1. Superconducting Microwave Dirac Billiards: From Graphene to Fullerene
MSU 2016
Graphene, band structure and relativistic Hamiltonian
Microwave billiards, photonic crystals and “artificial graphene”
Measured resonance spectra and the band structure, its topology and evidence for a topological phase transition (QPT)
Tight Binding Model description of photonic crystals, finite size scaling at the Van Hove singularities and the nature of the QPT
Some spectral properties in the Schrödinger and Dirac regions of the band structure
From plane “artificial graphene” to spherical “artificial fullerene”, (near) zero modes and the Atiyah-Singer theorem
Supported by DFG within SFB 634
B. Dietz, T. Klaus, M. Miski-Oglu, N. Pietralla, A. Richter, M. Wunderle, L. von Smekal, J. Wambach
2016 | SFB 634 | Achim Richter | 1

2. Graphene and its Band Structure
Monoatomic layer of carbon atoms arranged on a honeycomb lattice
conduction
band
valence  K+ K-
Two interpenetrating triangular sublattices → consequences: at half filling its Fermi surface reduces to two independent Dirac points K- and K+
Near each corner of the first hexagonal Brillouin zone the electron energy  has a conical dependence on the quasimomentum
Close to Dirac points: linear dispersion 𝜔= ℏ 𝑣 𝐹 𝑞 with a slope given by the low 𝑣 𝐹 ~ 𝑐 300
The low-energy spectrum of graphene around  = D closely resembles the Dirac spectrum for massless fermions (“massless spin-1/2 quasiparticles”)
2016 | SFB 634 | Achim Richter | 2
20/10/2017 |

3. Effective Hamiltonian near the Dirac Points
Close to Dirac points K the effective Hamiltonian is a 2x2 matrix
Substitution and leads to the Dirac equations
Independent contributions at K+ and K- thus yield a 4D Dirac equation
Extraordinary band structure of graphene stems solely from symmetry properties of the hexagonal lattice
Experimental realization of a model system to study the electronic properties of graphene: “artificial graphene”
→ superconducting microwave billiard containing a photonic crystal 
2016 | SFB 634 | Achim Richter | 3

4. Quantum Billiards and Microwave Billiards
Analogy for f < c/2d
eigenvalues E  resonance frequencies f
eigenfunctions Y  electric field strengths Ez
2016 | SFB 634 | Achim Richter | 4

5. Measurement Principle
Measurement of scattering matrix element S21
rf power in
out   d
Resonance spectrum
positions of the resonances
fn yield eigenvalues
2016 | SFB 634 | Achim Richter | 5

6. How it all began: Superconducting DArmstadt LINear ACcellerator (S-DALINAC)
2016 | SFB 634 | Achim Richter | 6

7. Open Microwave Photonic Crystal
A photonic crystal is a structure, whose electromagnetic properties vary periodically in space → array of metallic cylinders between two plates
For excitation frequencies f < c/2d the E-field is perpendicular to the plates
Propagating modes are solutions of the scalar Helmholtz equation → Schrödinger equation for a quantum multiple-scattering problem
2016 | SFB 634 | Achim Richter | 7
20/10/2017 |

8. Calculated Photonic Crystal Band Structure
Dispersion relation of a photonic crystal exhibits a band structure analogous to the electronic band structure in a solid f
second
band
first K- K+ f
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
2016 | SFB 634 | Achim Richter | 8

9. Microwave Dirac Billiard: Photonic Crystal in a Flat Resonator
Armchair edge
Zigzag edge
Zigzag edge
Graphene flake: the electron cannot escape → graphene billiard: “real graphene“
Photonic crystal: electromagnetic waves can escape from it
→ microwave Dirac billiard: “artificial graphene”
Model system for real graphene billiards (J. Wurm et al., PRB 84, (2011))
Classical dynamics of rectangular / Africa-shaped billiard is regular / chaotic
2016 | SFB 634 | Achim Richter | 9

10. Regular and Chaotic Superconducting Dirac Billiards
For the photonic crystal ~900 cylinders are milled out of a brass plate
Basin with scatteres and lid are lead plated
Lead is superconducting below Tc=7.2 K → high Q value at TLHe=4.2 K
Height d = 3 mm  for f < c/2d = 50 GHz → 2D system
2016 | SFB 634 | Achim Richter | 10

11. Transmission Spectrum of a Rectangular Dirac Billiard at 4 K
Measured S-matrix: |S21|2=P2 / P1
Quality factors > 5∙105
Pronounced band gaps and Dirac points
Extracted ~5000 resonance frequencies (eigenvalues)
2016 | SFB 634 | Achim Richter | 11

12. Density of States of the Measured Spectrum and the Band Structure
Positions of stop bands are in agreement with calculation
Dips correspond to Dirac points
DOS related to slope of a band
Flat band has very high DOS
High DOS at van Hove singularities → ESQPT?
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
2016 | SFB 634 | Achim Richter | 12

13. Experimental DOS and Topology of Band Structure
saddle point
 point
neck
K point
M point
saddle point
r ( f )
Each frequency f in the experimental DOS r ( f ) is related to an isofrequency line of the band structure in q space
Close to band edges isofrequency lines form circles around the G point
At the saddle point M the isofrequency lines become straight lines which cross each other and lead to the van Hove singularities where 𝛻 𝑞 𝜔 𝑞 =0
Parabolically shaped surface merges into the 6 Dirac cones around Dirac frequency (K point) → topological phase transition (“Neck-Disrupting Lifschitz Transition“) from non-relativistic to relativistic regime (B. Dietz et al., PRB 88, 1098 (2013))
2016 | SFB 634 | Achim Richter | 13
20/10/2017 |

14. Summary: Experimental DOS in the 1st and 2nd Band
Resonance density
Integrated resonance density
r( f ) a broad minimum at Dirac frequency / N( f ) exhibits a plateau
Dirac region bordered by sharp peaks in r( f ) / kinks in N( f )
Peaks evolve with increasing size of the photonic crystal / graphene sheet into van Hove singularities with a Dirac region in between and a Schrödinger region outside of them
Next: Comparison of experimental DOS with Tight Binding Model (TBM) calculations
2016 | SFB 634 | Achim Richter | 14

15. TBM Description for Spectrum of the Photonic Crystal
TBM for energies  and eigenstates
Generalized eigenvalue problem t3 t2
Tight-binding Hamiltonian t1
Wave function overlap matrix
is resonance frequency of an “isolated“ void
Nearest neighbour contribution  t1, s1
Next-nearest neighbour contribution  t2, s2
Second-nearest neighbour contribution  t3, s3
2016 | SFB 634 | Achim Richter | 15
20/10/2017 |
| 15

16. Fit of TBM with to Experiment
Excellent description of the data
Bump attributed to edge states at zigzag edges
Next: finite size scaling at the van Hove (VH) singularities and the nature of the topological phase transition
2016 | SFB 634 | Achim Richter | 16
20/10/2017 |

17. Finite-Size Scaling of DOS at the van Hove Singularities
The measured DOS of ideal graphene can be used to establish a VH singularity with a logarithmic divergence characteristic for 2D-structures in the thermodynamic limit (infinite extent of the system)
The TBM including only next-nearest hopping (t1=1, t2= t3=0) yields for finite size crystals formed by 𝑁 𝑐 hexagons
𝜌 𝑚𝑎𝑥 ≃ 3 2 𝜋 2 𝑁 𝑐 ( ln 𝑁 𝑐 −2 ln 𝜋+1+𝑂(1/ 𝑁 𝑐 ))
𝜌 𝑚𝑎𝑥 ~ 𝑎 𝑁 𝑐 ln 𝑁 𝑐
𝑎=0.141−0.155≈ 3 2 𝜋 2
Scaling law is universal and is typical for an ESQPT (Iachello et al., PRB 91, (2015))
2016 | SFB 634 | Achim Richter | 17

18. Schrödinger and Dirac Dispersion Relation in the Photonic Crystal
Dirac region
Schrödinger region
Dispersion relation along irreducible Brillouin zone 
Quadratic dispersion around the  point Schrödinger region 𝑓= 𝑞 2 /2 𝑚 𝑒𝑓𝑓
Linear dispersion around the  point  Dirac region 𝑓=𝑞 𝑣 𝐷
2016 | SFB 634 | Achim Richter | 18

19. Computed (TBM) Intensity Distributions Close to Band Edges (Schrödinger Region)
First ~250 (90) states counted from lower / upper band gap coincide with those of the empty rectangular (Africa-shaped) quantum billiard with Dirichlet boundary conditions
2016 | SFB 634 | Achim Richter | 19

20. Comparison of Spectral Properties of the Dirac Billiard and Quantum Billiard
: DB
: QB
Fluctuating part of integrated resonance density vs. eigenvalues ks of QB
Length spectrum from Gutzwiller’s trace formula
Very good agreement between DB and QB
2016 | SFB 634 | Achim Richter | 20

21. Computed (TBM) Intensity Distributions around Dirac Frequency
Intensity is spread evenly over the billiard area / localized at zigzag edges
A wave function structure is barely recognizable
Low-energy approximation, i.e., at Dirac point
2016 | SFB 634 | Achim Richter | 21

22. Spectral Properties: Nearest-Neighbour Spacing Distribution and D3 Statistics
200 resonance frequencies in region around Dirac point
Unfolding procedure: such that
Agreement with Poisson / GOE for regular rectangular / chaotic Africa-shaped billiard also very good in region near band edges
2016 | SFB 634 | Achim Richter | 22

23. Ratio Distribution of Adjacent Spacings at the van Hove Singularities
No analytical description for DOE around van Hove singularities
Spectrum can only be unfolded with polynomial ansatz for ~ 𝑓 𝑖 ’s
Consider ratio of two consecutive spacings 𝑟 𝑖 = 𝑠 𝑖 𝑠 𝑖+1
Advantage: ratios are independent of the DOS  no unfolding is required
Analytical prediction for Poisson and Gaussian RMT ensembles (Atas et al., 2013):
Poisson: 𝑃(𝑟)= 1 (1+𝑟) GOE: 𝑃 𝑟 = 𝑟+ 𝑟 2 (1+𝑟+𝑟 2 ) 5/2
2016 | SFB 634 | Achim Richter | 23

24. Ratio Distribution for Dirac Billiards
Ratio distribution for all 1656 (1823) resonance frequencies of rectangular (Africa) billiard agrees with Poisson (GOE) (Dietz et al., PRL 116, (2016))
2016 | SFB 634 | Achim Richter | 24

25. Computed (TBM) Intensity Distributions around Lower van Hove Singularity
Wave functions are scarred along zigzag rows within hexagonal lattice
2016 | SFB 634 | Achim Richter | 25

26. Summary: Spectral Properties of Plane Artificial Graphene Sheets (Photonic Crystals)
r( f )
Close to the G points, the quasimomenta coincide with the eigenvalues of corresponding quantum billiard and the dispersion relation is quadratic
→ non-relativistic Schrödinger region
Around the K points the Dirac billiards are described by the Dirac equation and the dispersion relation is linear
→ relativistic Dirac region
At the van Hove singularities happens a topological quantum phase transition between the S and D regions
2016 | SFB 634 | Achim Richter | 26

27. Curved Graphene: “Artificial Fulleren“
Graphene flake
Curved graphene flake
Fullerene C60
Plane graphene is composed of C atoms forming a hexagonal lattice
Curvature is introduced via replacing hexagons by pentagons
Spherical fullerene molecules possess 12 pentagons and a varying number of hexagons
Fullerene C60 consists of 12 pentagons and 20 hexagons arranged on a truncated icosahedron (“Buckyball“)
2016 | SFB 634 | Achim Richter | 27

28. Construction of Fullerene from a Plane Graphene Sheet
Mixing of sublattices
To introduce the 12 pentagons, p/3 sectors are cut out and glued together
Cones are created with the pentagon at the apex
Transformation from plane to sphere implies changes in the Dirac equation
Mixing of the sublattices along the seam is accounted for by a gauge field Am producing a fictitious flux due to a magnetic monopole at the center of the fullerene
The coordinate transformation associated with the curvature is accounted for by a quasi-spin connection Qm
2016 | SFB 634 | Achim Richter | 28

29. Comparison of the Dirac Operators in Covariant Form for Plane Graphene and Spherical Fullerene
Dirac operator for plane graphene around the Dirac point
Dirac operator for spherical fullerene around the Dirac point
curvature
sublattice mixing
Aim: experimental verification of the Atiyah-Singer index theorem
Relates the topology of the carbon lattice to the number of zero modes, i.e., of eigenvalues of the Dirac operator at the Dirac point
Plane graphene: no zero modes
Spherical fullerene: 2 pairs of triplets of (near) zero modes should exist
2016 | SFB 634 | Achim Richter | 29

30. Construction of the Superconducting Fullerene Microwave Billiard
The structure of C60 was milled out from a brass ball and lead plated
60 circular cavities with radius 12mm at the vertices of the truncated icosahedron
90 channels with width 14mm at edges of truncated icosahedron allow a weak coupling of modes excited inside the circular cavities
All cavities / channels closed with triangular / rectangular plates
Red caps mark antenna ports (altogether 8)
28 transmission spectra were measured
2016 | SFB 634 | Achim Richter | 30

31. Spectrum of the Fullerene Billiard
Band 1
Band 2
Band 3
Rest
Band 1 (60) centered around 1st (J0) mode of open circular billiard and below the 1st propagating mode → modes in cavities weakly coupled to neighbors
Band 2 (210) centered around 2nd (J1) mode of open circular billiard and above the 1st propagating mode → couples degenerate modes in cavities
Band 3 (90) centered around 3rd (J2) mode of open circular billiard and still below 2nd propagating mode → for symmetry reasons only 90 resonances
Rest: above 2nd propagating mode → no classification possible
2016 | SFB 634 | Achim Richter | 31

32. Band 1 3 5 4 9 1
Spectrum grouped into 15 multiplets and multiplicities agree with group theoretical predictions for the truncated icosahedral structure
The degeneracies are lifted due to inhomogeneities of lead coating, i.e., changes in radii of cavities
Dirac frequency is expected at the center of the first band, because of symmetry considerations (Manousakis PRB 44,10991 (1991))
In this region the Atiyah-Singer index theorem predicts 6 zero modes in the thermodynamic limit (infinite number of C atoms)
2016 | SFB 634 | Achim Richter | 32

33. Zoom into Region Around the Center of Band 1
Two triplets of nearly degenerate resonances
Triplets correspond to resonances closest to Dirac frequency
→ (near) zero modes predicted by Atiyah-Singer index theorem for a finite-size system
We corroborated this by matching the TBM to the experimental DOS and extrapolating it to larger fullerenes
Agreement between experimental and calculated DOS best when including 1st, 2nd and 3rd nearest-neighbor couplings
2016 | SFB 634 | Achim Richter | 33

34. Extrapolation of TBM Computations to Larger Fullerenes
DOS for Cn resembles more and more the DOS of the rectangular Dirac billiard with periodic boundary conditions
One important difference: DOS of the fullerenes exhibits a peak at fD due to the 2 triplets of (near) zero modes, that of graphene vanishes there
→ verification of the Atiyah-Singer theorem (Dietz et al., PRL 115, (2015))
With increasing n, i.e. towards the thermodynamic limit, the triplet frequencies converge towards the Dirac frequency fD = GHz
2016 | SFB 634 | Achim Richter | 34

35. Microwave Dirac Billiard
Outlook: Tailoring of Hamilton Operators in “Quantum Simulators“
Ultracold Atoms
in an Optical Lattice
Microwave Dirac Billiard
Examples for unique settings for quantum simulations of many-body systems
2016 | SFB 634 | Achim Richter | 35