Spawning Rings of Exceptional Points out of Dirac Cones

Abstract

The Dirac cone underlies many unique electronic properties of graphene 1 and topological insulators 2 , and its band structure-two conical bands touching at a single point-has also been realized for photons in waveguide arrays 3 , atoms in optical lattices 4 , and through accidental degeneracy 5, 6 . Deformations of the Dirac cone often reveal intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels 7 . A seemingly unrelated phenomenon is the exceptional point 8-11 , also known as the parity-time symmetry breaking point [12][13][14][15] , where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency 16 , unidirectional transmission or reflection 17-23 , and lasers with reversed pump dependence [24][25][26] or single-mode operation 27, 28 . These two fields of research are in fact connected: here we discover the ability of a Dirac cone to evolve into a ring of exceptional points, which we call an "exceptional ring." We experimentally demonstrate this concept 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 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 that exists in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain.Closed and lossless physical systems are described by Hermitian operators, which guarantee realness of the eigenvalues and a complete set of eigenfunctions that are orthogonal to each other. On the other hand, systems with open boundaries 10, 29 or with material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] are non-Hermitian 8 and have non-orthogonal eigenfunctions with complex eigenvalues where the imaginary part corresponds to decay or growth. The most drastic difference between Hermitian and non-Hermitian systems is that the latter exhibit exceptional points (EPs) where both the real and the imaginary parts of the eigenvalues coalesce. At an EP, two (or more) eigenfunctions collapse into one so the eigenspace no longer forms a complete basis, and this eigenfunction becomes orthogonal to itself under the unconjugated inner product 8-11 . To date, most studies of EP and its intriguing consequences concern parity-time symmetric systems that rely on material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] , but EP is a general property that requires only non-Hermiticity. Here, we show the existence of EPs in a photonic cryst...

Parity-time symmetry in a flat-band system

“…We note that different from Refs. [27,44,45], the system we consider here has a flat band before the PT -symmetric perturbation is introduced, instead of being the result of PT -symmetry breaking.…”

Exceptional points of degeneracy andPTsymmetry in photonic coupled chains of scatterers

“…In previous works, PT-symmetry has been shown in discrete arrangements of resonators and also using the so-called "tight-binding" (TB) approach [48,50]. Moreover, EPDs have been also observed in 2D and 3D geometries [7,37,51].…”

“…This pervasive concept may give rise to interesting phenomena in many branches of physics [1][2][3][4][5][6][7]. In connection with electromagnetic (EM) waves, of particular interest for this study, it is well known that the propagation in closed guiding structures such as metallic waveguides or periodic structures, in the absence of energy dissipation or gain, is mathematically described in terms of a Hermitian operator [8].…”

Tidal surface states as fingerprints of non-Hermitian nodal knot metals

“…In those contexts, metallicity refers not to a Fermi surface intersection, but essentially embodies the antithesis of a band insulator, i.e. the absence of spectral gaps 6 Besides parity-time (PT)-symmetric systems with real eigenspectrum due to balanced gain and loss 7 , non-Hermiticity, in combination with topology and surface terminations, has been recently shown to unfold a rich scope of experimentally robust phenomena far beyond mere dissipation [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] (see refs. [27][28][29][30][31] for excellent reviews).…”