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

Abstract

Dirac points (DPs) are topological singularities that determine the extraordinary properties of two-dimensional materials. They are generally classified by discrete topological invariants, which determine the possibility of DPs' annihilation upon their collision. Here, we study the behaviors of DPs within a photonic Kagome lattice created in atomic vapor. With optically engineering the potential difference among three sites constituting the Kagome unit cell while preserving time-reversal symmetry and the stability of an isolated DP, the DPs move in reciprocal space. By employing conical diffraction to measure their position and the topological invariant (Euler number), we demonstrate an obstruction to DPs' annihilation during collision and a transition to a case where the Euler number changes and annihilation occurs. Such topological transition is induced by a non-Abelian frame rotation of the eigenstates around the Brillouin zone torus. The associated conversion of the DP quaternionic charges during their motion explains the change of Euler number.

Figures (9)

• Figure 1: The Kagome photonic lattice and the Dirac points. (a) Scheme of the experiment. Top inset: the driven atomic energy-level configuration. CCD: camera for capturing the output probe beam. Bottom inset: scheme of the lattice (unit cell with 3 sites). (b) The Kagome-patterned coupling beam for "writing" the lattice. (c) Dispersion relation $E(k_x,k_y)$ in the tight-binding approximation. (d) The 1st Brillouin zone with principal Dirac point trajectories (blue and green lines) exhibiting collision (marked as "forbidden") and annihilation (marked with the star). The auxiliary Dirac points are marked with orange circles. (e) Top: 3D representation of the Brillouin zone as a torus, with the trajectories of the two principal Dirac points (green and blue lines) exhibiting collision with obstruction ("forbidden") followed by annihilation (star). Auxiliary Dirac points: two orange squares. Bottom: Comparison of the homotopy classes of two loops. The blue loop $\gamma$ initially encircles the two principal Dirac points at position 1 and is continuously deformed as the points move to position 2. The purple loop $\gamma'$ directly encircles the Dirac points at their final position 2, near the annihilation point. (f) Observed conical diffraction with exciting a single (auxiliary) Dirac point.
• Figure 2: Collision of Dirac points. (a-d) $E_A<0$; (e-h) $E_A>0$. (a,e) Scheme of the experimental implementation for covering different lattice sites. (b,f) Tight-binding dispersion (along $k_x$ and $k_y$, respectively) showing the positions of the Dirac points. (c,g) Experimental images of the conical diffraction experienced by the probe showing two Dirac points along horizontal/vertical directions, respectively. (d,h) Numerical simulations of conical diffraction corresponding to $E_A<0$ and $E_A>0$, respectively.
• Figure 3: Topological obstruction of the Dirac point annihilation. (a,c) Interference pattern of the conical diffraction with a reference beam exhibiting two fork-like dislocations (experiment, theory). (b,d) Extracted phase images showing two phase singularities of the same sign (experiment, theory).
• Figure 4: Annihilation of the Dirac points. Experimental demonstration of two Dirac points (a) and their disappearance (c). (b) and (d) theoretically reproduced the results in (a) and (c), respectively.
• Figure S1: Texture of the eigenstates of the Hamiltonian (3) from the main text: a before and b after collision of the Dirac points with the same winding.