Location and Topology of the Fundamental Gap in Photonic Crystals

The fundamental, or first, band gap is of unmatched importance in the study of photonic crystals. Here, we address precisely where this gap can be opened in the band structure of three-dimensional photonic crystals. Although strongly constrained by symmetry, this problem cannot be addressed directly with conventional bandsymmetry analysis due to the existence of a photonic polarization vortex at zero frequency. We develop an approach for overcoming the associated symmetry singularity by incorporating fictitious, auxiliary longitudinal modes. Our strategy also enables us to extend recent developments in symmetry-based topological analysis to the fundamental gap of three-dimensional photonic crystals. Exploiting this, we systematically study the topology of the minimal fundamental gaps. This reveals the existence of topological gap-obstructions that push the fundamental gap higher than what a conventional analysis would suggest. Our work demonstrates that topology can play a crucial role in the opening of the fundamental photonic gap and informs future theoretical and experimental searches for conventional and topological band gaps in three-dimensional photonic crystals.

[1]  Rose K. Cersonsky,et al.  The diversity of three-dimensional photonic crystals , 2021, Nature Communications.

[2]  Barry Bradlyn,et al.  Magnetic topological quantum chemistry , 2020, Nature Communications.

[3]  B. Bernevig,et al.  Application of induction procedure and Smith decomposition in calculation and topological classification of electronic band structures in the 230 space groups , 2020, Physical Review B.

[4]  Ling Lu,et al.  Discovering Topological Surface States of Dirac Points. , 2020, Physical review letters.

[5]  N. Regnault,et al.  Fragile Phases as Affine Monoids: Classification and Material Examples , 2019 .

[6]  E. Kaxiras,et al.  Quantum Theory of Materials , 2019 .

[7]  B. Bradlyn,et al.  Engineering fragile topology in photonic crystals: Topological quantum chemistry of light , 2019, Physical Review Research.

[8]  A. Vishwanath,et al.  Efficient topological materials discovery using symmetry indicators , 2019, Nature Physics.

[9]  D. Vanderbilt.  Berry Phases in Electronic Structure Theory , 2018 .

[10]  Bernd Souvignier,et al.  Crystallographic shelves: space-group hierarchy explained , 2018, Journal of Applied Crystallography.

[11]  Ling Lu,et al.  Space Group Theory of Photonic Bands. , 2018, Physical review letters.

[12]  A. Vishwanath,et al.  Efficient topological materials discovery using symmetry indicators , 2018, Nature Physics.

[13]  Bohm-Jung Yang,et al.  Band Topology and Linking Structure of Nodal Line Semimetals with Z_{2} Monopole Charges. , 2018, Physical review letters.

[14]  Teresa J. Feo,et al.  Structural absorption by barbule microstructures of super black bird of paradise feathers , 2018, Nature Communications.

[15]  Tiantian Zhang,et al.  Diagnosis for Nonmagnetic Topological Semimetals in the Absence of Spin-Orbital Coupling , 2017, Physical Review X.

[16]  Wim Desmet,et al.  Probability that a band-gap extremum is located on the irreducible Brillouin-zone contour for the 17 different plane crystallographic lattices , 2017 .

[17]  A. Vishwanath,et al.  Fragile Topology and Wannier Obstructions. , 2017, Physical review letters.

[18]  C. Felser,et al.  Band connectivity for topological quantum chemistry: Band structures as a graph theory problem , 2017, 1709.01937.

[19]  A. Vishwanath,et al.  Structure and topology of band structures in the 1651 magnetic space groups , 2017, Science Advances.

[20]  C. Felser,et al.  Double crystallographic groups and their representations on the Bilbao Crystallographic Server , 2017, 1706.09272.

[21]  Barry Bradlyn,et al.  Graph theory data for topological quantum chemistry. , 2017, Physical review. E.

[22]  M. I. Aroyo,et al.  Topological quantum chemistry , 2017, Nature.

[23]  A. Vishwanath,et al.  Symmetry-based indicators of band topology in the 230 space groups , 2017, Nature Communications.

[24]  J. Pendry,et al.  Metamaterials with index ellipsoids at arbitrary k-points , 2016, Nature Communications.

[25]  A. Vishwanath,et al.  Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups. , 2016, Physical review letters.

[26]  Hsin Lin,et al.  Colloquium : Topological band theory , 2016, 1603.03576.

[27]  Junwei Liu,et al.  Topological semimetals with helicoid surface states , 2015, Nature Physics.

[28]  A. Vishwanath,et al.  Filling-enforced quantum band insulators in spin-orbit coupled crystals , 2015, Science Advances.

[29]  Liang Fu,et al.  Topological nodal line semimetals with and without spin-orbital coupling , 2015, 1506.03449.

[30]  C. Kane,et al.  Dirac Line Nodes in Inversion-Symmetric Crystals. , 2015, Physical review letters.

[31]  M. Soljačić,et al.  Experimental observation of Weyl points , 2015, Science.

[32]  M. Soljačić,et al.  Topological photonics , 2014, Nature Photonics.

[33]  K. Busch,et al.  Generation of Wannier functions for photonic crystals , 2013 .

[34]  Harold T Stokes,et al.  Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions. , 2013, Acta crystallographica. Section A, Foundations of crystallography.

[35]  A. Soluyanov.  Topological aspects of band theory , 2013 .

[36]  B. Bernevig.  Topological Insulators and Topological Superconductors , 2013 .

[37]  Sébastien Guenneau,et al.  Dangers of using the edges of the Brillouin zone , 2012 .

[38]  Liang Fu,et al.  Weyl points and line nodes in gyroid photonic crystals , 2012, Nature Photonics.

[39]  X. Qi,et al.  Equivalent expression of Z 2 topological invariant for band insulators using the non-Abelian Berry connection , 2011, 1101.2011.

[40]  Zheng Wang,et al.  Observation of unidirectional backscattering-immune topological electromagnetic states , 2009, Nature.

[41]  J. M. Harrison,et al.  On occurrence of spectral edges for periodic operators inside the Brillouin zone , 2007, math-ph/0702035.

[42]  S. C. Tiwari.  Topological Photon , 2006, quant-ph/0612139.

[43]  S. Raghu,et al.  Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. , 2005, Physical review letters.

[44]  Martin Maldovan,et al.  Exploring for 3D photonic bandgap structures in the 11 f.c.c. space groups , 2003, Nature materials.

[45]  L. Michel.  Elementary energy bands in crystals are connected , 2001 .

[46]  Steven G. Johnson,et al.  Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis. , 2001, Optics express.

[47]  J. Zak,et al.  CONNECTIVITY OF ENERGY BANDS IN CRYSTALS , 1999 .

[48]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[49]  John,et al.  Strong localization of photons in certain disordered dielectric superlattices. , 1987, Physical review letters.

[50]  E. Yablonovitch,et al.  Inhibited spontaneous emission in solid-state physics and electronics. , 1987, Physical review letters.

[51]  Frank Wilczek,et al.  Appearance of Gauge Structure in Simple Dynamical Systems , 1984 .

[52]  D. Pines.  A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions , 1952 .

[53]  C. Herring.  Effect of Time-Reversal Symmetry on Energy Bands of Crystals , 1937 .

[54]  E. Wigner,et al.  Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals , 1936 .

[55]  R. Sec.,et al.  XXVI. On the remarkable phenomenon of crystalline reflexion described by Prof. Stokes , 1888 .

[56]  L. Rayleigh,et al.  XVII. On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure , 1887 .

[57]  A. Vishwanath,et al.  XFe_{4}Ge_{2}(X=Y,Lu) and Mn_{3}Pt: Filling-enforced magnetic topological metals , 2020 .

[58]  Petra Holtzmann,et al.  Optical Properties Of Photonic Crystals , 2016 .

[59]  L. Muñoz,et al.  ”QUANTUM THEORY OF SOLIDS” , 2009 .

[60]  J. Strutt.  Scientific Papers: On the Maintenance of Vibrations by Forces of Double Frequency, and on the Propagation of Waves through a Medium endowed with a Periodic Structure , 2009 .

[61]  On the remarkable phenomenon of crystalline reflexion described by Prof , 2008 .