Figure 1

(Color online) Contours for bilayer graphene (in the two-state representation). Broken curves (in red): the potential $V=0$ [in the Hamiltonian of Eq. (4)]. Dotted curve (in blue): the potential $U=0$. The circle (in green) denotes an evolution path. The positions of the Dirac points are at the crossings of the broken and dotted curves. In the text, the evolution starts at a positive value of the abscissa. Orthogonality points $t_v$, where $\chi =\pi$, are where the path intersects the broken (red) curve. For the evolution path shown ($\text{circling}\text{radius}=0.5$, $\text{shift}=0.3$), there is only a single orthogonal position $t_v$ on the circular path, which is located on the negative abscissa. For circles shifted to the left it is possible to have two more $t_v$’s.Reuse & Permissions

Figure 2

(Color online) Inverse relative slowness parameter $n^{\prime }$ for adiabatic circling plotted by open circles (in blue) against the starting distance from the central Dirac point (DP) in the direction of a trigonal DP [given in units (DP), measuring the distance of the trigonal DP from the central one]. $n^{\prime }$ is approximated by the integer $n$ in the Berry phase $\left(2n+1\right)\pi$ acquired at an orthogonality point $t_v$ during circling. A circular path with radius of 0.5 (DP) was assumed. The inverse slowness parameter $n^{\prime }$ is large when the path starts near the central DP or (numerically) at just half way to the trigonal DP. (Here $n^{\prime }$ turns negative.) The triangles (in red) show the results of numerically computing the phase jumps. Some triangles are missing because of numerical difficulties in achieving convergence.Reuse & Permissions

Figure 3

Adiabatic phase evolution in bilayered graphene (obtained from numerical solution of the time-dependent Schrödinger equation within the two-state formalism). The adiabatic revolution period $T$ is $2\pi \times {10}^5$ in inverse units of the Fermi velocity times the DP distance. Circular cycling meets three orthogonality points along its path (as explained in the caption of Fig. 1). In the upper drawing, at these points steep rises take place in the phase, with the middle one (on the left part of the abscissa in Fig. 1) being the longest rise. The near-horizontal wavy part becomes straightened out, as the motion becomes more adiabatic. In the lower drawing the first $t_v$ region (above the $k_x$ axis) is enlarged.Reuse & Permissions

Figure 4

(Color online) Comparison of the Berry phase behaviors of the four- and two-state models in graphene bilayer. Curves for the four state model are slightly displaced downward for clarity. Two different starting positions to the right of the central DP were chosen, expressed in units of the distance to the rightmost DP. Upper frame starting position: 0.8. Lower frame starting position: 0.25. The adiabatic revolution period $T$ is $2\pi \times {10}^4$ in inverse units of the Fermi velocity times the DP distance. Note the different ordinate scales.Reuse & Permissions