Graphene Quantum Scars Defy Relativistic Expectations

Researchers have demonstrated that quantum scars observed in graphene billiards do not exhibit the characteristic features of relativistic quantum systems, despite the fact that the electronic excitations in these structures are effectively governed by the Dirac equation. Published in Discover Quantum Science, the study by Barbara Dietz, Dung Xuan Nguyen, and Tilen Čadež compares the properties of scarred eigenstates in nonrelativistic quantum billiards, relativistic neutrino billiards, graphene billiards, and Haldane graphene billiards. The findings reveal a fundamental divergence in how these systems behave, showing that graphene billiards align with nonrelativistic models, while Haldane graphene billiards conform to relativistic predictions.

The research focuses on billiard systems shaped like full and quarter stadiums, a geometry known for its chaotic classical dynamics. In such systems, quantum wave functions can become localized along specific periodic orbits, a phenomenon known as quantum scarring. The most prominent of these orbits are bouncing-ball orbits, which bounce back and forth between the straight-line segments of the stadium boundary. The study investigates how these scarred states manifest in different physical contexts, specifically contrasting the nonrelativistic Schrödinger equation with the relativistic Dirac equation for massless spin-1/2 particles.

Graphene billiards are constructed by cutting a specific shape out of a honeycomb lattice. In the region of linear dispersion around the Fermi energy, the energy excitations of graphene are well described by the Dirac equation, leading to the expectation that their spectral properties would resemble those of relativistic neutrino billiards. However, previous numerical and experimental studies had already suggested that graphene billiards conform to the spectral properties of nonrelativistic quantum billiards, exhibiting Gaussian orthogonal ensemble statistics rather than the Gaussian unitary ensemble statistics expected for relativistic systems.

To resolve this discrepancy, the authors applied a semiclassical approach to analyze the spectral density and wave function properties. They computed eigenvalues and eigenfunctions for nonrelativistic and neutrino billiards using a boundary-integral method, while for graphene and Haldane graphene billiards, they employed a tight-binding model on a honeycomb lattice. The study utilized momentum distributions and Husimi functions to identify and classify scarred wave functions, providing a detailed map of the localization properties along various periodic orbits.

The analysis revealed that the scarred wave functions in graphene billiards are localized along the same types of periodic orbits as those in nonrelativistic billiards. In contrast, the momentum distributions for neutrino billiards lack symmetry with respect to theta_k = 0 due to the chirality and unidirectionality of the local current induced by the relativistic boundary conditions.

Furthermore, the spectral properties of the full-stadium graphene billiard agree well with those of the nonrelativistic quantum billiard. The spectrum of the graphene billiard can be described by block-diagonal random matrices consisting of four blocks of matrices from the Gaussian orthogonal ensemble, corresponding to the four mirror-symmetry classes. This stands in contrast to the neutrino billiard, where the Dirac Hamiltonian is not time-reversal invariant, leading to different symmetry properties and spectral statistics. The graphene billiard wave functions are real and exhibit mirror symmetry, further supporting their nonrelativistic classification.

In contrast, the Haldane graphene billiard, which incorporates a gap at one of the Dirac points, exhibits properties that align with relativistic neutrino billiards. By applying the Haldane model, the researchers introduced a mass term and a purely imaginary next-nearest-neighbor hopping term, which breaks time-reversal invariance explicitly. The resulting eigenstates of the Haldane graphene billiard are complex and do not exhibit the mirror symmetries observed in standard graphene billiards. The spectral properties of the Haldane graphene billiard agree well with those of the corresponding neutrino billiard, confirming its relativistic character.

The study also examined the impact of these scarred eigenstates on the spectral properties of the billiards. The authors found that only eigenstates scarred along bouncing-ball orbits have a perceivable impact on the spectral properties. By employing semiclassical trace formulas, they were able to extract the non-generic contributions from the spectrum, revealing that the remaining spectral statistics conform to random-matrix theory predictions. This extraction process confirmed that the slow oscillations in the integrated spectral density are well described by the semiclassical trace formulas for the respective billiard types.

The implications of these findings are significant for the field of quantum chaos. The results demonstrate that even though graphene billiards exhibit relativistic phenomena such as pseudodiffusive transport and Klein tunneling, their quantum scars do not provide a suitable model for the study of relativistic quantum chaos. Instead, Haldane graphene billiards, with their broken time-reversal symmetry and complex wave functions, offer a more appropriate platform for investigating relativistic quantum scars. This distinction is crucial for designing experiments and theoretical models aimed at exploring the interplay between quantum chaos and relativistic effects.

The authors note that the recent proposal to simulate the Haldane model with photonic crystals provides a potential avenue for realizing energy spectra with the particular features of neutrino billiards experimentally. While the Haldane model has been realized on several platforms, observing quantum chaos in these systems remains challenging due to limitations in system size and boundary control. Nevertheless, the study highlights the importance of considering the specific boundary conditions and symmetry properties when modeling relativistic quantum systems in condensed matter platforms.

This research contributes to a deeper understanding of the quantum-classical correspondence in chaotic systems. By meticulously comparing the properties of scarred eigenstates across different billiard types, the authors have clarified the conditions under which relativistic effects manifest in quantum scars. The study underscores the complexity of quantum chaos and the need for precise modeling of boundary conditions and symmetry properties.

In their computational framework, the authors modeled graphene sheets cut from a honeycomb lattice with boundary edges oriented along the armchair or zigzag axes, presenting specific numerical calculations for the zigzag configuration. To isolate relativistic behavior, they focused on the conical valley regions around the Fermi energy where valence and conduction bands touch at the hexagonal Brillouin zone corners. In these regions, massless spin-1/2 particles follow the relativistic Dirac equation. However, back-scattering at the boundary edges mixes valley states between the two inequivalent Dirac points, thereby restoring time-reversal invariance across the overall system. This intervalley mixing explains why standard graphene billiards reproduce the spectral patterns of nonrelativistic billiards governed by the Schrödinger equation rather than the chiral, time-reversal-violating behavior of neutrino billiards.

To test whether opening a spectral gap could recover true relativistic signatures, the researchers introduced the Haldane model onto the lattice structure. By combining opposite onsite potentials of magnitude 0.3 on the interpenetrating triangular sublattices to break inversion symmetry with purely imaginary next-to-nearest-neighbor tunneling at the critical threshold of 0.3 divided by three times the square root of three, they deliberately gapped out the Dirac cone at one valley while preserving a single cone at the other. Because the purely imaginary hopping term explicitly breaks time-reversal invariance, eigenstates in this energy window remained confined to a single Dirac cone, successfully reproducing the spectral characteristics and eigenspinor features of massless neutrino billiards subject to vanishing normal boundary current.

The study also analyzed localization widths and spectral scaling across the calculated wave functions. For both nonrelativistic billiards and neutrino billiards, the inverse participation numbers derived from on-shell momentum distributions reached their absolute maximum for two-bounce bouncing-ball orbits. An examination of the momentum distribution peak widths at half maximum showed that they decrease algebraically with wavenumber. Power-law fits demonstrated that these peak widths decay proportionally to the wavenumber raised to the power of negative 0.936 plus or minus 0.002 for nonrelativistic billiards and negative 0.957 plus or minus 0.001 for neutrino billiards, conforming closely to an inverse wavenumber scaling.

Calculations of the integrated spectral density further revealed how periodic orbit quantization governs these systems. In both the full-stadium and quarter-stadium geometries, the fluctuating spectral density showed slow oscillations matching analytical trace formulas for bouncing-ball orbits. For the nonrelativistic billiard and graphene billiard, the analytical cosine summation term incorporates a fixed phase shift of negative three pi over four. For the neutrino billiard and Haldane graphene billiard, the trace formula includes an additional phase of m times pi, shifting the quantization condition and the corresponding resonance wavenumbers. The authors showed that this condition depends linearly on wavenumber rather than energy dispersion, explaining why the discrete bouncing-ball wavenumbers in graphene billiards match nonrelativistic values while Haldane graphene billiards match relativistic neutrino billiards.

The authors identified clear computational and physical constraints in their comparative approach. While boundary-integral methods yielded 7,500 eigenvalues for the continuous billiards, tight-binding diagonalizations of lattice matrices spanning roughly 90,000 to 130,000 sites produced only about 500 valid eigenstates within the linear dispersion regime around the Dirac point. In standard graphene billiards, the presence of zigzag boundary edges introduced a slight perturbation of mirror symmetry, causing eigenvalue degeneracies to lift and preventing the complete extraction of non-generic scarred contributions. Furthermore, because accessible wavenumbers on the discrete lattice were relatively low, Husimi phase-space representations could not resolve scars in the lattice models, leaving momentum distributions and trace formulas as the primary tools for establishing correspondence.

Subject of Research: Nonrelativistic versus relativistic quantum scars in billiard systems

Article Title: Nonrelativistic versus relativistic quantum scars in billiard systems

Article References: Dietz, B., Nguyen, D. X., & Čadež, T. (2026). Nonrelativistic versus relativistic quantum scars in billiard systems. Discover Quantum Science, 2(1), Article 15. https://doi.org/10.1007/s44464-026-00018-8

Image Credits: AI Generated

DOI: 10.1007/s44464-026-00018-8

Keywords: Nonrelativistic, versus, relativistic, quantum, scars, billiard, systems, scientific research