Graphene-like microwave billiards: Van-Hove singularities and Excited-State Quantum Phase Transitions

Transcription

1 Graphene-like microwave billiards: Van-Hove singularities and Excited-State Quantum Phase Transitions Michal Macek Sloane Physics Laboratory, Yale University in collaboration with: Francesco Iachello (Yale) Experiments: Barbara Dietz, Maxim Miski-Oglu, and Achim Richter (Darmstadt) F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015) M. Macek, F. Iachello, in preparation 27th Indian-Summer School on Graphene - the Bridge between Low- and High-Energy Physics Sep 14-18, Praha

2 Goals: 1. Understand structure of eigenstates at Van Hove singularities (VHS) and near the Dirac point (DP) for transverse vibration of a finite honeycomb lattice. 2. Look for signatures of Excited-State Quantum Phase Transitions (ESQPTs) at these energies. Outline: Experiment: High-precision spectral measurements in microwave resonators simulating transverse vibrations of finite-size graphene flakes able to resolve ALL eigenstates in the bands encapsulating first Dirac point. Theoretical decription: algebraic Vibron Model on honeycomb lattice. Basics of ESQPTs and relation to Van Hove singularities and Dirac points. Numerical results: Structure of eigenstates of the Vibron Model at energies near the Van- Hove singularities and Dirac points.

3 Experiments with microwave resonators B. Dietz, T. Klaus, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015) F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015)

4 Experiments with microwave resonators B. Dietz, T. Klaus, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015) F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015)

5 Vibron model on 2D lattices C i - Casimir of SO i (k) on-site and M ij - Majorana operator of U ij (k) (site coupling) Transverse vibrations of 2D lattice: k = 2 v i = 0, 1, 2,., N i SO i (2) U i (2) F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015)

6 One phonon band: harmonic limit, tight binding Energy Dispersion Relations: Density of States (elliptic integrals): F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015)

7 Excited-State Quantum Phase Transition (ESQPT) at the Van Hove Singularities ESQPT = singularity in level density ρ(e) [DoS(E)] of a finite system with Hamiltonian dependent on a parameter λ - studied previously in atomic nuclei and molecules - signifies a structural change in the system at finite excitation energy E = EESQPT (example: water bent to linear transition at 7 th vibrational state) E = EESQPT QPT M. Caprio, P. Cejnar, F. Iachello, Ann. Phys. 323 (2008) 1106 λ

8 Excited-State Quantum Phase Transition (ESQPT) at the Van Hove Singularities ESQPT = singularity in level density ρ(e) [DoS(E)] of a finite system with Hamiltonian dependent on a parameter λ - studied previously in atomic nuclei and molecules - signifies a structural change in the system at finite excitation energy E = EESQPT - related to thermodynamic anomalies f degrees of freedom ESQPTs occur at stationary points of the classical Hamiltonian related to the system Pavel Stránský, Michal Macek, Pavel Cejnar, Ann. Phys., 345, June 2014, p Pavel Stránský, Michal Macek, Amiram Leviatan, Pavel Cejnar, Ann. Phys., 356, May 2015, p Michal Macek, Pavel Stránský, Pavel Cejnar, Amiram Leviatan, to be submitted Analogy in lattice systems: dispersion relation E(k) instead of H(x,p). Notice the difference in dimension of integration Singularities in DoS and its derivatives occur at stationary points of dispersion relation E(k)

9 Structure indicators for the eigenstates Finite lattice with 24 x 69 sites (~experimental resonator) We consider first purely NN-hopping interactions, then NN+2N+3N Open (non-cyclical) boundary conditions Shannon Entropy in local basis : Spatial distribution of field intensity: red blue

10 NN-hopping Density of States for the finite (24 x 69 site) system contains peak at DP -> edge states (vanishes in samples) pronounced peaks at VHSs (N logn -diverge in samples)

11 NN-hopping Density of States for the finite (24 x 69 site) system contains peak at DP -> edge states (vanishes in samples) pronounced peaks at VHSs (N logn -diverge in samples) Shannon entropy indicates -increased spatial localization of eigenstates at VHSs and DP

12 NN-hopping spatial distributions Low-E: delocalized states - simple standing waves triangular sublattices in-phase (accoustic branch) #0 #1 #2 #3 #4

13 NN-hopping spatial distributions High-E: delocalized states - simple standing waves triangular sublattices in anti-phase (optical branch) #1651 #1652 #1653 #1654 #1655

14 NN-hopping spatial distributions Dirac point: edge states occur along zig-zag edges - field intensity vanishes in the bulk and at the armchair edges. #829 #831 #833 #834 #835

15 NN-hopping spatial distributions Lower Van Hove singularity: stripes through the bulk - intensity constant in zigzag dir. - phase oscillates fast parallel to zigzag and slow parallel to armchair edge #616 #617 #618 #619 #620

16 NN-hopping spatial distributions Upper Van Hove singularity: stripes through the bulk - intensity constant in zigzag dir. - phase oscillates fast in both zigzag and armchair directions (antiphase for A/B) #1035 #1036 #1037 #1038 #1039

17 Beyond-NN hopping parameter values t NN = GHz t 2N = GHz t 3N = GHz taken from F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015) #594 #597 #598 #600 #605

18 Beyond-NN hopping parameter values t NN = GHz t 2N = GHz t 3N = GHz taken from F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, (2015) #1051 #1052 #1053 #1054 #1055

19 Conclusions: 1. Microwave resonators allow to access VHS energies of the honeycomb lattice experimentally, spectra well-described by Vibron or tight-binding models 2. VHSs represent examples of ESQPTs 3. Eigenstates with interesting spatially localized and directional structure predicted at VHSs

20 Thank you!

21 NN-hopping Density of States for the finite (24 x 69 site) system contains peak at DP -> edge states (vanishes in samples) pronounced peaks at VHSs (N logn -diverge in samples) Measured in the photonic crystal experiment B. Dietz et al. Phys. Rev. B 88 (2013)