[Paper Review] Higher-order topological superconductors based on weak topological insulators

Xun-Jiang Luo, Xiao-Hong Pan+1 more|arXiv (Cornell University)|Mar 2, 2021

Topological Materials and PhenomenaPhysics and Astronomy95 references35 citations

TL;DR

This paper proposes a mechanism to realize higher-order topological superconductors (TSCs) using weak three-dimensional topological insulators with band inversion at Γ and Z points. By applying s±-wave superconducting pairing and in-plane Zeeman fields, the authors demonstrate the emergence of robust helical or chiral Majorana hinge modes (HMHMs/CMHMs) and Majorana corner modes (MCMs), characterized by surface Z2 invariants or Chern numbers, with applications to iron-based superconductors like Li(OH)FeSe.

ABSTRACT

High-order topological phases host robust boundary states at the boundary of the boundary, which can be interpreted from their boundary topology. In this work, considering the interplay between superconductors and magnetic fields to gap the surface states of three-dimensional weak topological insulators, we show that second-order topological superconductors (TSCs) featuring helical or chiral Majorana hinge modes and third-order TSC featuring Majorana corner modes can be realized. Remarkably, the higher-order TSCs in our models can be attributed to their certain boundaries, surfaces, or hinges, which naturally behave as first-order TSC in DIII or D symmetry class. Correspondingly, these higher-order TSCs can be characterized by the boundary first-order topological invariants, such as surface Chern numbers or surface $Z_2$ topological invariants for surface TSCs. Our models can effectively capture the topology of iron-based superconductors with desired inverted band structures and superconducting pairings.

Motivation & Objective

• To explore whether weak topological insulators can host higher-order topological superconducting phases.
• To demonstrate the realization of second- and third-order TSCs with robust Majorana hinge and corner modes.
• To establish a connection between higher-order TSCs and first-order topological invariants on boundaries.
• To provide a theoretical framework applicable to iron-based superconductors with inverted band structures.

Proposed method

• Constructs a Bogoliubov–de Gennes (BdG) Hamiltonian for a 3D weak topological insulator with s±-wave pairing.
• Expands the Hamiltonian near Γ and Z points to derive effective continuum models for surface states.
• Applies in-plane Zeeman fields to drive surface states into chiral TSC phases via topological phase transitions.
• Uses surface Z2 invariants and Chern numbers to characterize the resulting TSCs.
• Performs numerical analysis to confirm coexistence of helical and chiral hinge modes.
• Proposes a 1D hinge TSC model to realize third-order TSCs with Majorana corner modes.

Experimental results

Research questions

• RQ1 Can weak topological insulators host second- and third-order topological superconducting phases?
• RQ2 How do s±-wave pairing and Zeeman fields induce Majorana hinge and corner modes in weak TIs?
• RQ3 What topological invariants characterize the higher-order TSCs arising from weak TI surfaces?
• RQ4 Can the surface states of weak TIs behave as first-order TSCs with nontrivial invariants?
• RQ5 Is the proposed model realizable in iron-based superconductors like Li(OH)FeSe?

Key findings

• Second-order TSCs with helical Majorana hinge modes (HMHMs) are realized via s±-wave pairing on side faces of weak TIs with band inversion at Γ and Z points.
• The side faces behave as time-reversal-invariant TSCs (TRITSCs), characterized by nontrivial surface Z2 invariants.
• An in-plane Zeeman field drives a topological phase transition, inducing chiral Majorana hinge modes (CMHMs) with nontrivial surface Chern numbers.
• Third-order TSCs hosting Majorana corner modes (MCMs) are realized through a 1D hinge TSC construction.
• The model captures the topological properties of iron-based superconductors such as Li(OH)FeSe, which is predicted to be a weak TI with band inversion at Γ and Z.
• Numerical verification confirms the coexistence of HMHMs and CMHMs under appropriate parameter tuning.