Fragile topology for six-fold rotation symmetry indicated by the concentric Wilson loop spectrum

This paper investigates topological phase transitions in Haldane and Kane-Mele models with six-fold rotation symmetry, demonstrating that while the Concentric Wilson Loop Spectrum successfully identifies topological properties where conventional invariants fail, the resulting topology for the six-fold symmetric system is fragile rather than robust.

The Gist

Imagine you are an architect designing a new kind of city. This city isn't built on a standard square grid like Manhattan; instead, it's a unique mosaic made of hexagons and triangles interlocked together. This specific pattern has a special rule: if you spin the whole city 60 degrees, it looks exactly the same. In physics, we call this six-fold rotation symmetry.

The scientists in this paper are exploring what happens to "traffic" (electrons) moving through this city. They want to know if the city acts like a normal insulator (where traffic stops completely) or a topological insulator (a special material that blocks traffic in the middle but lets it flow freely along the edges, like a one-way highway).

Here is the story of their discovery, broken down into simple concepts:

1. The Two Types of Cities

The researchers studied two versions of this hexagon-triangle city:

• The "One-Way" City (Haldane Model): In this version, the rules of the road are rigged so that traffic must go clockwise or counter-clockwise, but not both. This breaks "time-reversal symmetry" (you can't just play the movie backward and have the traffic make sense).

  • The Discovery: By tweaking how easily cars can jump between buildings (changing "hopping parameters"), they found they could create "super-highways" with very high traffic capacity. In physics terms, they found phases with high Chern numbers (values like 3 or 4), meaning the edge currents are incredibly strong and complex.

• The "Two-Way" City (Kane-Mele Model): In this version, traffic can go both ways, and the city respects time-reversal symmetry (if you play the movie backward, it still looks logical). This is more like a real-world material.

  • The Goal: They wanted to find a special "topological fingerprint" that proves this city is a topological insulator, even when the usual fingerprints fail.

2. The "Concentric Wilson Loop" (The Magic Ruler)

To check the topological fingerprint of the "Two-Way" city, the scientists used a clever tool called the Concentric Wilson Loop Spectrum (CWLS).

• The Analogy: Imagine you are standing in the center of a round city. You draw a small circle around yourself, then a slightly bigger one, then a bigger one, until you cover the whole city.
• The Measurement: As you expand your circle, you are measuring a "twist" in the traffic flow. If the traffic flow twists in a specific way as your circle grows, it leaves a mark.
• The "Crossing": The scientists look for a specific moment where the "twist" crosses a magical threshold (called a $\pi$-crossing). If this crossing happens an odd number of times, the city is topologically special.

For a long time, physicists believed this "Magic Ruler" (CWLS) was the Holy Grail—a strong, unbreakable rule that could identify topological insulators in any city with rotational symmetry, even when other tools failed.

3. The Big Surprise: The "Fragile" Topology

Here is the plot twist. When the scientists applied this Magic Ruler to their six-fold symmetric city, they found something shocking.

They discovered that the "twist" they were measuring was Fragile.

• The Metaphor: Imagine you have a stack of playing cards. If you look at just the top two cards, they might form a perfect, magical pattern (a "non-trivial" state). But, if you suddenly glue a boring, ordinary card (a "trivial" band) to the bottom of that stack, the magical pattern disappears.
• The Reality: The CWLS invariant they were measuring only worked if the specific pair of energy bands they were looking at were isolated from the rest. As soon as those bands mixed with other "boring" bands, the topological fingerprint vanished.

In physics terms, this is called Fragile Topology. It's like a house of cards: it looks stable until you breathe on it (mix it with other states), and then it collapses.

4. Why This Matters

Before this paper, there was a big debate in the physics community. Some experts thought the CWLS was the missing piece of the puzzle—a "Strong Invariant" that would finally complete the classification of all topological materials.

This paper says: "Not so fast."

The authors show that for this six-fold symmetric city, the CWLS is not a strong, permanent rule. It is fragile and can be destroyed by mixing. This means the search for the "true" missing invariant is not over. We still don't have the complete map of all possible topological states in nature.

Summary

• The Setup: A unique city made of hexagons and triangles.
• The Tool: A "Magic Ruler" (CWLS) that measures twists in electron flow.
• The Expectation: That this ruler would be a permanent, unbreakable proof of a topological state.
• The Reality: The ruler is "fragile." It works only if the system is perfectly isolated. If you mix the parts, the proof disappears.
• The Lesson: Nature is more subtle than we thought. The "missing link" in our understanding of topological materials is still out there, waiting to be found.

Technical Summary

1. Problem Statement

The paper addresses the classification of topological phases in crystalline systems with specific spatial symmetries. While the "ten-fold way" classifies topological insulators based on time-reversal (TRS), particle-hole, and chiral symmetries, recent extensions incorporate lattice symmetries (e.g., rotation).

• The Gap: A recent K-theory classification proposed that for systems with TRS and $n$-fold rotational symmetry (but lacking other lattice symmetries), there exists a unique topological invariant derived from the Concentric Wilson Loop Spectrum (CWLS). This invariant was previously shown to detect topological phases in 3-fold and 4-fold symmetric models that conventional invariants (like the $Z_2$ invariant) might miss.
• The Question: Does this CWLS invariant represent a "strong" topological invariant (robust against band hybridization) in 6-fold symmetric systems, or does it exhibit fragile topology (where the invariant changes when the system is hybridized with trivial bands)? The authors investigate this by extending the analysis to a 6-fold ($p6$) symmetric lattice.

2. Methodology

The authors constructed a tight-binding model on a 2D lattice composed of hexagons and triangles, designed to possess $p6$ (six-fold) rotational symmetry but no other spatial symmetries.

• Model Hamiltonians:
  • Haldane Model (Broken TRS): A spinless model with nearest-neighbor (NN) hopping ($t_h$ in hexagons, $t_t$ in triangles), complex next-nearest-neighbor (NNN) Haldane hopping ($\gamma_h, \gamma_t$) to break TRS, and next-next-nearest-neighbor (NNNN) hopping ($t'_h$).
  • Kane-Mele Model (Preserved TRS): A spinful extension where the NNN terms become intrinsic spin-orbit coupling (SOC), preserving TRS.
• Topological Invariants Calculated:
  • Chern Number ($C$): Calculated for the Haldane model using a discrete multi-band non-Abelian Wilson loop formalism. The Brillouin zone (BZ) is discretized, and Berry flux is summed over plaquettes.
  • $Z_2$ Invariant: Calculated for the Kane-Mele model using the parallel Wilson loop approach.
  • Concentric Wilson Loop Spectrum (CWLS): The core method of the study. The authors define a family of Wilson loops starting from the BZ center and expanding to enclose fractional sectors ($1/6$ of the BZ) consistent with $C_6$ symmetry. They analyze the eigenvalues of these loops (specifically $\pi$-crossings) for individual Kramers pairs.
• Parameter Space: The study systematically varies hopping parameters ($t_h, t_t, t'_h$) and SOC strengths ($\gamma_h, \gamma_t$) to map out phase diagrams.

3. Key Contributions and Results

A. Haldane Model (Broken TRS)

• Rich Phase Diagram: By independently tuning hopping in triangles and hexagons, the authors generated a diverse set of topological phases.
• High Chern Numbers: The introduction of NNNN hopping ($t'_h$) significantly enriched the phase diagram. While standard parameters yielded Chern numbers up to $\pm 2$ or $\pm 3$, the inclusion of $t'_h$ allowed for Chern numbers up to $\pm 4$.
• Band Structure Evolution: The model exhibits various gap-closing and reopening scenarios, including the formation of Dirac cones at high-symmetry points ($\Gamma, K, K', M$) and their subsequent gapping.

B. Kane-Mele Model (Preserved TRS)

• Application of CWLS: The authors successfully applied the CWLS formalism to a 6-fold symmetric system, calculating the number of $\pi$-crossings for individual Kramers pairs.
• Fragile Topology Discovery: This is the paper's most significant finding.
  • The CWLS invariant was found to be fragile.
  • Evidence: The value of the CWLS invariant (specifically the parity of $\pi$-crossings) changes when the occupied Kramers pairs hybridize with other occupied bands (i.e., when internal band gaps close and reopen).
  • Phase Diagrams: The phase diagrams constructed by summing $\pi$-crossings were partitioned by all internal gap-closing lines within the subspace, not just the gap above the occupied subspace. This behavior contrasts with "strong" invariants, which should remain robust against such hybridization.
• Comparison with $Z_2$: The authors compared the CWLS results with the total $Z_2$ invariant (calculated via Berry flux). They demonstrated that while the CWLS provides detailed information about individual Kramers pairs, its value is not a stable strong invariant for the entire system in the presence of band mixing.

4. Significance and Implications

• Refutation of a "Missing Strong Invariant": The study challenges the previous claim (based on K-theory arguments for 3- and 4-fold systems) that the CWLS invariant represents a missing strong topological invariant for rotationally symmetric systems. In the 6-fold case, it is shown to be a fragile invariant.
• Clarification of Fragile Topology: The work provides a concrete physical realization where the CWLS detects topology that is sensitive to the specific band structure (spectral isolation) rather than being a robust property of the occupied subspace as a whole.
• Completeness of Classification: The results suggest that the search for a complete classification of topological insulators with rotational symmetry is not yet settled. The "missing" strong invariant for 6-fold symmetric systems remains elusive, and the CWLS alone is insufficient to define a robust topological phase in this context.
• Methodological Extension: The paper successfully extends the CWLS formalism to 6-fold symmetry, providing a template for analyzing similar systems, while simultaneously highlighting the limitations of the invariant in the presence of band hybridization.

Conclusion

The authors mapped the topological phase transitions of a $p6$ symmetric lattice, finding high Chern numbers in the Haldane limit and a rich variety of phases in the Kane-Mele limit. Crucially, they demonstrated that the Concentric Wilson Loop Spectrum, previously proposed as a robust strong invariant for rotationally symmetric systems, acts as an indicator of fragile topology in 6-fold symmetric models. This finding necessitates a re-evaluation of the K-theory classification for such systems and indicates that the search for a complete set of strong topological invariants for 6-fold symmetric topological insulators is ongoing.