Localized states, topology and anomalous Hall conductivity on a 30 degrees twisted bilayer honeycomb lattice

This paper investigates a $30^{\circ}$ twisted bilayer Haldane model, revealing that while weak interlayer coupling preserves the topological properties of isolated layers, strong coupling induces a non-topological bulk gap, multifractal states, and distinct corner states that are unrelated to the original edge states, thereby demonstrating that topological entanglement entropy and anomalous Hall conductivity effectively characterize the system's topology.

The Gist

Imagine you have two sheets of graph paper. On each sheet, you've drawn a honeycomb pattern (like a beehive), and you've painted little arrows on the lines to show that electrons (the tiny particles carrying electricity) prefer to move in a specific, swirling direction. This is a "topological" material: it's like a one-way street for electricity that is incredibly robust. Even if you scratch the paper or make a small hole, the electricity keeps flowing along the edges without getting stuck.

Now, imagine taking these two sheets and stacking them on top of each other, but you twist one of them by exactly 30 degrees.

This creates a weird, non-repeating pattern called a quasicrystal. It's like looking at two overlapping honeycombs and trying to find a repeating pattern—you can't. It's a beautiful mess that follows rules but never repeats itself.

This paper is about what happens to our "one-way street" for electricity when we twist these two sheets and then start gluing them together (turning up the "interlayer coupling").

Here is the story of what the author, Grigory Bednik, discovered:

1. The Gentle Glue (Weak Coupling)

When the two sheets are just barely touching, everything works exactly as you'd expect. The "one-way street" (the topological property) survives perfectly. The electrons still flow smoothly along the edges of the material, ignoring the messy twist in the middle. It's like having two separate highways that happen to be stacked; traffic flows fine on both.

2. The Heavy Glue (Strong Coupling)

Now, imagine we press the sheets together really hard, or use a super-strong glue. The electrons start hopping back and forth between the top and bottom sheets so fast that the neat, repeating rules of the original sheets get scrambled.

• The Highway Collapses: The main "bulk" gap (the safe zone where the one-way street exists) closes up. The smooth edge states disappear. It looks like the topological magic has vanished.
• The New Gap: But wait! If you press hard enough, a new gap opens up. It looks like a safe zone again. You might think, "Aha! The topological magic is back!"

3. The Surprise: It's a Trap, Not a Treasure

The author found that this new gap is a decoy. Even though there is a gap, the material is not topological anymore.

• The Corner Traps: In the old days, electrons would flow along the edges. Now, at strong coupling, the electrons get stuck in weird places. They don't just stay on the edge; they get trapped in the corners of the square sheet, and surprisingly, some get trapped right in the center of the sheet!
• The "Multifractal" Mess: The electrons aren't flowing smoothly, nor are they just sitting still. They are in a weird state called "multifractal." Imagine a cloud of smoke that is neither a solid block nor a thin gas, but has a complex, self-similar structure that looks different no matter how much you zoom in. The electrons are everywhere and nowhere at once.

4. How Did They Know? (The Detective Work)

Since the material is a messy quasicrystal, you can't use the standard math tools (which rely on perfect repeating patterns) to check if it's topological. So, the author used three clever "detective tools":

1. Topological Entanglement Entropy: Think of this as measuring how "knotted" the electrons are with each other. In a topological material, there's a specific, non-zero "knot" value. In this strong-glue phase, the knot value dropped to zero. Verdict: Not topological.
2. Local Chern Marker: This is like a map that tells you where the "one-way street" is. In a normal crystal, the map is blue in the middle and red at the edges. In this twisted, strong-glue material, the map became a chaotic static noise. The "blue" and "red" areas were mixed up everywhere. Verdict: No clear one-way street.
3. Anomalous Hall Conductivity: This measures how electricity curves when you push it. The author showed that in this messy system, the electricity doesn't curve in a uniform, predictable way like it does in a perfect crystal. It behaves locally, just like the "Chern marker" map. Verdict: Confirms the chaos.

The Big Picture

The paper teaches us a valuable lesson: Just because a material has a gap (a safe zone) and electrons stuck in corners, it doesn't mean it's a topological insulator.

In the world of perfect crystals, corners usually mean "topology." But in the messy, twisted world of quasicrystals, corners can just be a result of the geometry getting weird, not a sign of deep topological magic.

The Takeaway:
The author successfully created a "topological quasicrystal" by twisting two perfect sheets. But as they twisted and glued them harder, the topological protection broke down, leaving behind a chaotic, multifractal mess where electrons get stuck in corners and the center, but without the special "one-way street" protection we love.

It's a reminder that in physics, appearance can be deceiving. A gap doesn't always mean safety, and a corner doesn't always mean topology. Sometimes, it's just a very complex, beautiful mess.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing topological phases in quasicrystals, specifically those lacking translational symmetry. While topological concepts like Chern numbers are well-defined in momentum space for periodic crystals, they are not directly applicable to quasicrystals. The authors investigate whether a system constructed by smoothly deforming a known topological crystal (the Haldane model) into a quasicrystalline structure (a 30° twisted bilayer) retains its topological properties. Specifically, they explore the evolution of the system's energy spectrum, localization properties, and topological invariants as the interlayer coupling strength ($t_{inter}$) is varied.

2. Methodology

The study employs a tight-binding model of a 30° twisted bilayer honeycomb lattice, where each monolayer is described by the Haldane model (a Chern insulator).

• Hamiltonian: The total Hamiltonian $H$ consists of two Haldane Hamiltonians ($H_1, H_2$) for the individual layers and an interlayer interaction term ($V_{1-2}$) that decays exponentially with distance.
• Parameters: The system is characterized by the Haldane mass ($m$), the interlayer coupling strength ($t_{inter}$), and system size ($L_x = L_y$). Open boundary conditions are used.
• Numerical Techniques:
  • Exact Diagonalization: Used for systems up to $L \approx 40$ to compute the full energy spectrum and wavefunctions.
  • Lanczos Diagonalization: Used for larger systems ($L \approx 800$) to study the scaling of fractal dimensions.
  • Fractal Dimensions ($D_q$): Calculated to distinguish between extended, localized, and multifractal states.
  • Topological Entanglement Entropy ($\gamma$): Computed using a specific subtraction scheme (four-subsystem configuration) to isolate the topological contribution from area-law terms.
  • Local Chern Marker ($C(\vec{r}_i)$): An empirical real-space measure of topology used to map local topological properties without momentum space.
  • Anomalous Hall Conductivity (AHC): Derived using the Kubo formula in real space (without momentum representation) to calculate the local Hall conductivity $\sigma_{ij}$ and its sum over the lattice.

3. Key Contributions

• Realization of a Topological Quasicrystal: The authors demonstrate that a 30° twisted bilayer of Chern insulators acts as a topological quasicrystal at weak coupling, retaining the topological edge states and bulk gap of the constituent layers.
• Discovery of Non-Topological Corner States: They identify localized states at the corners and the center of the lattice at strong coupling. Crucially, they argue these are not topological edge states derived from the bulk gap, as their energies do not track the gap and they persist even when the gap closes or reopens.
• Multifractality in Strong Coupling: The paper establishes that at strong interlayer coupling, the system enters a multifractal phase where eigenstates are neither fully extended nor fully localized.
• Validation of Real-Space Topological Probes: The study confirms that Topological Entanglement Entropy, Local Chern Markers, and Anomalous Hall Conductivity (computed via Kubo formula in real space) are consistent and effective tools for characterizing topology in non-periodic systems.

4. Key Results

A. Phase Diagram and Energy Spectrum

The system exhibits distinct phases based on $m$ and $t_{inter}$:

1. Weak Coupling ($t_{inter} \approx 0$): The system behaves as a superposition of two Haldane models. It possesses a bulk energy gap (for $m < 1$) and topological edge states. The topological entanglement entropy $\gamma$ is non-zero.
2. Intermediate Coupling: As $t_{inter}$ increases, the bulk gap shrinks and eventually closes (gapless phase). Topological edge states disappear.
3. Strong Coupling ($t_{inter} \gg 1$):
   • A new bulk gap may reopen at small $m$ (and $m \gtrsim 1$).
   • However, this new gap is non-topological. The topological entanglement entropy $\gamma$ approaches zero, and the Local Chern marker fluctuates wildly within the bulk rather than maintaining a constant value.
   • The system hosts multifractal states ($0 < D_q < 1$).

B. Localized States

• Corner and Center States: At strong coupling, states localize at the corners and, surprisingly, at the center of the lattice.
• Non-Topological Nature: The location of these states is not tied to the bulk gap. For example, at $L=30$, corner states lie within the bulk gap for a range of $t_{inter}$ but merge with the bulk spectrum at other values. This contrasts with topological corner states in higher-order topological insulators, which are strictly tied to the bulk gap.
• Origin: These states arise from the lack of translational symmetry, which allows for isolated localized states in the bulk, unlike periodic crystals.

C. Topological Characterization

• Local Chern Marker: In the weak coupling regime, the marker is constant in the bulk (equal to the Chern invariant) and cancels out at the edges. In the strong coupling regime, it fluctuates randomly throughout the bulk, indicating the loss of a well-defined topological phase.
• Anomalous Hall Conductivity (AHC): The authors derive a real-space Kubo formula for AHC. They find that for a finite, isolated sample, the total Hall conductivity sums to zero (due to the cancellation of bulk and edge contributions). However, the local AHC behaves identically to the Local Chern marker: it is constant in the topological phase and fluctuates in the multifractal phase. This suggests AHC can serve as a local topological probe in quasicrystals.

5. Significance

• Theoretical Framework: The paper provides a systematic approach to studying topology in quasicrystals by bridging the gap between crystalline topological insulators and disordered/quasiperiodic systems.
• New Phase of Matter: It identifies a "strongly coupled multifractal" phase where topology is lost, yet the system retains complex localization properties (multifractality) distinct from standard Anderson localization.
• Experimental Relevance: While 30° twisted bilayer graphene typically has weak coupling, the authors suggest that similar physics could be realized in other 2D material stacks with incommensurate angles or different lattice constants, or via ultracold atom simulations.
• Methodological Advance: The successful application of AHC and Entanglement Entropy in real space validates these metrics as robust tools for diagnosing topology in systems where momentum-space definitions fail.

In conclusion, the work demonstrates that while weak coupling preserves the topological nature of the Haldane model in a twisted bilayer, strong coupling drives the system into a non-topological, multifractal phase characterized by unique localized states that are not protected by bulk topology.