$SO(5)$ Landau Models and Nested Nambu Matrix Geometry

Abstract

The $SO(5)$ Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the $SO(5)$ Landau models and the associated matrix geometries through the Landau level projection. With the $SO(5)$ monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized $SO(5)$ gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau levels the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an $SO(4)$ Pauli-Schrödinger model and the $SO(4)$ Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the $SO(5)$ Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.

1+52 papes, 8 figures, minor modifications

References in corpus (4)

• Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras

Cited by in corpus (7)

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• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background
• Five-dimensional Floquet topological semimetals with emergent Yang monopoles and linked Weyl surfaces
• Spin-entangled Squeezed State on a Bloch Four-hyperboloid
• A Unified Construction of Skyrme-type Non-linear sigma Models via The Higher Dimensional Landau Models
• Deformation of Matrix Geometry via Landau Level Evolution