Euler band topology and multiple hinge modes in three-dimensional insulators

This paper establishes that three-dimensional $C_{2z}T$-symmetric insulators characterized by the topological invariant $\bar{e}_2$ support a corresponding number of chiral hinge modes at surface domain walls, distinguishing these phases from stacked Chern insulators through the derivation of effective surface Hamiltonians and numerical tight-binding model demonstrations.

The Gist

Imagine you are holding a block of cheese. In a normal block of cheese, if you cut off the crust (the surface), the inside is just plain cheese. But in the world of topological insulators, the "cheese" is special. It's an electrical insulator on the inside (it doesn't conduct electricity), but the "crust" is a superhighway for electrons.

This paper takes that idea a step further. It's not just about the crust; it's about the edges where the crust meets the crust. Think of the sharp corner where the top of the cheese meets the side. In this new type of material, electricity doesn't just flow on the flat faces; it gets trapped and zooms along these sharp corners (or "hinges").

Here is the breakdown of what the authors discovered, using simple analogies:

1. The "Real" Magic (Euler Band Topology)

Usually, in quantum physics, things are described by complex numbers (like $i$, the square root of -1). But in this specific type of material, the math simplifies. The electrons behave as if they are living in a "real" world (no imaginary numbers).

The authors use a mathematical tool called the Euler Class to count how "twisted" the electron paths are.

• Analogy: Imagine a ribbon. If you twist it once and tape the ends together, you get a Möbius strip. If you twist it twice, it's different. The Euler class is like a counter that tells you exactly how many times the ribbon is twisted.
• In this paper, they look at a 3D block of material and check the "twist" on the top slice and the bottom slice. The difference between these two twists is a special number they call $\bar{e}_2$.

2. The "Traffic Jam" on the Surface

The authors realized that this "twist difference" ($\bar{e}_2$) creates a very specific problem on the surface of the material.

• The Surface Mass: Imagine the surface of the material is a road. Usually, this road is open. But because of the twist, the road gets "blocked" or "massive" (like a heavy truck parked on the road) in some areas.
• The Domain Wall: However, the direction of this blockage flips as you move across the surface. On the left side of a corner, the truck is facing North; on the right side, it's facing South.
• The Result: Right in the middle of that flip (the corner), the road clears up! The "truck" disappears, and a clear lane opens up. This clear lane is where the electrons can flow without resistance.

3. The Big Discovery: Multiple Lanes!

Previous research showed that if the twist difference was 1, you got one clear lane (one hinge mode) running along the corner.

This paper says: "What if the twist difference is 2? Or 3? Or 10?"

• The Finding: If the twist difference is 2, you don't just get one lane; you get two parallel lanes of electricity zooming along the corner. If the difference is 3, you get three lanes.
• The Rule: The number of "superhighway lanes" on the corner is exactly equal to the number of twists ($\bar{e}_2$).

4. Why This is Different (The "Stacked" vs. "Fused" Analogy)

You might think, "Okay, if I want 2 lanes, I just stack two normal 1-lane materials on top of each other."

• The Catch: The authors show this is not the same thing.
• Analogy: Imagine stacking two separate sheets of paper. You have two sheets. Now, imagine gluing them together so tightly they become a single, new sheet of paper that has properties neither of the original sheets had.
• This new material is a single, fused object. If you try to add more "normal" layers to it, the magic lanes might disappear or change. This makes it a unique, fragile, but very special state of matter.

5. How They Proved It

The team did two things:

1. The Math (Theory): They built a theoretical model using "low-energy continuum Hamiltonians" (fancy math for describing how electrons move at slow speeds). They showed that if you have a twist of $N$, the math forces $N$ lanes to appear at the corners.
2. The Simulation (Computer Models): They built digital versions of these materials on a computer (using "tight-binding models"). They simulated the electrons moving and saw exactly what the math predicted:
   • A model with twist 2 showed 2 corner lanes.
   • A model with twist 3 showed 3 corner lanes.

Why Should We Care?

This isn't just about cheese blocks. These materials could be the future of low-power electronics.

• Because the electricity flows only on the sharp corners, it's extremely efficient and doesn't lose energy as heat.
• Having multiple lanes (2, 3, or more) means we could potentially carry more data or current through a single tiny corner, which is great for making smaller, faster computer chips.
• The authors also mention that while this is hard to find in natural rocks, we can build these materials using sound waves (acoustics) or light (photonic crystals) right now in the lab.

In a nutshell:
The authors discovered a new rule of nature for 3D materials: The number of "superhighways" running along the sharp corners of a material is directly determined by a hidden "twist" in the material's internal structure. If you twist it twice, you get two highways; twist it three times, you get three. It's a new way to control electricity, sound, or light with extreme precision.

Technical Summary

1. Problem Statement

The paper addresses the gap in understanding the role of Euler band topology in three-dimensional (3D) insulators.

• Context: In 2D systems with space-time inversion symmetry ($C_{2z}T$), wave functions can be chosen to be real, leading to a $\mathbb{Z}$-valued topological invariant known as the Euler class ($e_2$). This characterizes the topology of a pair of isolated real bands.
• The Gap: While previous studies established that 3D $C_{2z}T$-symmetric insulators with a difference in Euler classes between the $k_z=0$ and $k_z=\pi$ planes (denoted as $\bar{e}_2 = e_2(0) - e_2(\pi)$) support surface states, it was unclear whether these systems exhibit higher-order boundary states (specifically, hinge modes) and how the number of such modes relates to the topological invariant $\bar{e}_2$.
• Specific Challenge: Unlike standard topological insulators where invariants are stable under the addition of trivial bands, Euler band topology is "fragile" (defined only for a specific number of isolated bands). The authors aim to determine if 3D Euler insulators with $\bar{e}_2 = N$ support $N$ chiral hinge modes and to clarify the physical mechanism behind this correspondence.

2. Methodology

The authors employ a dual approach combining analytical continuum theory and numerical tight-binding modeling:

• Analytical Continuum Theory:

  • They construct generic low-energy continuum Hamiltonians for 3D Euler insulators characterized by $\bar{e}_2 = 1, 2, 3$, and generally $N$.
  • Surface Derivation: They derive effective surface Hamiltonians by introducing a spatially varying mass term ($\lambda \to \lambda(x)$) to model a boundary (e.g., the (100) surface).
  • Domain Wall Analysis: They analyze the "surface mass" profile. By breaking time-reversal symmetry with a specific mass term ($m_n \Gamma_5$), they create domain walls where the mass changes sign between adjacent surfaces (e.g., (100) vs. (010)).
  • Boundary State Counting: They solve the resulting differential equations for the boundary states. Crucially, they analyze the order of spatial derivatives ($m$-th order) required to generate gapless solutions, showing that the number of independent gapless solutions corresponds to the winding number (Euler class) of the bulk Hamiltonian.

• Numerical Tight-Binding Models:

  • They construct specific lattice models on simple cubic and stacked triangular lattices corresponding to the continuum Hamiltonians for $\bar{e}_2 = 2$ and $\bar{e}_2 = 3$.
  • Topological Invariant Calculation: They compute the Euler class using the Wilson loop matrix (specifically the winding number of the Wannier center phase) on the $k_z=0$ and $k_z=\pi$ planes to confirm the bulk topology.
  • Spectral Analysis: They calculate energy spectra under various boundary conditions (Periodic Boundary Conditions vs. Open Boundary Conditions) to visualize the emergence of hinge modes localized at the edges of the 3D crystal.

3. Key Contributions

• Generalization of Bulk-Hinge Correspondence: The paper establishes a direct bulk-boundary correspondence for 3D Euler insulators: a topological invariant $\bar{e}_2 = N$ guarantees the existence of $N$ chiral hinge modes along the $z$-direction.
• Mechanism of Multiple Hinge Modes: The authors reveal that multiple hinge modes arise from multiple gapless boundary solutions at the domain walls of the surface mass. For $\bar{e}_2 = N$, the bulk Hamiltonian's dispersion (e.g., quadratic for $N=2$, cubic for $N=3$) leads to multiple distinct boundary state wavefunctions that remain gapless at the hinge.
• Distinction from Stacked Chern Insulators: The paper clarifies that these phases are topologically distinct from simply stacking $N$ layers of 2D Chern insulators. The hinge modes in Euler insulators rely on the specific "fragile" nature of the Euler class (requiring exactly two occupied bands) and the non-Abelian braiding of band degeneracy points, rather than a simple additive Chern number.
• Continuum Theory for Arbitrary $N$: The authors generalize their theory to arbitrary positive integers $N$, proving that the number of chiral hinge modes scales linearly with the Euler class difference $\bar{e}_2$.

4. Results

• $\bar{e}_2 = 1$: Revisited to confirm the existence of a single chiral hinge mode, consistent with previous literature linking $\bar{e}_2 \mod 2$ to the Chern-Simons invariant.
• $\bar{e}_2 = 2$:
  • Theory: The continuum Hamiltonian with quadratic dispersion terms yields two distinct boundary state solutions ($\psi_1, \psi_2$) localized at the surface mass domain walls.
  • Simulation: The tight-binding model on a cubic lattice confirms the presence of two chiral hinge modes localized at the hinges when open boundary conditions are applied in $x$ and $y$. Wilson loop calculations verify $|e_2(0)|=2$ and $e_2(\pi)=0$.
• $\bar{e}_2 = 3$:
  • Theory: The Hamiltonian with cubic dispersion terms generates three gapless solutions. One arises from the dominant third-order spatial derivative, and two arise from lower-order terms, totaling three modes.
  • Simulation: The tight-binding model on a stacked triangular lattice successfully demonstrates three chiral hinge modes.
• $\bar{e}_2 = N$: The generalized theory proves that for any integer $N$, the system supports exactly $N$ chiral hinge modes. The analysis of $m$-th order spatial derivatives shows that the number of valid boundary solutions matches the winding number of the bulk.

5. Significance

• New Paradigm for Higher-Order Topology: This work extends the concept of higher-order topological insulators (HOTIs) beyond the standard classification based on stable invariants (like $\mathbb{Z}_2$ or Chern numbers) to fragile topological phases. It demonstrates that fragile topology can robustly support multiple hinge modes as long as the band isolation condition is met.
• Experimental Relevance: The authors highlight that Euler band topology has already been realized in artificial platforms (acoustic metamaterials, photonic crystals, transmission line networks). The prediction of multiple hinge modes ($N > 1$) provides a clear experimental signature for detecting 3D Euler insulators in these systems.
• Material Candidates: While isolating two real bands in 3D electronic systems is challenging, the paper suggests that materials like twisted bilayer graphene, ZrTe, and RE$_8$CoX$_3$ families may host these phases, particularly when the Euler class is odd (where at least one mode remains robust due to connection to the Chern-Simons invariant).
• Theoretical Framework: The derivation of effective surface Hamiltonians for arbitrary $N$ provides a robust theoretical tool for analyzing complex topological phases where standard K-theory classifications fail.

In summary, Tanaka and Kobayashi successfully bridge the gap between abstract Euler band topology and observable physical phenomena in 3D, demonstrating that the difference in Euler classes between high-symmetry planes directly dictates the multiplicity of chiral hinge modes in 3D insulators.