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

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, 214307 (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

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.

Experiments with microwave resonators B. Dietz, T. Klaus, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, 035411 (2015) F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, 214307 (2015)

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, 214307 (2015)

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, 214307 (2015)

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. 73-97 Pavel Stránský, Michal Macek, Amiram Leviatan, Pavel Cejnar, Ann. Phys., 356, May 2015, p. 57-82 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)

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

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)

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

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

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

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

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

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

Beyond-NN hopping parameter values t NN = 4.573 GHz t 2N = -0.284 GHz t 3N = 0.104 GHz taken from F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, 214307 (2015) #594 #597 #598 #600 #605

Beyond-NN hopping parameter values t NN = 4.573 GHz t 2N = -0.284 GHz t 3N = 0.104 GHz taken from F. Iachello, B. Dietz, M. Miski-Oglu, and A. Richter Phys. Rev. B 91, 214307 (2015) #1051 #1052 #1053 #1054 #1055

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

Thank you!

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) 104101