- Editors' Suggestion
- Open Access
Landau levels on the surface of a cube
APS Open Sci. 1, 000198 – Published 3 November, 2026
DOI: https://doi.org/10.1103/j2tl-p8y1
Abstract
We study the quantum mechanics of a charged particle confined to the surface of a cube enclosing a magnetic monopole. The magnetic field is chosen to have a constant magnitude on each face and to point along the outward normal, preserving the rotational symmetry of the cube. We formulate the continuum problem using two gauge patches on the cube surface and show that consistency of the wavefunction gives the Dirac quantization condition. Since an explicit vector potential does not remain invariant under ordinary rotations, we construct gauge-modified rotation operators and use them to classify the eigenstates. Even monopole charges are described by the irreducible representations of the cubic rotation group , while odd monopole charges require the spinorial representations of the binary octahedral group . We compute the spectrum with a gauge-covariant finite-difference discretization and find Landau-level-like manifolds whose degeneracies are split by the discrete cubic symmetry. We also study the corresponding tight-binding Hofstadter problem on the discretized cube. The resulting spectrum contains the usual magnetic subband structure together with additional gap states localized near the cube corners.
Physics Subject Headings (PhySH)
Article Text
References (21)
- J. Bělín, S. A. R. Horsley, and T. Tyc, Quantum mechanics and Talbot revivals on a tetrahedron, Phys. Rev. A 100, 033806 (2019).
- D. Cidlinský and T. Tyc, Quantum particle on the surface of a cube, Phys. Rev. A 110, 062218 (2024).
- Y. Avishai and J. M. Luck, Tight-binding electronic spectra on graphs with spherical topology: I. The effect of a magnetic charge, J. Stat. Mech. (2008) P06007.
- M. O. Oktel, Spectrum of a particle on a polyhedron enclosing a synthetic magnetic monopole, Eur. Phys. J. D 66, 88 (2012).
- G. M. Kemp and A. P. Veselov, Discrete analogues of Dirac's magnetic monopole and binary polyhedral groups, in Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012–2014, edited by V. M. Buchstaber, B. A. Dubrovin, and I. M. Krichever, American Mathematical Society Translations, Series 2, Vol. 234 (American Mathematical Society, 2014), pp. 223–240.
- P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc. London, Ser. A 133, 60 (1931).
- T. T. Wu and C. N. Yang, Dirac monopole without strings: Monopole harmonics, Nucl. Phys. B 107, 365 (1976).
- F. D. M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51, 605 (1983).
- N. Goldman, G. Juzeliūnas, P. Öhberg, and I. B. Spielman, Light-induced gauge fields for ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014).
- N. Navon, R. P. Smith, and Z. Hadzibabic, Quantum gases in optical boxes, Nat. Phys. 17, 1334 (2021).
- D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
- R. C. T. da Costa, Quantum mechanics of a constrained particle, Phys. Rev. A 23, 1982 (1981).
- M. Tinkham, Group Theory and Quantum Mechanics (Dover Publications, Mineola, NY, 2003).
- E. Brown, Bloch electrons in a uniform magnetic field, Phys. Rev. 133, A1038 (1964).
- J. Zak, Magnetic translation group, Phys. Rev. 134, A1602 (1964).
- P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., SciPy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
- A. Kura, B. Tekin, and M. O. Oktel, Landau levels on the surface of a cube, Zenodo, 2026, https://doi.org/10.5281/zenodo.21640419.
- H. Atakişi and M. Ö. Oktel, Landau levels in lattices with long-range hopping, Phys. Rev. A 88, 033612 (2013).
- R. R. Biswas and D. T. Son, Fractional charge and inter-Landau–level states at points of singular curvature, Proc. Natl. Acad. Sci. USA 113, 8636 (2016).
- K. Kudo, J. Schirmer, and J. K. Jain, Crossover from integer to fractional quantum Hall effect, Phys. Rev. B 109, 075157 (2024).
- G. Angelone, M. Asorey, P. Facchi, D. Lonigro, and Y. Martinez, Boundary conditions for the quantum Hall effect, J. Phys. A: Math. Theor. 56, 025301 (2023).
