Random matrices and quantum chaos in weakly-disordered graphene nanoflakes

Statistical distribution of energy levels for Dirac fermions confined in a quantum dot is studied numerically on the examples of triangular and hexagonal graphene flakes with random electrostatic potential landscape. When increasing the disorder strength, level distribution evolves from Poissonian to Wigner, indicating the transition to quantum chaos. The unitary ensemble (with the twofold valley degeneracy) is observed for triangular flakes with zigzag or Klein edges and potential varying smoothly on the scale of atomic separation. For small number of edge defects, the unitary-to-orthogonal symmetry transition is found at zero magnetic field. For remaining systems, the orthogonal ensemble appears. These findings are rationalized by means of additive random-matrix models for the cases of weak and strong intervalley scattering of charge carriers in graphene. The influence of weak magnetic fields, as well as the strong-disorder-induced wavefunction localization, on the level distribution is also briefly discussed.