Annihilation of Dirac points and its topological obstruction in a photonic Kagome lattice

This study experimentally demonstrates the annihilation of Dirac points in a photonic Kagome lattice and the associated topological obstruction, revealing how non-Abelian frame rotations and quaternionic charge conversions govern the transition between different topological phases characterized by the Euler number.

The Gist

Imagine you are walking through a magical, invisible forest made of light. In this forest, there are special "potholes" in the ground called Dirac points. These aren't just empty spots; they are like tiny whirlpools or tornadoes in the fabric of space itself. In the world of physics, these whirlpools give materials their super-cool, strange properties (like being able to conduct electricity without resistance).

Usually, if two of these whirlpools crash into each other, they cancel out and disappear—like a positive and negative charge meeting. But in this specific experiment, the scientists discovered a rule that says: "Not so fast!" Sometimes, these whirlpools are forbidden from disappearing. They have to bounce off each other and keep going.

Here is the story of how they figured this out, using simple analogies:

1. The Playground: A Photonic Kagome Lattice

The scientists built a playground for light using a special gas (rubidium vapor) and lasers. They arranged the light into a pattern called a Kagome lattice.

• The Analogy: Imagine a net made of triangles, like a honeycomb but with triangles instead of hexagons. This is the "floor" the light walks on.
• The Trick: They used lasers to "paint" this pattern onto the gas. By tweaking the lasers, they could change the "height" of the floor at specific spots, making the light behave as if it were rolling over hills and valleys.

2. The Whirlpools (Dirac Points)

In this light-net, the energy levels of the light create "roads." At certain spots, two roads cross each other perfectly. These crossings are the Dirac points.

• The Analogy: Think of them as two rivers merging into one. If you shine a flashlight (a laser beam) through this spot, the light doesn't just go straight; it spreads out into a hollow cone, like a ring of light. This is called Conical Diffraction.
• The Clue: In the center of that ring of light, there is a dark spot. This dark spot is the "pothole" (the Dirac point).

3. The Collision: The "Forbidden" Bounce

The scientists wanted to see what happened when they pushed two of these whirlpools together.

• The Setup: They slowly changed the shape of their light-net, pushing the two whirlpools toward each other along a straight line.
• The Expectation: You'd expect them to smash together and vanish (annihilate).
• The Reality: They got close, touched, and then bounced off each other like two magnets with the same pole facing each other. They couldn't disappear!
• Why? This is the Topological Obstruction. Imagine the whirlpools are wearing special "topological backpacks." As long as the backpacks are the same color, they can't merge. The universe has a rule that says, "You can't cancel these out unless you change the backpacks first."

4. The Secret Weapon: The "Quaternionic Charge"

How did they know the backpacks were the same? They looked at the winding of the light.

• The Analogy: Imagine the light swirling around the dark spot like water going down a drain. The scientists measured how many times the light twisted. They found that both whirlpools were twisting in the same direction.
• The Math: In the language of advanced math, these whirlpools carry "charges" (called quaternionic charges). If two whirlpools have the same charge, they are like two left shoes; you can't make a pair out of them, so they can't disappear. They must bounce.

5. The Escape: Changing the Rules

But the scientists didn't just stop at the bounce. They wanted to see if they could make them disappear eventually.

• The Solution: They realized that if they could rotate the "frame" of the light around the entire playground (the Brillouin zone), they could flip the "backpack" of one of the whirlpools.
• The Analogy: Imagine taking one of the whirlpools on a long journey around the edge of the world. When it comes back, its "charge" has flipped (it's now a right shoe). Now, when it meets the other whirlpool (a left shoe), they are compatible! They can finally merge and vanish.
• The Result: By carefully tuning their lasers, they induced this "frame rotation." The obstruction vanished, the whirlpools met, and poof! They annihilated, leaving the light path clear.

Why Does This Matter?

This isn't just a cool light show.

1. New Physics: It proves that in systems with more than two energy bands, the rules of topology are more complex and "non-Abelian" (meaning the order in which you do things matters).
2. Future Tech: Understanding how to control these "whirlpools" and make them appear or disappear is crucial for building topological lasers and ultra-fast, unbreakable optical computers.
3. Measurement: They showed a new way to "see" these invisible mathematical rules by simply looking at the interference patterns of light rings.

In a nutshell: The scientists built a light-trap where two magical whirlpools refused to die when they collided because they were "wearing the same shoes." By sending one on a trip around the world to change its shoes, they finally allowed them to merge and disappear, proving a deep, hidden rule of the universe.

Technical Summary

1. Problem Statement

Dirac points (DPs) are topological singularities in the band structure of two-dimensional materials that dictate their extraordinary physical properties. In systems with three or more bands and real Hamiltonians (time-reversal symmetric), the topology of these DPs is governed by non-Abelian invariants, specifically the Euler number and quaternionic charges.

A central question in this field is determining the conditions under which two Dirac points can collide and annihilate. While simple band crossings often allow annihilation, recent theoretical work suggests that in multi-band systems, a topological obstruction can prevent this annihilation even when the points collide. This obstruction depends on the interplay between principal Dirac points and auxiliary Dirac points, or on the global topology of the Brillouin zone (BZ). The challenge lies in experimentally verifying these non-Abelian topological transitions, measuring the associated invariants (Euler numbers), and demonstrating the mechanism (frame rotation) that allows or prevents annihilation.

2. Methodology

The authors employed a versatile photonic platform based on Electromagnetically Induced Transparency (EIT) in a Rubidium (Rb) atomic vapor cell to simulate a Kagome lattice.

• System Implementation:
  • A Kagome-patterned coupling field ($E_{c1}$) creates the lattice potential.
  • A second, tunable coupling field ($E_{c2}$) is superimposed to selectively modulate the potential energy of specific lattice sites (A, B, or C) within the unit cell.
  • This optical "writing" allows for the engineering of the Hamiltonian parameter $E_A$ (energy on site A relative to others) while preserving time-reversal symmetry.
• Theoretical Model:
  • The system is modeled using a tight-binding approximation with a real $3 \times 3$ Hamiltonian.
  • The Hamiltonian exhibits principal Dirac points (between the top and middle bands) and auxiliary Dirac points (between middle and bottom bands).
  • Topological properties are analyzed using quaternionic charges and non-Abelian frame rotation around the Brillouin zone torus.
• Experimental Probes:
  • Conical Diffraction: A Gaussian probe beam is sent through the lattice. At Dirac points, the beam transforms into a hollow cone (ring-like intensity distribution). The "dark spots" (intensity minima) in the ring correspond to the Dirac points.
  • Interferometry: To measure topological invariants, the conical diffraction pattern is interfered with a Gaussian reference beam. This reveals phase singularities (fork-like dislocations) whose winding numbers correspond to the topological charges.
  • Fourier Filtering: Used to remove lattice-induced intensity modulations, isolating the envelope of the conical diffraction.

3. Key Contributions

1. Experimental Observation of Topological Obstruction: The study provides the first experimental demonstration of a topological obstruction preventing the annihilation of a pair of principal Dirac points upon collision.
2. Direct Measurement of Euler Numbers: The authors successfully extracted the patch Euler number (a topological invariant) by measuring the winding numbers of phase singularities in the interference patterns of conical diffraction.
3. Demonstration of Non-Abelian Frame Rotation: The paper experimentally verifies that the transition from a state of "obstructed annihilation" to "allowed annihilation" is induced by a non-Abelian frame rotation of the eigenstates around the Brillouin zone torus, rather than the presence of auxiliary Dirac points.
4. Quaternionic Charge Conversion: The work elucidates how the quaternionic charges of the Dirac points change sign during their motion across the Brillouin zone, enabling the topological transition.

4. Key Results

A. Collision with Topological Obstruction

• By tuning the parameter $E_A$ (via the frequency of $E_{c2}$), the authors moved two principal Dirac points along the $k_x$ axis toward each other.
• Observation: Upon collision, the points did not annihilate. Instead, they "bounced" off each other, forming a second-order degeneracy (parabolic band touching), and continued moving along the perpendicular $k_y$ axis.
• Topological Proof: Interference measurements revealed two phase singularities with the same sign (winding). In the language of quaternionic charges, the loop encircling both points had a charge of $-1$ (indicating a pair of same-type points that cannot annihilate). This confirmed the topological obstruction.

B. Annihilation via Frame Rotation

• The authors further tuned the system to a different regime where the Dirac points were positioned near the annihilation point on the $k_y$ axis.
• Observation: By adjusting $E_A$ across a critical value, the two Dirac points collided and annihilated, resulting in the disappearance of the dark spots in the conical diffraction pattern.
• Mechanism: This transition was not caused by auxiliary Dirac points. Instead, it was driven by the non-Abelian frame rotation of the eigenstates as the points traversed the Brillouin zone torus.
• Topological Change: The continuous deformation of the integration loop around the moving Dirac points caused a change in the homotopy class. The quaternionic charge of the loop flipped sign ($c(\gamma) = -c(\gamma')$), effectively converting the pair from a non-annihilating state to an annihilating state. This was confirmed by the change in the measured Euler number.

5. Significance

• Fundamental Physics: This work bridges the gap between abstract non-Abelian topology theories (quaternionic charges, Euler numbers) and experimental reality. It confirms that the global topology of the Brillouin zone (the torus) plays a crucial role in local band degeneracies.
• Photonic Applications: The ability to control Dirac point annihilation and measure topological invariants in real-time using photonic lattices opens new avenues for topological photonics. This includes the development of robust topological lasers, optical isolators, and devices that rely on topological protection.
• Metrology: The method of using conical diffraction interference to extract topological invariants (winding numbers/Euler numbers) offers a powerful tool for high-precision metrology and the characterization of topological materials without requiring complex momentum-resolved spectroscopy.
• Generalizability: The findings suggest that similar topological obstructions and transitions could be engineered in other physical systems, including cold atoms and solid-state materials, providing a deeper understanding of topological phase transitions in multi-band systems.