schrödinger- and dirac- microwave billiards, photonic crystals and graphene supported by dfg within...

Schrödinger- and Dirac-Microwave Billiards, Photonic Crystals and Graphene

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

2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 1

• Microwave billiards, graphene and photonic crystals • Band structure and relativistic Hamiltonian• Dirac billiards• Spectral properties• Periodic orbits• Quantum phase transitions• Outlook

ECT* 2013

d

Microwave billiardQuantum billiard

eigenfunction Y electric field strength Ez

Schrödinger- and Microwave Billiards

Analogy

eigenvalue E wave number

2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 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

conductionband

valenceband


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 (TM0 mode)

• Propagating modes are solutions of the scalar Helmholtz equation

→ Schrödinger equation for a quantum multiple-scattering problem

→ Numerical solution yields the band structure2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 4

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

conductionband

valenceband


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 crystalS. Bittner et al., PRB 82, 014301 (2010)


Reflection Spectrum of an Open Photonic Crystal

• Characteristic cusp structure around the Dirac frequency• Van Hove singularities at the band saddle point• Next: experimental realization of a relativistic (Dirac) billiard

• Measurement with a wire antenna a put through a drilling in the top plate → point like field probe


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)

Zigzag edge

Arm

chai

r e

dge


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


• Measured S-matrix: |S21|2=P2 / P1

• Pronounced stop bands and Dirac points

• Quality factors > 5∙105

• Altogether 5000 resonances observed

Transmission Spectrum at 4 K


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

stop band

stop band

Dirac point

Dirac point


• Level density

• Dirac point • Van Hove singularities of the bulk states at• Next: TBM description of experimental DOS

Tight-Binding Model (TBM) for ExperimentalDensity of States (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

t1

t3

t2


determined from experiment

Fit of the TBM to Experiment

• Good agreement

• Fluctuation properties of spectra


Schrödinger and Dirac Dispersion Relation in the Photonic Crystal

Dirac regimeSchrödinger 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

• Schrödinger regime:

• Dirac Regime: (J. Wurm et al., PRB 84, 075468 (2011))

• Fit of Weyl’s formula to the data and

Schrödinger regime

Dirac regime


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“

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

wavenumbers length spectrum

FT

Dirac billiard


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?


• 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

Experimental DOS and Topology of Band Structure

saddle point

saddle point r ( f )


Neck-Disrupting Lifshitz Transition

topological transitionin two dimensions

• Gradually lift Fermi surface across saddle point, e.g., with a chemical potential m → topology of the Fermi surface changes

• 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

with and the nearest-neighbor vectors•

• Overlap of wave functions for intraband (=l’) and interband (λ=-l’) transitions within, respectively, between cones

• Use TBM taking into account only nearest-neighbor hopping


• 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

Static Susceptibility and Spectral Distribution of Particle-Hole Excitations I

• Only interband contributions and excitations for w > 2 m (Pauli blocking)

• Same logarithmic behavior observed for ground and excited states → ESQPT

q


• 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

Static Susceptibility and Spectral Distribution of Particle-Hole Excitations II

Diracregime

Schrödingerregime


• Normalization is fixed due to charge

conservation via f - sum rule

f-Sum Rule as Quasi-Order Parameter

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

• Transition due to change of topology of Fermi surface → no order parameter


• ”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

50

mm


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