist even when all surfaces are gapped with the same sign.

Majorana vortex phases are present in both topologically nontrivial and trivial insulators.

Abstract

Majorana vortex phases have been extensively studied in topological materials with conventional superconducting pairing. Inspired by recent experimental progress in realizing time-reversal invariant higher-order topological insulators (THOTIs) and inducing superconducting proximity effects, we investigate Majorana vortex phases in these systems. We construct THOTIs as two copies of a topological insulator (TI) with time-reversal symmetry-preserving mass terms that anisotropically gap the surface states. We find that these mass terms have a negligible impact on the vortex phase transitions of double TIs when treated as perturbations, and no additional topological phase transitions are induced. Consequently, $\mathbb{Z}_2$-protected Majorana vortex end modes (MVEMs) emerge when the chemical potential lies between the critical chemical potentials $\mu_c^{(1)}$ and $\mu_c^{(2)}$ of the two TI vortex phase transitions. We demonstrate this behavior across multiple THOTI models, including rotational symmetry-protected THOTI, inversion symmetry-protected THOTI, rotational and inversion symmetries-protected THOTI bismuth, and extrinsic THOTI. Remarkably, MVEMs persist even when all surfaces are gapped with the same sign, rendering the system topologically trivial in both first- and second-order classifications. Our findings establish that MVEMs can be realized in time-reversal invariant systems with fully gapped surfaces, encompassing both topologically nontrivial and trivial insulators, thus significantly broadening the solid state material platforms for hosting Majorana vortex phases.