SUPPORTED BY: NATIONAL SCIENCE CENTER OF POLAND (NCN) GRANT NO. N–N202–031440 Foundation for Polish Science (FNP) Program TEAM Random matrices and quantum chaos in graphene nanoflakes Adam Rycerz Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Introduction Transitions between symmetry classes The model Potential disorder in the tight-binding model on a honeycomb lattice The lattice Hamiltonian for disordered graphene in weak magnetic field reads where H TBA was diagonalized numerically for each quasirandom realization of the impurity potential. Numerical results Random matrices and spectral statistics of disordered systems When generic integrable system undergoes the transition to quantum chaos, its spectral properties may be reproduced by the random Hamiltonian where H 0 is is diagonal random matrix, which elements follow a Gaussian distribution with zero mean and the unit variance, whereas V is a member of GOE or GUE. (For N=2, the level-spacings distribution can be found analytically for any λ.) The transition to quantum chaos in disordered graphene flakes is rationalized using additive random-matrix models as the above. The functional relation between the best-fitted model parameter λ fit and the disorder strength has a form of a power law. The unitary symmetry class is observed in spectral statistics, providing almost all terminal atoms belong to one sublattice. This is satisfied for equilateral triangles with zigzag or Klein boundaries, which also show an approximate valley degeneracy of each energy level. The degeneracy is lifted at weak magnetic fields. For a fixed disorder strength in the chaotic range and increasing the number of edge vacancies we have observed the transition to GOE distribution. DIRAC FERMIONS IN WEAKLY-DISORDERED GRAPHENE SYMMETRIES OF THE HAMILTONIAN The effective Hamiltonian for low-energy excitations reads and is symmetric (at B=0) with respect to standard time reversal and two special time reversals: where C denotes complex conjugation. The mass term breaks symplectic symmetry leading to the two possible scenarios listed below. 1. WEAK INTERVALLEY SCATTERING commutes with so the system consists of two independent subsystems (one for each valley). Each subsystem lacks TRS as commutes only with full . Because the Kramer’s degeneracy ( ) the Hamiltonian of a chaotic system consists of two degenerate blocks (one per each valley), each of which may be modeled by a random matrix belonging to the Gaussian Unitary Ensemble (GUE). 2. STRONG INTERVALLEY SCATTERING For irregular and abrupt system edges (or a potential abruptly varying on the scale of atomic separation) the two sublattices are nonequivalent, so both special time- reversal symmetries and become irrelevant. For B=0, commutes with leading to the orthogonal symmetry class and statistical properties following from the Gaussian Orthogonal Ensemble (GOE) of random matrices. When increasing |B|, transition GOE- GUE appears. ARXIV:1201.3909 [ PRB, TO BE PUBLISHED ] M (x, y )σ z ⊗ τ 0 T v H Dirac T H Dirac T 2 v = −I T sl T v T H Dirac
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SUPPORTED BY: NATIONAL SCIENCE CENTER OF POLAND (NCN) GRANT NO. N–N202–031440 Foundation for Polish Science (FNP) Program TEAM
R andom matrices and quantum chaos in graphene nanoflakes
Adam RycerzMarian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
IntroductionTransitions between symmetry classes
The modelPotential disorder in the tight-binding model on a honeycomb lattice
The lattice Hamiltonian for disordered graphene in weak magnetic field reads
where
HTBA was diagonalized numerically for each quasirandom realization of the impurity potential.
Numerical resultsRandom matrices and spectral statistics of disordered systems
When generic integrable system undergoes the
transition to quantum chaos, its spectral properties may be reproduced by the random Hamiltonian
where H0 is is diagonal random matrix, which
elements follow a Gaussian distribution with zero mean and the unit variance, whereas V is a member of GOE or GUE. (For N=2, the level-spacings
distribution can be found analytically for any λ.)The transition to quantum chaos in disordered
graphene flakes is rationalized using additive random-matrix models as the above. The functional
relation between the best-fitted model parameter λfit and the disorder strength has a form of a power law. The unitary symmetry class is observed in spectral
statistics, providing almost all terminal atoms belong to one sublattice. This is satisfied for equilateral
triangles with zigzag or Klein boundaries, which also show an approximate valley degeneracy of each energy level. The degeneracy is lifted at weak
magnetic fields.For a fixed disorder strength in the chaotic
range and increasing the number of edge vacancies we have observed the transition to GOE distribution.
DIRAC FERMIONS IN WEAKLY-DISORDERED GRAPHENE
SYMMETRIES OF THE HAMILTONIANThe effective Hamiltonian for low-energy excitations reads
and is symmetric (at B=0) with respect to standard time reversal and two special time reversals:
where C denotes complex conjugation.
The mass term breaks symplectic symmetry leading to the two possible scenarios listed below.
1. WEAK INTERVALLEY SCATTERING commutes with so the system consists of two independent subsystems (one for each valley). Each subsystem lacks TRS as commutes only with full . Because the Kramer’s degeneracy ( ) the Hamiltonian of a chaotic system consists of two degenerate blocks (one per each valley), each of which may be modeled by a random matrix belonging to the Gaussian Unitary Ensemble (GUE).
2. STRONG INTERVALLEY SCATTERINGFor irregular and abrupt system edges (or a potential abruptly varying on the scale of atomic separation) the two sublattices are nonequivalent, so both special time-reversal symmetries and become irrelevant. For B=0, commutes with leading to the orthogonal symmetry class and statistical properties following from the Gaussian Orthogonal Ensemble (GOE) of random matrices. When increasing |B|, transition GOE-GUE appears.
ARXIV:1201.3909 [ PRB, TO BE PUBLISHED ]
M(x, y)σz⊗τ0
Tv HDirac
THDirac
T 2v = −I
Tsl Tv
THDirac
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