Generalized GMP Algebra in Three-Dimensional Landau Levels

Three-dimensional Landau levels (3DLLs) emerge in systems with spin-orbit coupling and time-reversal symmetry, extending the quantum Hall effect into three dimensions with full-rotational symmetry. The system exhibits highly degenerate Landau levels with so(3,2) symmetry and quaternion analytic wavefunctions. We find that the exactly solvable semiclassical cyclotron orbits' centers correspond to the quantum guiding centers, whose non-commutative coordinates lead to non-trivial commutation relations of density operators projected to the lowest Landau level (LLL) - a 3D generalization of the Girvin-MacDonald-Platzman (GMP) algebra - crucial for understanding collective excitations in 3DLLs.