Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.26642

Massless Dirac Fermions in curved surfaces with localized curvature

A. R. N. Lima, D. F. S. Veras, J. E. G. Silva

quant-phcond-mat.mes-hallSubmitted Mar 27, 2026• Announced Mar 30, 2026
arXivHTMLPDF

Abstract

We investigate how a localized curvature affects the dynamics of massless Dirac fermions in a curved surface. We consider a smooth bump with axial symmetry, adopting two specific geometric models, namely a Gaussian and a volcano-like bumps. By considering a minimal coupling between the spinor and the surface geometry, described by the vielbeins and the spin connection, we study the behavior of the wave function over the surface. By using appropriate numerical methods, we find a linear discrete energy spectrum for the Dirac fermions and its corresponding wavefunctions when the Fermi velocity is considered. It turns out that, since the curvature vanishes asymptotically, the electron states are free waves far from the bumps, but around the curved points, the wave function increases its probability density.

Comments
12 pages, 19 figures

Related papers

Scattering due to geometry: Case of a spinless particle moving on an asymptotically flat embedded surface

Neslihan Oflaz, Ali Mostafazadeh, Mehrdad Ahmady

A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low-density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.

arXiv:1807.10099•Physical Review A•Aug 20, 2018•8 citations

No existence of the geometric potential for a Dirac fermion on a two-dimensional curved surface of revolution

Z.Q. Yang, X. Y. Zhou, Z. Li +2 more

For a free particle that moves non-relativistically on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states. However, the experimental results in topological insulators, whether curved or not, indicate no evidence of such a potential for the relativistic particles constrained on the curved surface. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-induced quantum potential is permissible.

arXiv:1912.02462•Physics Letters A•May 29, 2020•4 citations

Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface

Li Zhao, L. Q. Lai

Abstract Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.

arXiv:2403.15982•Physica Scripta•Sep 18, 2025•0 citations

Unified Exact Cylindrical Dirac Modes and Symmetry-Resolved Quantum Geometry

Zhongze Guo, Bei Xu, Qiang Gu

Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.

quant-pharXiv:2608.10563•Aug 12, 2026

Tails of the density of states of two-dimensional Dirac fermions

Simon Villain-Guillot, G. Jug, K. Ziegler

The density of states of Dirac fermions with a random mass on a two-dimensional lattice is considered. We give the explicit asymptotic form of the single-electron density of states as a function of both energy and (average) Dirac mass, in the regime where all states are localized. We make use of a weak-disorder expansion in the parameter g/m^2, where g is the strength of disorder and m the average Dirac mass for the case in which the evaluation of the (supersymmetric) integrals corresponds to non-uniform solutions of the saddle point equation. The resulting density of states has tails which deviate from the typical pure Gaussian form by an analytic prefactor.

arXiv:cond-mat/9908318•Annalen der Physik•Jan 1, 2000•2 citations

Modular Hamiltonians for the massless Dirac field in the presence of a boundary

Mihail Mintchev, Erik Tonni C

A bstract We study the modular Hamiltonians of an interval for the massless Dirac fermion on the half-line. The most general boundary conditions ensuring the global energy conservation lead to consider two phases, where either the vector or the axial symmetry is preserved. In these two phases we derive the corresponding modular Hamiltonian in explicit form. Its density involves a bi-local term localised in two points of the interval, one conjugate to the other. The associated modular flows are also established. Depending on the phase, they mix fields with different chirality or charge that follow different modular trajectories. Accordingly, the modular flow preserves either the vector or the axial symmetry. We compute the two-point correlation functions along the modular flow and show that they satisfy the Kubo-Martin-Schwinger condition in both phases. The entanglement entropies are also derived.

arXiv:2012.00703•Journal of High Energy Physics•Mar 1, 2021•42 citations
We investigate how a localized curvature affects the dynamics of massless Dirac fermions in a curved surface. We consider a smooth bump with axial symmetry, adopting two specific geometric models, namely a Gaussian and a volcano-like bumps. By considering a minimal coupling between the spinor and the surface geometry, described by the vielbeins and the spin connection, we study the behavior of the wave function over the surface. By using appropriate numerical methods, we find a linear discrete energy spectrum for the Dirac fermions and its corresponding wavefunctions when the Fermi velocity is considered. It turns out that, since the curvature vanishes asymptotically, the electron states are free waves far from the bumps, but around the curved points, the wave function increases its probability density.
A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low-density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.
For a free particle that moves non-relativistically on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states. However, the experimental results in topological insulators, whether curved or not, indicate no evidence of such a potential for the relativistic particles constrained on the curved surface. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-induced quantum potential is permissible.
Abstract Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.
Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.
The density of states of Dirac fermions with a random mass on a two-dimensional lattice is considered. We give the explicit asymptotic form of the single-electron density of states as a function of both energy and (average) Dirac mass, in the regime where all states are localized. We make use of a weak-disorder expansion in the parameter g/m^2, where g is the strength of disorder and m the average Dirac mass for the case in which the evaluation of the (supersymmetric) integrals corresponds to non-uniform solutions of the saddle point equation. The resulting density of states has tails which deviate from the typical pure Gaussian form by an analytic prefactor.
A bstract We study the modular Hamiltonians of an interval for the massless Dirac fermion on the half-line. The most general boundary conditions ensuring the global energy conservation lead to consider two phases, where either the vector or the axial symmetry is preserved. In these two phases we derive the corresponding modular Hamiltonian in explicit form. Its density involves a bi-local term localised in two points of the interval, one conjugate to the other. The associated modular flows are also established. Depending on the phase, they mix fields with different chirality or charge that follow different modular trajectories. Accordingly, the modular flow preserves either the vector or the axial symmetry. We compute the two-point correlation functions along the modular flow and show that they satisfy the Kubo-Martin-Schwinger condition in both phases. The entanglement entropies are also derived.