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Abstract

We consider the problem of magnetic charges in (2+1) dimensions for a torus geometry in real space, subjected to an inverted Lorentz force due to an external electric field applied normal to the surface of the torus. We compute the Hall conductivity associated with transport of these charges for the case of finite energy gap between the ground state and excitations and global U(1) charge conservation symmetry, and find it is proportional to an integer-valued topological invariant Q, corresponding to the magnetic quantum Hall effect (MQHE). We identify a lattice model realizing this physics in the absence of an external electric field. Based on this, we identify a generalization of the MQHE to be quantized transport of magnetic skyrmions, the quantum skyrmion Hall effect (QSkHE), with a U(1) easy-plane anisotropy/spin rotation symmetry of magnetic skyrmions and effective conservation of charge associated with magnetic skyrmions yielding incompressibility, provided a hierarchy of energy scales is respected. As the lattice model may be characterized both by a total Chern number and the topological invariant Q, we furthermore outline a possible field theory for electric charges, magnetic charges, and correlations between magnetic and electric charges approximated as composite particles, on a two-torus, to handle the scenario of intermediate-strength correlations between electric and magnetic charges modeled as composite particles. We map this problem to a generalized (4+1)D theory of the quantum Hall effect for the composite particles. © 2024 authors. Published by the American Physical Society.