1. Dirac-Microwave Billiards, Photonic Crystals and Graphene Supported by DFG within SFB 634 S. Bittner, C. Cuno, B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. R., F.Iachello, N. Pietralla, L. von Smekal, J. Wambach Warsaw 2013 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 1 Graphene, Schrödinger-microwave billiards and photonic crystals Band structure and relativistic Hamiltonian Dirac-microwave billiards Spectral properties Periodic orbits Edge states Quantum phase transitions Outlook

2. Graphene Two triangular sublattices of carbon atoms Near each corner of the first hexagonal Brillouin zone the electron energy E has a conical dependence on the quasimomentum but low Experimental realization of graphene in analog experiments of microwave photonic crystals “What makes graphene so attractive for research is that the spectrum closely resembles the Dirac spectrum for massless fermions.” M. Katsnelson, Materials Today, 2007 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 2 conduction band valence band

3. Closed Flat Microwave Billiards: Model Systems for Quantum Phenomena x y z scalar Helmholtz equation  Schrödinger equation for quantum billiards vectorial Helmholtz equation cylindrical resonators 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 3

4. Open Flat Microwave Billiard: Photonic Crystal A photonic crystal is a structure, whose electromagnetic properties vary periodically in space, e.g. an array of metallic cylinders → open microwave resonator Flat “crystal” (resonator) → E-field is perpendicular to the plates (TM 0 mode) Propagating modes are solutions of the scalar Helmholtz equation → Schrödinger equation for a quantum multiple-scattering problem → Numerical solution yields the band structure 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 4

5. Calculated Photonic Band Structure Dispersion relation of a photonic crystal exhibits a band structure analogous to the electronic band structure in a solid The triangular photonic crystal possesses a conical dispersion relation → Dirac spectrum with a Dirac point where bands touch each other conduction band valence band 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 5

6. Effective Hamiltonian around Dirac Point Close to Dirac point the effective Hamiltonian is a 2x2 matrix Substitution and leads to the Dirac equation Experimental observation of a Dirac spectrum in open photonic crystal S. Bittner et al., PRB 82, 014301 (2010) Next: experimental realization of a relativistic billiard 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 6

7. Microwave Dirac Billiard: Photonic Crystal in a Box→ “Artificial Graphene“ Graphene flake: the electron cannot escape → Dirac billiard Photonic crystal: electromagnetic waves can escape from it → microwave Dirac billiard: “Artificial Graphene“ Relativistic massless spin-one half particles in a billiard (Berry and Mondragon,1987) 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 7 Zigzag edge Armchair edge

8. Microwave Dirac Billiards with and without Translational Symmetry Boundaries of B1 do not violate the translational symmetry → cover the plane with perfect crystal lattice Boundaries of B2 violate the translational symmetry → edge states along the zigzag boundary Almost the same area for B1 and B2 billiard B2 billiard B1 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 8

9. Superconducting Dirac Billiard with Translational Symmetry The Dirac billiard is milled out of a brass plate and lead plated 888 cylinders Height h = 3 mm  f max = 50 GHz for 2D system Lead coating is superconducting below 7.2 K  high Q value Boundary does not violate the translational symmetry  no edge states 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 9

10. Measured S-matrix: |S 21 | 2 =P 2 / P 1 Quality factors > 5∙10 5 Altogether 5000 resonances observed Pronounced stop bands and Dirac points Transmission Spectrum at 4 K 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 10

11. Density of States of the Measured Spectrum and the Band Structure Positions of stop bands are in agreement with calculation DOS related to slope of a band Dips correspond to Dirac points High DOS at van Hove singularities  ESQPT? Flat band has very high DOS Qualitatively in good agreement with prediction for graphene (Castro Neto et al., RMP 81,109 (2009)) Oscilations around the mean density  finite size effect stop band Dirac point 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 11

12. Level density Dirac point  Van Hove singularities of the bulk states  Next: TBM description of experimental DOS Tight-Binding Model (TBM) for Experimental Density of States (DOS) 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 12

13. Tight Binding Description of the Photonic Crystal The voids in a photonic crystal form a honeycomb lattice resonance frequency of an “isolated“ void nearest neighbour contribution  t 1 next nearest neighbour contribution  t 2 second-nearest neighbour contribution  t 3 Here the overlap is neglected t1t1 t3t3 t2t2 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 13

14. Fit of the Tight-Binding Model to Experiment Numerical solution of Helmholtz equation Fit of the tight-binding model to the experimental frequencies,, yields the unknown coupling parameters f 0,t 1,t 2,t 3 Experimental DOS 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 14

15. Fit of the TBM to Experiment obvious deviationsgood agreement Fluctuation properties of spectra 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 15

16. Schrödinger and Dirac Dispersion Relation in the Photonic Crystal Dirac regime Schrödinger regime Dispersion relation along irreducible Brillouin zone  Quadratic dispersion around the  point  Schrödinger regime Linear dispersion around the  point  Dirac regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 16

17. Integrated Density of States Schrödinger regime: with w, where m is the “effective mass“ discribing the parabolic dispersion Dirac Regime: with, where is the group velocity at the Dirac frequency Unfolding is necessary in order to obtain the length spectra Schrödinger regime Dirac regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 17

18. Integrated DOS near Dirac Point Weyl’s law for Dirac billiard: (J. Wurm et al., PRB 84, 075468 (2011)) group velocity is a free parameter Same area A for two branches, but different group velocities    electron-hole asymmetry like in graphene (different opening angles of the upper and lower cone) 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 18

19. Spectral Properties of a Rectangular Dirac Billiard: Nearest Neighbour Spacing Distribution 159 levels around Dirac point Rescaled resonance frequencies such that Poisson statistics Similar behavior in the Schrödinger regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 19

20. Periodic Orbit Theory (POT) Gutzwiller‘s Trace Formula Description of quantum spectra in terms of classical periodic orbits Periodic orbits spectrum spectral density Peaks at the lengths l of PO’s wave numbers length spectrum FT Dirac billiard Effective description around the Dirac point 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 20

21. Experimental Length Spectrum: Schrödinger regime Very good agreement Next: Dirac regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 21

22. Experimental Length Spectrum: Around the Dirac Point Some peak positions deviate from the lengths of POs Comparison with semiclassical predictions for a relativistic Dirac billiard ( J. Wurm et al., PRB 84, 075468 (2011)) Possible reasons for deviations: - Short sequence of levels (80 levels only) - Anisotropic dispersion relation around the Dirac point (trigonal warping, i.e. deformation of the Dirac cone) 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 22

23. Superconducting Dirac Billiard without Translational Symmetry Boundaries violate the translational symmetry  edge states Additional antennas close to the boundary Zigzag edge Armchair edge 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 23

24. Transmission Spectra of B1 and B2 around the Dirac Frequency Accumulation of resonances above the Dirac frequency Resonance amplitude is proportional to the product of field strengths at the position of the antennas  detection of localized states 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 24

25. Comparison of Spectra Measured with Different Antenna Combinations Modes living in the inner part (black lines) Modes localized at the edge (red lines) have higher amplitudes Antenna positions 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 25

26. Smoothed Experimental Density of States Clear evidence of the edge states Position of the peak for the edge states deviates from the theoretical prediction (K. Sasaki, S. Murakami, R. Saito (2006)) Modification of tight-binding model including the overlap is needed 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 26 TB prediction

27. Summary I Measured the DOS in a superconducting Dirac billiard with high resolution Observation of two Dirac points and associated van Hove singularities: qualitative agreement with the band structure for graphene Description of the experimental DOS with a Tight-Binding Model yields perfect agreement Fluctuation properties of the spectrum agree with Poisson statistics Evaluated the length spectra of periodic orbits around and away from the Dirac point and made a comparison with semiclassical predictions Edge states are detected in the spectra Outlook: Do we see quantum phase transitions? 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 27

28. Experimental density of states in the Dirac billiard Features of the DOS related to the topology of the isofrequency lines in k-space Van Hove singularities at saddle point: density of states diverges logarithmically for quasimomenta near the M point Topological phase transition Spectroscopic Features of the DOS in a Dirac Billiard saddle point 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 28

29. Why is this a Topological Phase Transition? phase transition in two dimensions 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 29 Consider real graphene with tunable Fermi energies, i.e. variable chemical potential → topology of the Fermi surface changes with Disruption of the “neck“ of the Fermi surface This is called a Lifshitz topological phase transition with as a control parameter (Lifshitz 1960) What happens when is close to the Van Hove singularity?

30. Finite-Size Scaling of DOS at the Van Hove Singularities TBM for infinitely large crystal yields Logarithmic behaviour as seen in - transverse vibration of a hexagonal lattice (Hobson and Nierenberg, 1952) - vibrations of molecules (Pèrez-Bernal, Iachello, 2008) - two-level fermionic and bosonic pairing models (Caprio, Scrabacz, Iachello, 2011) Finite size photonic crystals or graphene flakes formed by hexagons, i.e. logarithmic scaling of the VH peak determined using Dirac billiards of varying size: 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 30 

31. DOS, Static Susceptibility and Particle-Hole Excitations: Lindhard Function Polarization of the medium by a photon → bubble diagram Summation over all momenta of virtual electron-hole pairs → Lindhard function Static susceptibility defined as It can be shown within the tight-binding approximation that, i.e. evolves as function of the chemical potential like the DOS → logarithmic divergence at (Van Hove singularity) Divergence of at caused by the infinite degeneracy of ground state: ground state QPT 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 31

32. Spectral Distribution of Particle-Hole Excitations Spectral distribution of the particle-hole excitations Same logarithmic behavior as for the ground-state observed for the excited states: ESQPT Logarithmic singularity separates the relativistic excitations from the nonrelativistic ones Dirac regime Schrödinger regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 32

33. ”Artificial” Fullerene Understanding of the measured spectrum in terms of TBM Superconducting quantum graphs Test of quantum chaotic scattering predictions (Pluhař + Weidenmüller 2013) 200 mm Outlook 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 33 50 mm