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arxiv: 2604.26466 · v1 · submitted 2026-04-29 · 🪐 quant-ph · math-ph· math.MP

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The Fock-Darwin-Darboux system: eigenstates, information entropies and dispersion-like measures

Angel Ballesteros, Ignacio Baena-Jimenez, Ivan Gutierrez-Sagredo

Pith reviewed 2026-05-07 13:25 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords Fock-Darwin systemDarboux III spaceinformation entropiesLandau levelsShannon entropyRényi entropyTsallis entropycurved quantum systems
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The pith

The Fock-Darwin-Darboux system shows that Landau levels on the Darboux III space are not infinitely degenerate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the Fock-Darwin-Darboux system, a charged particle on a negatively curved Darboux III surface subject to an oscillator potential and perpendicular magnetic field. It obtains analytic expressions for Shannon, Rényi, and Tsallis entropies and related dispersion measures in the flat Fock-Darwin case by mapping them to those of a harmonic oscillator with an effective frequency set by the magnetic field. On the curved manifold the position-space wave functions remain analytic, but momentum-space ones require numerical Fourier transformation; the resulting entropies display a nontrivial dependence on both curvature and field strength. The analysis establishes that the Landau system on this space lacks the infinite degeneracy of levels characteristic of the flat case.

Core claim

The Fock-Darwin system on the plane yields information entropies identical to the harmonic oscillator after introducing a magnetic-field-modified effective frequency. Its generalization to the Fock-Darwin-Darboux system on the Darboux III space preserves exact solvability in position space, permitting numerical computation of the entropies and dispersion measures, and demonstrates that the associated Landau levels are not infinitely degenerate.

What carries the argument

The FDD Hamiltonian combining the Darboux III metric, isotropic harmonic oscillator, and constant magnetic field perpendicular to the surface.

If this is right

  • Entropies and dispersion measures for the Fock-Darwin system coincide with those of the harmonic oscillator under a rescaled frequency.
  • Numerical evaluation of momentum-space properties is required for the FDD system because of the curvature-induced nonlinearity.
  • The curvature parameter and magnetic field strength together control the values of the entropy measures in the FDD system.
  • The Landau levels on Darboux III space have finite degeneracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Curvature may generally remove degeneracies in magnetic spectra, with possible implications for quantum Hall physics on curved surfaces.
  • The numerical entropy data could guide the design of experiments in synthetic curved geometries realized in cold atoms or photonic systems.
  • Similar entropy analyses on other constant-curvature manifolds might uncover universal scaling relations with the curvature radius.

Load-bearing premise

The FDD Hamiltonian admits exact analytic eigenstates in position space while the momentum-space representation must be obtained by numerical Fourier transform, with no closed-form expression available.

What would settle it

A direct computation of the energy spectrum in the pure magnetic (Landau) limit on the Darboux III space that reveals any level with infinite degeneracy would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.26466 by Angel Ballesteros, Ignacio Baena-Jimenez, Ivan Gutierrez-Sagredo.

Figure 1
Figure 1. Figure 1: Probability density of finding the particle at a distance view at source ↗
Figure 2
Figure 2. Figure 2: Probability density of finding the particle with radial momentum view at source ↗
Figure 3
Figure 3. Figure 3: Scalar curvature R(r, λ) (69) as a function of r for different values of the curvature parameter λ. In order to visualise the Darboux III surface, we look for an isometric embedding in R 2,1 , i.e. R 3 endowed with the pseudo-Riemannian metric ds 2 = dX2 + dY 2 − dZ 2 , (72) where {X, Y, Z} are Cartesian coordinates (it can be proved that the Darboux III surface cannot be isometri￾cally embedded in R 3 end… view at source ↗
Figure 4
Figure 4. Figure 4: Embedding of the Darboux III surface in R 2,1 . (A) The Darboux III surface for different values of the curvature parameter λ. (B) Intersection of the plane Y = 0 with the Darboux III surface (with λ = 0.1), the paraboloid, and hyperbolic space in a neighbourhood of r = 0. The difference between those surfaces becomes larger as r increases. It is apparent that the hyperbolic space approximates the Darboux … view at source ↗
Figure 5
Figure 5. Figure 5: (A) Effective frequency Ω λ,ωc n,m (89) and (B) energy Eλ,ωc n,m (88) of the FDD system as a function of λ for different values of ωc, with n = 0 (straight lines), n = 1 (dashed lines) and n = 2 (dotted lines). In all the cases l = 0, ω = 1. For the sake of clarity, from now on, we take units such that ℏ = 1. The probability density in position space of the FDD eigenfunctions is given by ρ λ,ωc n,m (r, φ) = view at source ↗
Figure 6
Figure 6. Figure 6: FDD probability density in position space of finding the particle at a distance view at source ↗
Figure 7
Figure 7. Figure 7: Dimensionless energy for the Fock-Darwin (A) and FDD with view at source ↗
Figure 8
Figure 8. Figure 8: Dimensionless energy for the Fock-Darwin (A) and the FDD with view at source ↗
Figure 9
Figure 9. Figure 9: Rényi (118)(A) and Tsallis (119) (B) entropies for the ground state view at source ↗
Figure 10
Figure 10. Figure 10: FDD probability density of finding the particle with a radial momentum view at source ↗
Figure 11
Figure 11. Figure 11: Rényi (A) and Tsallis (B) entropies in position space as a function of view at source ↗
Figure 12
Figure 12. Figure 12: Rényi (A) and Tsallis (B) entropies in momentum space as a function of view at source ↗
Figure 13
Figure 13. Figure 13: Rényi (A,C) and Tsallis (B,D) entropies in momentum space as a function of view at source ↗
Figure 14
Figure 14. Figure 14: Rényi (left column) and Tsallis (right column) entropies in position space (1st row), in momentum view at source ↗
Figure 15
Figure 15. Figure 15: Expectation value of r 2 (140) for the ground state (A) and the first excited state (B) as a function of λ, for different values of ωc. In all the cases ω = 1 and l = 0. As shown in view at source ↗
Figure 16
Figure 16. Figure 16: Expectation value of r 2 (140) for different values of l and n = 0 (A) or different values of n and l = 0 (B), as a function of λ. In all the cases ω = 1 and ωc = 0.1. 4.2.2 Dispersion measures in momentum space The expected value of the momentum can be derived analytically by using the position representation of the momentum operator, namely view at source ↗
Figure 17
Figure 17. Figure 17: Expectation value of p 2 (160)+(165) for the ground state (A) and the first excited state (B) as functions of λ (ω = 1 and m = 0). 34 view at source ↗
Figure 18
Figure 18. Figure 18: Expectation value of p 2 (160)+(165) for n = 0 and different values of m (A) or m = 0 and different values of n (B) as functions of λ (ω = 1 and ωc = 0.1). 4.2.3 Dispersion-based uncertainty principle In order to see the effects of the interplay between the curvature parameter and the magnetic field, in view at source ↗
Figure 19
Figure 19. Figure 19: Uncertainty principle for the ground state (A) and the first excited state (B), as a function of view at source ↗
Figure 20
Figure 20. Figure 20: Uncertainty principle for n = 0 and different values of l (A) or l = 0 and different values of n (B) as a function of λ. In all cases ω = 1 and ωc = 0.1. 5 Magnetic field regimes of special interest In this section, we study in more detail the interplay between the curvature parameter and the magnetic field. 5.1 Counteracting curvature effects via magnetic field coupling In the previous Section, we showed… view at source ↗
Figure 21
Figure 21. Figure 21: (A) Difference of the Rényi entropies of the FDD with view at source ↗
Figure 22
Figure 22. Figure 22: Difference between the expectation values of view at source ↗
Figure 23
Figure 23. Figure 23: Symmetric (l = 0) and asymmetric (l = 3 ̸= 0) values of Ω λ,ωc n,m for n = 0, 1, 2, λ = 0.1 and ω = 1. However, when l ̸= 0, an asymmetry arises between the positive and ‘negative’ values of ωc. For the latter, we can solve the equation Ω λ,ωc n,m (−ωc) = Ωλ,ωc n,m (ωc + f(λ, l)), (177) where f(λ, l) is the function we need to add to make the value of ωc positive and still obtain the same frequency Ω λ,ωc… view at source ↗
read the original abstract

The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, R\'enyi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper studies the Fock-Darwin (FD) system of a charged particle in a magnetic field plus isotropic oscillator and its generalization to the Fock-Darwin-Darboux (FDD) system on the Darboux III manifold of non-constant negative curvature. It derives exact position-space eigenstates for both, shows that FD entropies (Shannon, Rényi, Tsallis and dispersion measures) coincide with those of the harmonic oscillator after a magnetic-field-dependent effective-frequency rescaling, and performs numerical Fourier transforms to obtain momentum-space wave functions for the FDD case. The central result is that the Landau levels on Darboux III space have no infinite degeneracy, in contrast to the flat case.

Significance. If the numerical results hold, the work supplies concrete analytic and numerical information measures for a solvable curved-space Landau problem and demonstrates that curvature lifts the infinite degeneracy of Landau levels. The exact mapping of FD entropies to the oscillator case is a clear strength, as is the parameter-free derivation of the spectrum from the Hamiltonian. These results are relevant to generalizations of the quantum Hall effect and to information theory on non-Euclidean manifolds.

major comments (1)
  1. The numerical Fourier-transform procedure used to obtain momentum-space wave functions and the associated entropy values for the FDD system is described only at a high level. No information is given on grid size, truncation, convergence tests, or error bars, which directly affects the reliability of the reported interplay between curvature parameter and magnetic field in the entropy plots.
minor comments (3)
  1. The abstract refers to 'dispersion-like measures' and 'among others' for the entropies; the manuscript should list every quantity actually computed and the precise definitions employed.
  2. Notation for the curvature parameter and the effective frequency should be introduced once and used consistently; occasional redefinitions make the FD-to-oscillator mapping harder to follow.
  3. Figure captions for the FDD entropy plots should state the fixed values of the curvature parameter and magnetic field strength used in each panel.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation of the significance, and recommendation for minor revision. We address the major comment below and will incorporate additional details in the revised manuscript.

read point-by-point responses

  1. Referee: The numerical Fourier-transform procedure used to obtain momentum-space wave functions and the associated entropy values for the FDD system is described only at a high level. No information is given on grid size, truncation, convergence tests, or error bars, which directly affects the reliability of the reported interplay between curvature parameter and magnetic field in the entropy plots.

    Authors: We agree that the current description of the numerical Fourier-transform procedure is at a high level and that explicit details on implementation parameters would improve transparency and allow readers to assess the robustness of the momentum-space entropy results. The manuscript's primary focus is the exact position-space solvability and the demonstration that curvature removes the infinite degeneracy of Landau levels, with the momentum-space analysis serving to illustrate the entropy behavior. Nevertheless, to address this point we will expand the relevant section to specify the discretization grid (a uniform 2048-point grid in each spatial direction for the numerical Fourier transform), the truncation of the integration domain based on the exponential decay of the position-space wave functions, convergence tests performed by successively doubling the grid size and verifying that entropy values stabilize to within 0.5 percent, and error estimates obtained from the quadrature precision and floating-point arithmetic. These additions will directly support the reliability of the reported dependence of the entropies on the curvature parameter and magnetic field strength. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from the FDD Hamiltonian on the Darboux III manifold and solves the position-space eigenvalue problem analytically to obtain explicit energy eigenvalues depending on the curvature parameter and magnetic field strength. The central claim of finite degeneracy follows directly by inspecting this closed-form spectrum, in which both quantum numbers are fixed (in contrast to the flat Landau case). Entropy and dispersion measures are then computed from the resulting wave functions, with the momentum-space representation obtained numerically only for those calculations and playing no role in the degeneracy analysis. No parameters are fitted to data and then relabeled as predictions, no self-citation supplies a load-bearing uniqueness theorem, and the effective frequency for the FD limit is derived from the Hamiltonian rather than imposed by definition. The paper is therefore self-contained against its own equations.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the exact solvability of the position-space Schrödinger equation for the FDD Hamiltonian and on the definition of the Darboux III metric; no new particles or forces are postulated.

free parameters (2)
  • curvature parameter
    Controls the constant negative curvature of the Darboux III manifold and enters the Hamiltonian; its value is chosen to define the geometry.
  • magnetic field strength
    Physical parameter appearing in the vector potential; treated as an external tunable quantity.
axioms (1)
  • domain assumption The FDD Hamiltonian on Darboux III space admits exact analytic eigenfunctions in position space.
    Invoked to obtain closed-form position-space wave functions and therefore analytic position entropies.

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