Shaping chaos in bilayer graphene cavities

TL;DR

This work demonstrates that boundary–lattice misalignment in AB-stacked BLG hexagonal cavities drives a quantum transition from near-integrable to chaotic spectral behavior, driven by trigonal warping of the Fermi surface. By combining atomistic tight-binding calculations with a semiclassical ray-dynamics picture, the authors show sector-resolved level statistics evolve from Poisson/semi-Poisson toward GOE/GUE mixtures as the boundary is rotated by $\theta$, while eigenstates acquire random-wave-like momentum content and reduced correlation lengths approaching the wavelength. The study links the loss of commensurability between the warped Fermi surface and the polygonal boundary to enhanced ergodicity in phase space, offering a tunable route to engineer quantum-chaotic behavior in graphene-based devices. These insights establish BLG cavities as a controllable platform for exploring quantum chaos with potential applications in electron transport and device engineering, and they provide experimental pathways via lithographic patterning or electrostatic confinement to probe level statistics and wavefunction morphology in graphene.

Abstract

Bilayer graphene cavities where electrons are confined within finite graphene flakes provide an alluring platform not only for the future nanoelectronic devices owing to the tunable energy gap but also for investigating the quantum nature of chaos due to the trigonal warping of their Fermi surface. Here we demonstrate that rotating the cavity boundary relative to the underlying lattice structure drives a quantum transition from nearly integrable dynamics to chaotic regime, observed as a concomitant crossover of eigenvalue statistics and eigenstate profiles. Complementing the full quantum treatment, we examine the classical backbone of this onset of chaos by employing semiclassical ray dynamics. Our results position bilayer graphene cavities as a promising venue for investigating and engineering quantum-chaotic behavior in graphene-based devices.

Figures (5)

• Figure 1: Illustration of the tight-binding model, Fermi surface, and cavity geometry. (a) Schematic diagram of the BLG tight-binding model showing atoms and hopping. (b) K valley of BLG's band structure, where the white curve represents the Fermi surface at $E=0.2\,\mathrm{eV}$, showing trigonal warping due to interlayer hopping $\gamma_3$. (c) and (d) show diagrams of BLG cut into a hexagonal shape, where purple indicates the dimer atoms $B_1$ and $A_2$, i.e., the vertically aligned sites that are directly coupled between the two layers, and red and blue indicate the non-dimer atoms $A_1$ and $B_2$. In (c), the boundary is aligned with the inner lattice, and the point group of the atoms is $D_{3d}$. In (d), the boundary is rotated by $15^\circ$ relative to the inner lattice, and the point group of the atoms is $S_6$. The cavities we study contain millions of atoms.
• Figure 2: Level Statistics of unrotated cavities and rotated BLG cavities. (a) and (b) show the level spacing statistics and spectral rigidity for the unrotated cavity, with the corresponding $\langle \tilde{r} \rangle$ values and averaged rigidity indicated in each panel. The red dashed line denotes the Poisson distribution, the blue solid line represents the Wigner–Dyson distribution, and the green dash-dotted line shows the fitted semi-Poisson distribution. (c) and (d) present the same analyses for the cavity rotated by $15^\circ$. The purple solid line represents the GUE distribution, and the orange dash-dotted line denotes the fitted GOE–GUE distribution.
• Figure 3: Angle dependence of $\langle \tilde{r} \rangle$ and $\langle \Delta_3 \rangle$ on $\theta$. (a) and (b) present the dependence of the $r$-value and the average spectral rigidity on the rotation angle $\theta$ for the $A$ and $E$ subspaces, respectively. In the $A$ subspaces, two lines are shown: for the unrotated cases $\theta = 0^\circ$ and $30^\circ$, the blue solid line corresponds to the pseudointegrable sector, while the purple dashed line represents the integrable sector.
• Figure 4: The wavefunctions and their associated statistical properties. (a1)–(a3) representative probability density distributions $|\psi(\mathbf{r})|^2$ of eigenstates in unrotated cavities for different symmetry subspaces. (a4) and (a5) the corresponding results for the rotated case. (b1)–(b5) the associated distributions $|\tilde{\psi}(\mathbf{k})|^2$ in momentum space with trigonal warped Fermi surface at $K$ and $K'$. (c1)–(c5) the statistical behavior of the correlation length $l$ for the unrotated and rotated cavities across the symmetry subspaces.
• Figure 5: Poincaré sections constructed by recording, for a single trajectory crossing a chosen reference edge, the boundary coordinate $s$ (position along the edge) and the incidence angle $\theta$ (relative to the boundary normal). (a) With the boundary aligned to crystalline $C_3$ axis, the map collapses onto a few invariant curves characteristic of pseudo-integrable motion. (b) After rotating the cavity by $15^\circ$, the map spreads quasi-ergodically over the accessible phase space.(c) For comparison, a mirror reflecting irrational hexagon shows uniformly filled Poincaré points, demonstrating true ergodicity without crystalline commensurability. (d) The angular magnification $|d\phi_{\rm out}/d\phi_{\rm in}|$ for BLG’s anisotropic reflection (blue) compared with the constant unit magnification of specular mirror reflection (red dashed). The variations in BLG magnification quantify the underlying warping responsible for the structures in Figs. \ref{['FIG5']} (a) and (b).