Two-dimensional higher-order Weyl semimetals

This paper proposes a theoretical scheme to realize two-dimensional higher-order Weyl semimetals with topological corner states by coupling a trilayer topological insulator with a d-wave altermagnet, which gaps helical edge states while preserving Weyl points governed by subspace topology.

The Gist

Imagine the world of electronics as a bustling city. In this city, electrons are the commuters, and the materials they travel through are the roads. Usually, these roads are either wide highways (conductors) where traffic flows freely, or dead-end cul-de-sacs (insulators) where traffic stops completely.

But in recent years, physicists discovered a special kind of "magic road" called a Topological Insulator. On the inside, it's a dead-end (insulator), but on the very edges, it's a super-highway where electrons can zip along without hitting any bumps or traffic jams.

This paper proposes a way to build an even stranger, more advanced version of this magic road system: a 2D Higher-Order Weyl Semimetal.

Here is the story of how they did it, explained simply:

1. The Starting Point: A Three-Layer Cake

The researchers started with a "sandwich" made of three layers of a special material called Bismuth Selenide (a topological insulator). Think of this as a three-layer cake.

• The Result: Without any extra help, this cake acts like a 2D "Weyl Semimetal."
• What does that mean? Imagine the surface of the cake has four special "traffic circles" (called Weyl points) where the roads cross over each other in a very specific way. Because of this, electrons can travel along the edges of the cake in a perfect, one-way loop. These are called helical edge states. It's like a highway that circles the entire perimeter of the cake, but the middle of the cake is empty.

2. The Secret Ingredient: The "D-Wave Altermagnet"

To get to the next level, the researchers introduced a new ingredient: a d-wave altermagnet.

• What is an altermagnet? Think of it as a magnetic material that is a bit of a chameleon. It's not a standard magnet (like a fridge magnet) where all the north poles point one way. It's also not a standard anti-magnet where north and south cancel each other out perfectly. Instead, it has a pattern where the magnetic "spin" flips direction depending on the angle you look at it (like a flower petal pattern).
• The Magic Trick: They placed this altermagnet on the top and bottom layers of their cake, pointing straight up and down.

3. The Transformation: From Edge Roads to Corner Hubs

When they added this magnetic ingredient, something fascinating happened:

• The Edge Roads Closed: The magnetic field acted like a construction crew that closed down the circular highway running along the edges of the cake. The electrons could no longer travel along the sides.
• The Traffic Circles Stayed: However, two of the four "traffic circles" (Weyl points) in the middle of the cake remained open. The bulk of the material is still a semimetal.
• The New Hubs Appear: Here is the "Higher-Order" part. Since the edge roads are closed, the electrons have nowhere to go... except to the corners.
  • Imagine a square room. If you block the walls, the only place left to stand is the four corners.
  • In this new state, the electrons get trapped and concentrated exactly at the four corners of the material. These are called Topological Corner States.

Why is this a Big Deal?

Think of it like a game of musical chairs.

1. Normal Topological Insulator: The music stops, and everyone sits on the chairs along the walls (edge states).
2. This New Discovery: The music stops, the wall chairs are removed, but suddenly, four special VIP chairs appear in the corners of the room, and the electrons sit there perfectly protected.

The "Why" Behind the Magic

The paper explains that this happens because of the symmetry of the system.

• The three-layer cake has a specific geometric balance.
• When the magnetic ingredient is added, it breaks some of the rules but keeps others.
• This creates a "winding number" (a mathematical way of counting how many times a path twists). In the specific directions of the crystal, this number is non-zero, which forces the electrons to gather at the corners. It's like a mathematical law that says, "If the edge is blocked, the charge must go to the corner."

What Can We Do With This?

The researchers suggest that we could actually build this in a lab using Manganese Fluoride (a magnetic material) and Bismuth Selenide.

• How to see it? They propose using a Scanning Tunneling Microscope (STM), which is like a super-powerful camera that can see individual atoms.
• The Proof: If you scan the corners of this new material, you should see a sharp, bright spike in the energy signal, proving that the electrons are indeed hanging out in the corners.

Summary

In short, this paper describes a recipe to turn a standard 3-layer electronic cake into a "corner-hugging" machine. By adding a special magnetic layer, they shut down the edge highways and force the electrons to congregate at the four corners. This creates a new type of matter that could be incredibly useful for future quantum computers, as these corner states are very stable and hard to disturb.

Technical Summary

1. Problem Statement

The study addresses the gap in understanding the connection between higher-order topology (specifically corner states) and two-dimensional (2D) semimetals.

• Context: While higher-order topological insulators (HOTIs) with corner states are well-established, and higher-order topological semimetals (like hinge states in 3D) have been predicted, the realization of 2D higher-order Weyl semimetals hosting both bulk Weyl points and topological corner states remains an open theoretical challenge.
• Goal: To propose a feasible theoretical scheme to realize a 2D system that simultaneously exhibits bulk Weyl points and gapped edge states leading to localized corner states, utilizing the emerging class of materials known as altermagnets.

2. Methodology

The authors employ a tight-binding model approach combined with symmetry analysis and topological invariant calculations.

• System Model:
  • Base Material: A trilayer film of the topological insulator Bi$_2$Se$_3$. The Hamiltonian includes kinetic energy terms and inter-layer coupling.
  • Perturbation: The system is coupled with a $d$-wave altermagnet oriented along the $z$-direction (out-of-plane). This is modeled via a magnetic proximity effect acting on the top and bottom layers, described by the term $H_{AM} = J(\cos k_x - \cos k_y)s_z \otimes \text{diag}(1, 0, 1)$.
• Analytical Tools:
  • Symmetry Decomposition: The bulk Hamiltonian is decomposed into symmetric subspaces based on the mirror symmetry operator $M$. This allows the system to be treated as three decoupled sub-Hamiltonians ($H_{+1}, H_{-1}, H_0$).
  • Topological Invariants:
    • Chern Numbers: Calculated for gapped subspaces to characterize edge states.
    • Winding Numbers: Calculated along high-symmetry lines ($k_x = \pm k_y$) within specific subspaces to characterize the corner states.
  • Numerical Simulations: Energy spectra were computed for nanoribbons (open boundary conditions) and finite-size nanoflakes (rectangular geometry) to visualize edge and corner states.

3. Key Contributions

1. Proposal of a New Phase: The paper theoretically proposes the first scheme for a 2D Higher-Order Weyl Semimetal (HOWSM). This phase is distinct because it hosts bulk Weyl points and topological corner states simultaneously.
2. Role of Altermagnetism: It demonstrates that a $d$-wave altermagnet is the critical ingredient. Unlike ferromagnets or antiferromagnets, the specific symmetry of the altermagnet (breaking time-reversal and spatial inversion but preserving certain rotation/mirror symmetries) allows it to gap out the helical edge states while preserving specific bulk Weyl points.
3. Symmetry-Governed Topology: The work elucidates how the topology of symmetric subspaces governs the formation of both Weyl points and corner states. The persistence of Weyl points in the $H_0$ subspace and the non-zero winding numbers in the gapped $H_{\pm 1}$ subspaces are the mathematical origins of the phase.

4. Results

• Phase 1: Pristine Trilayer (No Altermagnet, $J=0$):

  • The system behaves as a standard 2D Weyl Semimetal.
  • Bulk: Contains four Weyl points located at high-symmetry points ($\Gamma, M, X_1, X_2$) in the Brillouin zone.
  • Boundary: Exhibits gapless helical edge states connecting the Weyl points.
  • Topology: Characterized by Chern numbers $C = \pm 1$ in the gapped subspaces, confirming the existence of chiral edge modes.

• Phase 2: With $d$-wave Altermagnet ($J \neq 0$):

  • Edge States: The altermagnet opens a gap in the helical edge states, destroying the first-order boundary modes.
  • Bulk States: The Weyl points at $X_1$ and $X_2$ become gapped, but the Weyl points at $\Gamma$ and $M$ persist.
  • Corner States: In finite-size nanoflakes, four zero-energy corner states emerge within the edge gap. The wavefunctions of these states are localized at the four corners of the sample.
  • Topological Origin:
    • The system retains $M$ symmetry along the high-symmetry lines $k_x = \pm k_y$.
    • The subspace Hamiltonians $H_{\pm 1}$ along these lines possess a non-zero winding number ($v = \pm 1$).
    • This non-zero winding number is the topological invariant protecting the corner states.

• Topological Phase Diagram:

  • The authors mapped the phase diagram as a function of hybridization gap ($m_0$) and parabolic band component ($m_1$).
  • HOWSM Phase: Exists when $|m_0| < 8|m_1|$ and $m_0 m_1 < 0$. This region supports both bulk Weyl points and corner states.
  • WSM Phase: Exists in other parameter regimes where only bulk Weyl points exist without corner states.

5. Significance and Implications

• Paradigm Shift: This work bridges the gap between 2D semimetals and higher-order topology, proving that 2D systems can host "higher-order" features (corner states) without being insulators.
• Experimental Feasibility: The authors propose a realistic experimental setup: sandwiching a Bi$_2$Se$_3$ trilayer between MnF$_2$ (an altermagnetic material). The magnetic proximity effect would induce the required $d$-wave altermagnetism.
• Detection: The paper suggests that Scanning Tunneling Microscopy (STM) can detect these states. While the pristine system shows broad peaks, the induced corner states should manifest as sharp, localized peaks at the sample corners in the energy spectrum.
• Future Applications: The ability to manipulate corner states in semimetals opens avenues for "cornertronics" and novel quantum devices that leverage the unique transport properties of higher-order topological semimetals.

In summary, this paper provides a comprehensive theoretical framework for realizing 2D higher-order Weyl semimetals, identifying the specific symmetry requirements and material platforms needed to observe these exotic topological phases.