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Journal of Physics A Mathematical and General

Flux quantization and quantum mechanics on Riemann surfaces in an external magnetic field

Jens Bolte, Frank Steiner

Published Aug 21, 1991• 12 citations
Publisher

Abstract

The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.

DOI
10.1088/0305-4470/24/16/019

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quant-pharXiv:2607.25684•Jul 29, 2026
The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.
The motion of a charged particle moving on a flat surface is studied through the constants of motion associated with the system, given the magnetic gauge. The usual Landau’ solution and the non-separable solution for the Landau’s gauge are found, and a new non-separable solution for the symmetric gauge is given. As a consequence of this, the quantization of the magnetic flux results from the invariance of the solutions under the unitary transformations which arise from the operator’s constants of motion.
Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime example are spin-coherent states which are used e.g. to describe single molecule magnets (SMM). We use this example to develop instanton calculus which does not rely on explicit solutions of the classical equations of motion. Energy conservation restricts the complex phase space to a Riemann surface of complex dimension one, allowing to deform integration paths according to Cauchy's integral theorem. As a result, the semiclassical actions can be evaluated without knowing actual classical paths. Furthermore we show that in many cases such actions may be solely derived from monodromy properties of the corresponding Riemann surface and residue values at its singular points. As an example, we consider quenching of tunneling processes in SMM by an applied magnetic field.
The author considers the quantum mechanics coupled to the gauge field which has the Chern-Simons term as the action. The generalized Hamiltonian formalism of the system is constructed and canonical and path-integral quantizations are performed. In solving the Gauss law constraint, the gauge field is represented by a multivalued function and the charge density operator of the particle. The gauge potential is singular and also induces a singular magnetic field through the magnetic flux is finite. By a phase transformation, the singular gauge field is eliminated. Then the transformed wavefunction absorbing the gauge field induces a novel phase factor under 2 pi -rotation in space. Thus the author shows that the transformed wavefunction describes exotic spin state. The extension of these analyses to the relativistic case is also presented.
Hydrodynamical formalism of quantum mechanics is revisited for a particle under external electromagnetic field, with detailed explanations as to its subtle aspects such as the role of singular vortex lines with quantized circulation. The theory is applied to several problems where a charged particle moves under certain new configurations of external magnetic field, which are suitable to provide examples of Aharonov·Bohm effect in different situations. A remarkable consequence is given about the flow in the case of string magnetic field and is employed to obtain solutions explicitly for the movement under the field of a circular magnetic string.
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 O, while odd monopole charges require the spinorial representations of the binary octahedral group 2O. 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.