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
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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
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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
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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 structure
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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
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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)
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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
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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 edg
e
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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
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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?
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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 ? 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 19
; 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
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D:
S:
Effective description
Experimental Length Spectrum:Schrödinger Regime
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• 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
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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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• 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 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 pointr( f )
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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)
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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)
• 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:
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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 (=’) and interband (λ=-’) transitions within, respectively, between cones
• Use TBM taking into account only nearest-neighbor hopping
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• 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> 2m(Pauli blocking)
• Same logarithmic behavior observed for ground and excited states → ESQPT
q
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• Sharp peaks of at m=1,w=0 and for -1≤ m ≤ 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
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• 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 m < 1• At m = 1 its derivative diverges logarithmically • Z decreases approximately linearly in non-relativistic regime m > 1
• Transition due to change of topology of Fermi surface → no order parameter
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• ”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 m
m
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