1. 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

2. 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

3. 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.
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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
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
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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
The voids form a honeycomb lattice like atoms in graphene
conduction
band
valence

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

7. 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
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8. 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)

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  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

10. 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

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

12. 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

13. 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
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14. Fit of the TBM to Experiment
Good agreement
Fluctuation properties of spectra
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15. 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

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

17. 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?

18. 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, (2013) )

19. 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“

20. 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

21. 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

22. 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, (2011))
Effective description ( ) has a relative error of 20% at the frequency of the highest eigenvalue in the regime

23. 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?
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24. 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

25. 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)

26. 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: 

27. 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

28. 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

29. 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

30. 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

31. 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