Higher-order topological superconductivity in type-II time-reversal-symmetric Weyl semimetals with a hybrid pairing

This study demonstrates that type-II time-reversal-symmetric Weyl semimetals can host a self-consistent hybrid $s$-wave and $p$-wave superconducting state, where surface Fermi-arc configurations dictate spatial pairing dominance and give rise to second-order topological superconductivity with hinge states.

The Gist

Imagine a world where electricity doesn't just flow like water in a pipe, but dances in complex, topological patterns. This is the world of Weyl Semimetals, a special class of materials that physicists are currently obsessed with.

This paper by Huang, Wang, and Zhou is like a detective story. They investigated a specific type of these materials (called Type-II Time-Reversal-Symmetric Weyl Semimetals) to see what happens when you make them superconducting (a state where electricity flows with zero resistance).

Here is the story of their discovery, explained simply:

1. The Setting: A Twisted Highway

Think of the electrons in this material as cars on a highway.

• Normal Materials: The highway is flat and straight.
• Type-I Weyl Semimetals: The highway has a few steep, narrow ramps (called "Weyl nodes").
• Type-II Weyl Semimetals (The Star of the Show): The highway is tilted! It's so tilted that the "cars" (electrons) can move in two directions at once, creating huge, sprawling "traffic jams" of energy called Fermi Arcs. These arcs are like glowing bridges that only exist on the surface of the material, not in the middle.

2. The Mystery: How Do They Pair Up?

In a superconductor, electrons usually pair up like dance partners to move without friction. Usually, they pick one style of dance:

• S-wave: A simple, round hug (singlet pairing).
• P-wave: A more complex, spinning dance (triplet pairing).

The researchers asked: If we turn on the superconductivity in this tilted, surface-heavy material, which dance will the electrons choose?

3. The Discovery: A Split Personality

Using a powerful computer simulation (the "self-consistent method"), they found something surprising. The material didn't pick just one dance style. It developed a Hybrid Pairing, but with a twist: It depends on which side of the material you are looking at.

• The Bottom Surface: The electrons here decided to do the simple S-wave dance.
• The Top Surface: The electrons on the opposite side decided to do the complex P-wave dance.

The Analogy: Imagine a two-story building where the people on the bottom floor only like to hold hands in a circle, while the people on the top floor only like to spin in a line. The building itself forces this split because the "traffic patterns" (Fermi arcs) on the bottom floor look different from those on the top floor. The shape of the surface dictates the dance.

4. The Grand Finale: The "Hinge" States

This is where the physics gets really cool. Because the top and bottom are dancing differently, the edges where the sides meet (the "hinges" of the material) become special.

• The Bulk (Middle): The middle of the material is safe and quiet; it has a full "energy gap" (no traffic).
• The Surface (Top/Bottom): The surfaces are busy with the mixed dances.
• The Hinges (The Corners): This is the magic. Because the top and bottom are so different, the corners of the material become superhighways for electrons. These are called Higher-Order Topological Hinge States.

The Metaphor: Imagine a cube. The faces of the cube are busy with different parties. But the edges where the faces meet are the only places where a secret, frictionless tunnel exists. Electrons can zip along these corners without getting stuck, even though the rest of the material is "closed."

5. Why Does This Matter?

The researchers concluded that this specific type of material (Type-II Weyl Semimetals) is a perfect playground for creating "unconventional" superconductors.

• No Extra Parts Needed: Usually, to get these weird quantum states, you have to glue different materials together (like a sandwich). Here, the material does it all by itself because of its internal geometry.
• Tunable: You can change the "dance floor" by squeezing the material (pressure) or stretching it (strain), which moves the Weyl nodes around. This means we could potentially design materials that become superconductors at higher temperatures.

Summary

In simple terms, this paper shows that if you take a specific, tilted crystal and cool it down, the electrons on the top and bottom will naturally split into two different superconducting styles. This conflict creates a magical, frictionless highway running along the corners (hinges) of the crystal. It's a new way to build quantum computers and ultra-efficient electronics, all by understanding the shape of the electron's dance floor.

Technical Summary

1. Problem Statement

The paper addresses the challenge of realizing unconventional and topological superconductivity in time-reversal-symmetric (TR) Weyl semimetals (WSMs). While TR-breaking WSMs and Type-I TRWSMs have been studied, Type-II TRWSMs (characterized by tilted Weyl cones and large Fermi surfaces) remain under-explored as a platform for intrinsic superconductivity.

• Key Question: Can the unique electronic structure of Type-II TRWSMs, specifically the surface-localized Fermi arcs and bulk electron/hole pockets, induce a stable superconducting state with non-trivial topological properties (specifically higher-order topology)?
• Context: Previous studies focused largely on surface superconductivity in Type-I systems or proximity effects. There is a need to understand the intrinsic pairing mechanisms and the resulting topological phases in Type-II systems without external interfaces.

2. Methodology

The authors employed a self-consistent mean-field approach based on a minimal tight-binding model.

• Model Hamiltonian:
  • A two-orbital model describing a Type-II TRWSM.
  • The Hamiltonian includes onsite interorbital coupling ($M$), offsite interorbital hopping ($t_x, t_y, t_z, \lambda_z$), and intraorbital hopping ($\lambda_y$).
  • A long-range intraorbital parameter ($\gamma$) controls the transition from Type-I to Type-II WSM. For $|\gamma| > 0.5$, the system exhibits tilted Weyl cones and electron/hole Fermi pockets.
  • The superconducting interaction is modeled via a nearest-neighbor attractive exchange interaction ($H_I$).
• Boundary Conditions:
  • Normal State: Partially open boundary conditions in the $z$-direction to expose surface Fermi arcs.
  • Superconducting State: Partially open boundary conditions in both $x$ and $z$ directions to investigate hinge states (higher-order topology).
• Calculation Technique:
  • Self-Consistent Method: The superconducting order parameters ($\Delta$) were determined iteratively using the gap equations derived from the mean-field approximation.
  • Analysis Tools:
    • Spectral functions ($A(k, \omega)$) and Local Density of States (LDOS) to visualize surface states and gaps.
    • Quadrupole Moment ($q_{xz}$): Calculated to confirm the second-order topological nature of the bulk superconductor.

3. Key Contributions

1. Discovery of Hybrid Pairing: The study reveals that the ground state in Type-II TRWSMs is a hybrid pairing of singlet $s$-wave and triplet $p$-wave symmetries.
2. Surface-Dependent Pairing Dichotomy: A novel finding is that the dominant pairing symmetry is spatially separated based on the surface orientation:
   • Bottom Surface ($z=1$): Dominated by $s$-wave pairing.
   • Top Surface ($z=N_z$): Dominated by $p$-wave pairing.
   • This dichotomy is driven by the distinct geometric configurations of the Fermi arcs on opposite surfaces.
3. Higher-Order Topological Superconductivity (HOTSC): The system is identified as a second-order topological superconductor. While the bulk and surfaces are fully gapped, zero-energy hinge states emerge at the intersections of the edges (the hinges).
4. Mechanism Identification: The paper establishes that the geometry of the Fermi arcs (specifically their orientation relative to the Brillouin zone boundaries) acts as an intrinsic selector for pairing symmetry, stabilizing the hybrid state without external proximity effects.

4. Key Results

A. Normal-State Electronic Structure

• The system features tilted Weyl cones creating large electron and hole Fermi pockets, leading to a high density of states at the Fermi level.
• Fermi Arcs: Two distinct types of Fermi arcs exist on the surfaces:
  • Short horizontal arcs connecting Weyl points.
  • Long vertical arcs stretching across the Brillouin zone.
• Orbital Texture: The LDOS analysis shows that the sharp zero-energy peaks on the $z=1$ surface are dominated by orbital 2, while those on the $z=N_z$ surface are dominated by orbital 1, due to broken inversion symmetry.

B. Superconducting State and Hybrid Pairing

• Gap Structure: The bulk and surfaces are fully gapped (U-shaped LDOS).
• Spatial Separation:
  • On the $z=1$ surface, the intralayer $s$-wave order parameter (orbital 2) is dominant.
  • On the $z=N_z$ surface, the intralayer $p_x$-wave order parameter (orbital 1) is dominant.
• Origin of Asymmetry: The $s$-wave pairing ($\propto \cos k_x$) is stabilized where Fermi arcs are near $k_x=0$, while $p$-wave pairing ($\propto \sin k_x$) is stabilized where arcs are near $k_x=\pm \pi/2$. The different Fermi arc geometries on opposite surfaces select different pairing symmetries.
• Absence of $p+ip$: The system does not form a chiral $p+ip$ state because the $C_4$ symmetry is absent, and the Fermi surface passes through the nodal points of a $p+ip$ gap, which would leave the system gapless.

C. Topological Properties

• Hinge States: When boundaries are opened in both $x$ and $z$ directions, the bulk and edge bands are gapped, but zero-energy modes appear at the four hinges (corners of the $x-z$ plane).
• Spectral Evidence: The spectral functions at the corners show distinct edge bands crossing at $k_y = \pm \pi/2$, confirming the hinge states.
• Quadrupole Moment: The calculated quadrupole moment $q_{xz}(k_y)$ exhibits windings at $k_y = \pm \pi/2$, providing a topological invariant proof of the second-order nature. The opposite winding directions at $k_y = \pm \pi/2$ indicate the hinge states are non-chiral (no net current).

5. Significance

• New Platform for HOTSC: The paper identifies Type-II TRWSMs as a robust, intrinsic platform for realizing higher-order topological superconductivity, eliminating the need for complex heterostructures or external magnetic fields.
• Tunability: The authors suggest that the pairing symmetry and topological properties can be tuned via surface decoration, pressure, or strain, which can reconfigure the Weyl points and Fermi arcs.
• High-$T_c$ Potential: The large Fermi surface and high density of zero-energy states in Type-II WSMs suggest a pathway to achieving higher critical temperatures ($T_c$) for topological superconductivity compared to Type-I or TR-breaking systems.
• Fundamental Insight: The work provides a deep understanding of how surface Fermi arc geometry dictates the symmetry of the superconducting order parameter, offering a design principle for engineering unconventional superconductors.