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Spawning rings of exceptional points out of Dirac cones


The Dirac cone underlies many unique electronic properties of graphene1 and topological insulators, and its band structure—two conical bands touching at a single point—has also been realized for photons in waveguide arrays2, atoms in optical lattices3, and through accidental degeneracy4,5. Deformation of the Dirac cone often reveals intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels. A seemingly unrelated phenomenon is the exceptional point6,7, also known as the parity–time symmetry breaking point8,9,10,11, where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency12, unidirectional transmission or reflection11,13,14, and lasers with reversed pump dependence15 or single-mode operation16,17. Dirac cones and exceptional points are connected: it was theoretically suggested that certain non-Hermitian perturbations can deform a Dirac cone and spawn a ring of exceptional points18,19,20. Here we experimentally demonstrate such an ‘exceptional ring’ in a photonic crystal slab. Angle-resolved reflection measurements of the photonic crystal slab reveal that the peaks of reflectivity follow the conical band structure of a Dirac cone resulting from accidental degeneracy, whereas the complex eigenvalues of the system are deformed into a two-dimensional flat band enclosed by an exceptional ring. This deformation arises from the dissimilar radiation rates of dipole and quadrupole resonances, which play a role analogous to the loss and gain in parity–time symmetric systems. Our results indicate that the radiation existing in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain.

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Figure 1: Accidental degeneracy in Hermitian and non-Hermitian photonic crystals.
Figure 2: Experimental reflectivity spectrum and accidental Dirac dispersion.
Figure 3: Experimental demonstration of an exceptional ring.


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We thank T. Savas for fabrication of the samples, and F. Wang, Y. Yang, N. Rivera, S. Skirlo, O. Miller and S. G. Johnson for discussions. This work was partly supported by the Army Research Office through the Institute for Soldier Nanotechnologies under contract nos W911NF-07-D0004 and W911NF-13-D-0001. B.Z., L.L. and M.S. were partly supported by S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under grant no. DE-SC0001299. L.L. was supported in part by the Materials Research Science and Engineering Center of the National Science Foundation (award no. DMR-1419807). I.K. was supported in part by Marie Curie grant no. 328853-MC-BSiCS.

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All authors discussed the results and made critical contributions to the work.

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Correspondence to Bo Zhen or Chia Wei Hsu.

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Zhen, B., Hsu, C., Igarashi, Y. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).

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