Fig. 1. Dynamics of whispering gallery modes in circular graphene resonators.
(A) The parallel transport of a vector around a closed path C on a curved surface. For parallel transport the vector t remains perpendicular to r, and the orthogonal frame containing t and r does not twist about r (3). The transport results in the angular difference α between initial and final t vectors. (B) (lower) Schematic momentum-space contours for magnetic fields below (blue) and above (red) the critical magnetic field Bc, corresponding to the vectors shown in (D); (upper) the same contours projected on the Dirac cone by evaluating the kinetic energy T(Π). Above Bc (red), the contour encloses the Dirac point leading to a π Berry phase. (C) Schematics of the potential profile in the circular graphene resonator formed by a p-doped graphene center region and n-doped background, and classical orbits for positive m states below (left, blue) and above (right, red) Bc. (D) Schematic phase-space tori corresponding to the orbits shown in (C). The kinematic momenta Π (arrows), uniquely defined on the torus, are shown along the topologically distinct loops Cθ and CR. Lower panel: For CR below Bc (blue), the winding number of Π is zero and has a zero Berry phase; above Bc (red), the winding number is one, leading to a π Berry phase.