Schrödinger- and Dirac-Microwave Billiards, Photonic Crystals and Graphene ECT* 2013 Microwave billiards, graphene and photonic crystals Band structure and relativistic Hamiltonian Dirac billiards Spectral properties Periodic orbits Quantum phase transitions Outlook Supported by DFG within SFB 634 C. Bouazza, C. Cuno, B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. Richter, F. Iachello, N. Pietralla, L. von Smekal, J. Wambach

Schrödinger- and Microwave Billiards Quantum billiard d Microwave billiard Analogy Part of the subproject C4 focuses on the spectral properties of classically regular or fully chaotic systems and of systems with mixed regular and chaotic dynamics. Billiards are well suited for such studies as their classical dynamics is solely determined by their shape. The eigenvalues and eigenfunctions of these so-called quantum billiards are determined experimentally by using the analogy between them and flat microwave resonators. Below a certain frequency, that is, for wave lengths larger than double the height of the resonator, the electric field strength is perpendicular to the top and bottom of the resonator. There the underlying Helmholtz equation is mathematically equivalent to the Schrödinger equation of the quantum billiard of corresponding shape. In both systems the solutions of the wave equation fulfill Dirichlet boundary conditions along the wall of the billiard. Accordingly the eigenvalues of the quantum billiard are directly related to the resonance frequencies, the eigenfunctions to the electric field strengths of the microwave resonator, which is also called microwave billiard. The eigenvalues are obtained in superconducting measurements of resonance spectra. eigenvalue E  wave number eigenfunction Y  electric field strength Ez 2

Graphene “What makes graphene so attractive for research is that the spectrum closely resembles the Dirac spectrum for massless fermions.” M. Katsnelson, Materials Today, 2007 conduction band valence 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 Ich beginne mein Vortrag mit einen Zitat aus der Arbeit …… Graphen ist eine flache periodische Anordnung der Kohlenstoff Atomen. Den Graphene hat eine Gitterstruktur die aus zwei überlappenden dreieckigen Untergittern besteht. Durch blau und rot sind die Atomen aus diesen Untergittern gekenzeichnet. Das Bild zeigt die Energiespektrum des Elektrons als Funktion von Quasimomentum. Der Hexagon zeigt die erste Brilliouine zone. Das untere Band ist eine Valenzband, die obere ist die Leitungsband. Die beider Bänder berühren sich an der Ecken der Ersten Brilloiune Zone und formen zeigen konische Dispersionrelation. Die Dispersionrelation ist dem der von massenlosen fermionen Ähnlich ist. 23/04/2017 |

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 Ein photonische Kristall ist eine Gebilde deren elektromagnetische Eigenschaften ändern sich periodisch, zum Beispiel die periodische Anordnung der Metall Zylindern die zwischen zwei Kupfer platten eingespannt ist. Wie es wurde in dem Vortrag von Dr. Barbara Dietz gezeigt in diesem flachen Kristall unter einen bestimmten Frequenz nur die Moden existieren mit dem elektrischen Feld senkrecht zu metal Platten. Die propagierenden Moden sind Lösungen der skalaren Helmholz Gleichung, die mathematisch equivalent zu der Srödinger Gleichung für das multiple Streuproblem. Flat “crystal” (resonator) → E-field is perpendicular to the plates (TM0 mode) Propagating modes are solutions of the scalar Helmholtz equation → Schrödinger equation for a quantum multiple-scattering problem → Numerical solution yields the band structure 23/04/2017 |

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 The voids form a honeycomb lattice like atoms in graphene conduction band valence

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 an open photonic crystal S. Bittner et al., PRB 82, 014301 (2010)

Reflection Spectrum of an Open Photonic Crystal Measurement with a wire antenna a put through a drilling in the top plate → point like field probe Characteristic cusp structure around the Dirac frequency Van Hove singularities at the band saddle point Next: experimental realization of a relativistic (Dirac) billiard 23/04/2017 |

Microwave Dirac Billiard: Photonic Crystal in a Box→ “Artificial Graphene“ Zigzag edge Armchair edge 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)

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  fmax = 50 GHz for 2D system Lead coating is superconducting below Tc=7.2 K  high Q value Boundary does not violate the translational symmetry  no edge states

Transmission Spectrum at 4 K Measured S-matrix: |S21|2=P2 / P1 Pronounced stop bands and Dirac points Quality factors > 5∙105 Altogether 5000 resonances observed

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)) Oscillations around the mean density  finite size effect stop band Dirac point stop band Dirac point stop band

Tight-Binding Model (TBM) for Experimental Density of States (DOS) Level density Dirac point  Van Hove singularities  of the bulk states at Next: TBM description of experimental DOS

TBM Description of the Photonic Crystal The voids in a photonic crystal form a honeycomb lattice resonance frequency of an “isolated“ void nearest neighbour contribution  t1 next-nearest neighbour contribution  t2 second-nearest neighbour contribution  t3 Here the overlap is neglected t3 t2 t1 determined from experiment 23/04/2017 |

Fit of the TBM to Experiment Good agreement Fluctuation properties of spectra 23/04/2017 |

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

Integrated Density of States Dirac regime Schrödinger regime Schrödinger regime: Dirac Regime: (J. Wurm et al., PRB 84, 075468 (2011)) Fit of Weyl’s formula to the data  and

Spectral Properties of a Rectangular Dirac Billiard: Nearest Neighbour Spacing Distribution Spacing between adjacent levels depends on DOS Unfolding procedure: such that 130 levels in the Schrödinger regime 159 levels in the Dirac regime Spectral properties around the Van Hove singularities?

Ratio Distribution of Adjacent Spacings DOS is unknown around Van Hove singularities Ratio of two consecutive spacings Ratios are independent of the DOS  no unfolding necessary Analytical prediction for Gaussian RMT ensembles (Y.Y. Atas, E. Bogomolny, O. Giraud and G. Roux, PRL, 110, 084101 (2013) )

Ratio Distributions for Dirac Billiard Poisson: ; GOE: Poisson statistics in the Schrödinger and Dirac regime GOE statistics to the left of first Van Hove singularity Origin ? ;  e.m. waves “see the scatterers“

Peaks at the lengths l of PO’s Periodic Orbit Theory (POT) Gutzwiller‘s Trace Formula Description of quantum spectra in terms of classical periodic orbits spectrum wave numbers spectral density length spectrum FT Peaks at the lengths l of PO’s Dirac billiard Periodic orbits D: S: Effective description

Experimental Length Spectrum: Schrödinger Regime Effective description ( ) has a relative error of 5% at the frequency of the highest eigenvalue in the regime Very good agreement Next: Dirac regime

Experimental Length Spectrum: Dirac Regime upper Dirac cone (f>fD) lower Dirac cone (f<fD) Some peak positions deviate from the lengths of POs Comparison with semiclassical predictions for a Dirac billiard (J. Wurm et al., PRB 84, 075468 (2011)) Effective description ( ) has a relative error of 20% at the frequency of the highest eigenvalue in the regime

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 both in the Schrödinger and the Dirac regime, but not around the Van Hove singularities Evaluated the length spectra of periodic orbits around and away from the Dirac point and made a comparison with semiclassical predictions Next: Do we see quantum phase transitions? 23/04/2017 | | 23

Experimental DOS and Topology of Band Structure saddle point saddle point r ( f ) Each frequency f in the experimental DOS r ( f ) is related to an isofrequency line of band structure in k space Close to band gaps isofrequency lines form circles around G point Sharp peaks at Van Hove singularities correspond to saddle points Parabolically shaped surface merges into Dirac cones around Dirac frequency → topological phase transition from non-relativistic to relativistic regime

Neck-Disrupting Lifshitz Transition Gradually lift Fermi surface across saddle point, e.g., with a chemical potential m → topology of the Fermi surface changes topological transition in two dimensions Disruption of the “neck“ of the Fermi surface at the saddle point At Van Hove singularities DOS diverges logarithmically in infinite 2D systems → Neck-disrupting Lifshitz transition with m as a control parameter (Lifshitz 1960)

Finite-Size Scaling of DOS at the Van Hove Singularities TBM for infinitely large crystal yields Logarithmic behaviour as seen in bosonic systems - 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: 

Particle-Hole Polarization Function: Lindhard Function Polarization in one loop calculated from bubble diagram → Lindhard function at zero temperature Use TBM taking into account only nearest-neighbor hopping with and the nearest-neighbor vectors Overlap of wave functions for intraband (=l’) and interband (λ=-l’) transitions within, respectively, between cones

Static susceptibility at zero-momentum transfer Static Susceptibility and Spectral Distribution of Particle-Hole Excitations I Static susceptibility at zero-momentum transfer Nonvanishing contributions only from real part of Lindhard function Divergence of at m =  1 caused by the infinite degeneracy of ground states when Fermi surface passes through Van Hove singularities → GSQPT Imaginary part of Lindhard function at zero-momentum transfer yields for spectral distribution of particle-hole excitations q Only interband contributions and excitations for w > 2m (Pauli blocking) Same logarithmic behavior observed for ground and excited states → ESQPT

Static Susceptibility and Spectral Distribution of Particle-Hole Excitations II Schrödinger regime Dirac regime Sharp peaks of at m=1, w=0 and for -1≤  ≤ 1, w=2 clearly visible Experimental DOS can be quantitatively related to GSQPT and ESQPT arising from a topological Lifshitz neck-disrupting phase transition Logarithmic singularities separate the relativistic excitations from the nonrelativistic ones

f-Sum Rule as Quasi-Order Parameter Transition due to change of topology of Fermi surface → no order parameter Normalization is fixed due to charge conservation via f - sum rule Z- Z+ Z=Z++Z- Z  const. in the relativistic regime  < 1 At  = 1 its derivative diverges logarithmically Z decreases approximately linearly in non-relativistic regime  > 1

Outlook ”Artificial” Fullerene 200 mm Understanding of the measured spectrum in terms of TBM Superconducting quantum graphs Test of quantum chaotic scattering predictions (Pluhař + Weidenmüller 2013) 200 mm 50 mm