[Paper Review] Mixed-parity octupolar pairing and corner Majorana modes in three dimensions

Bitan Roy, Vladimir Juričić|arXiv (Cornell University)|Jun 2, 2021

Topological Materials and PhenomenaPhysics and Astronomy58 references36 citations

TL;DR

This paper proposes a novel three-dimensional third-order topological superconductor (TOTSC) hosting eight robust Majorana corner modes, enabled by mixed-parity pairing: an odd-parity isotropic p-wave pairing coexisting with an even-parity octupolar dx²−y² + id₃z²−r² pairing in cubic crystals. The resulting intrinsic TOTSC state, protected by cubic symmetry and time-reversal symmetry breaking, supports corner-localized Majorana modes that are stable against weak s-wave pairing, with a doped octupolar Dirac insulator identified as a viable material platform, including under high pressure in NaCl and similar compounds.

ABSTRACT

We identify time-reversal symmetry breaking mixed-parity superconducting states that feature eight Majorana corner modes in properly cleaved three-dimensional cubic crystals. Namely, when an odd-parity isotropic $p$-wave pairing coexists with cubic symmetry preserving even-parity octupolar $d_{x^2-y^2}+i d_{3z^2-r^2}$ pairing, the gapless surface Majorana modes of the former get localized at the eight corners, thus yielding an \emph{intrinsic} third-order topological superconductor (TOTSC). A cousin $d_{xy}+id_{3z^2-r^2}$ pairing also accommodating eight corner Majorana modes, by virtue of breaking the cubic symmetry, in contrast, yields an \emph{extrinsic} TOTSC. We identify a doped octupolar (topological or trivial) Dirac insulator as a suitable platform to sustain such unconventional superconductors, realized from an intraunit cell pairing. Finally, we argue that the proposed TOTSC can be experimentally realizable in NaCl and other structurally similar compounds under high pressure.

Motivation & Objective

• To identify the pairing symmetry enabling robust Majorana corner modes in three-dimensional topological superconductors.
• To distinguish between intrinsic and extrinsic third-order topological superconducting states based on symmetry protection.
• To identify viable material platforms—specifically doped octupolar Dirac insulators—where such unconventional superconductivity can emerge from local intraunit cell pairing.
• To propose experimentally realizable systems, including NaCl and structurally similar compounds under high pressure, for observing these corner Majorana modes.

Proposed method

• Constructs an effective Bogoliubov-de Gennes (BdG) Hamiltonian around the Fermi surface with Kramers and two-fold sublattice degeneracy, incorporating p-wave (odd-parity) and dx²−y² + id₃z²−r² (even-parity octupolar) pairing terms.
• Uses group theory to identify the dx²−y² + id₃z²−r² pairing as transforming under the irreducible Eg representation, preserving cubic symmetry while breaking time-reversal symmetry.
• Analyzes the interplay between the p-wave and octupolar d-wave pairings, showing that the latter acts as a gap-opening mass for surface modes, leaving only corner modes gapless.
• Demonstrates that the gapless surface modes of the p-wave superconductor become localized at the eight corners due to vanishing Wilson-Dirac mass along ⟨111⟩ directions.
• Performs numerical diagonalization on a cubic lattice to confirm the presence of eight near-zero-energy corner modes, robust against small s-wave pairing.
• Identifies doped octupolar Dirac insulators as a minimal platform for realizing such pairing via fully gapped local intraunit cell pairing, with a quantized octupolar moment distinguishing intrinsic from extrinsic TOTSCs.

Experimental results

Research questions

• RQ1 What pairing symmetry in three-dimensional superconductors supports eight Majorana corner modes?
• RQ2 How do mixed-parity p-wave and octupolar d-wave pairings interact to stabilize corner modes while preserving topological protection?
• RQ3 What distinguishes intrinsic from extrinsic third-order topological superconductors in terms of symmetry and topological invariants?
• RQ4 Which material platforms can host such unconventional superconducting states with local pairing mechanisms?
• RQ5 Can these TOTSC phases be realized experimentally in cubic compounds under high pressure?

Key findings

• The p ⊕ (dx²−y² + id₃z²−r²) pairing state in a cubic crystal supports exactly eight zero-energy Majorana corner modes, localized at the corners along the ⟨111⟩ directions.
• The dx²−y² + id₃z²−r² pairing preserves cubic symmetry and breaks time-reversal symmetry, forming an intrinsic TOTSC with a quantized octupolar moment Qxyz = 0.5.
• The corner modes remain stable even in the presence of a small s-wave pairing, which is naturally induced by the d-wave component.
• Numerical simulations on a cubic lattice confirm eight near-zero-energy corner modes, sharply localized at the corners, with no bulk states nearby.
• The doped octupolar Dirac insulator is identified as a minimal platform for realizing this pairing, with local intraunit cell pairing leading to fully gapped superconductivity.
• The proposed TOTSC can be experimentally realized in NaCl and structurally similar compounds (e.g., InTe, SnAs, SnSb) under high pressure, with predicted superconducting transition temperatures Tc ∼ 2–7 K for NaCl and ∼1–3 K for others.