Quantum Hall Effect on the Grassmannians ${\mathbf{Gr}}_2({\mathbb{C}}^N)$

Abstract

Quantum Hall Effects (QHEs) on the complex Grassmann manifolds ${\mathbf{Gr}}_2({\mathbb{C}}^N)$ are formulated. We set up the Landau problem in ${\mathbf{Gr}}_2({\mathbb{C}}^N)$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner $\mathcal{D}$-functions for charged particles on ${\mathbf{Gr}}_2({\mathbb{C}}^N)$ under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. In particular, for the simplest case of ${\mathbf{Gr}}_2({\mathbb{C}}^4)$ we explicitly write down the $U(1)$ background gauge field as well as the single and many-particle eigenstates by introducing the Plücker coordinates and show by calculating the two-point correlation function that the Lowest Landau Level (LLL) at filling factor $\nu =1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for QHE on $\mathbb{C}{P}^N$ and generalize them to all ${\mathbf{Gr}}_2({\mathbb{C}}^N)$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on ${\mathbf{Gr}}_2({\mathbb{C}}^4)$ and the Hall effect on the odd sphere $S^5$, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on $\mathbb{C}{P}^3$ and $\mathbb{C}{P}^7$ and those on the spheres $S^4$ and $S^8$, respectively.

34 pages, revtex 4-1, Minor Corrections

References in corpus (20)

• M Theory As A Matrix Model: A Conjecture
• A Large-N Reduced Model as Superstring
• A Four Dimensional Generalization of the Quantum Hall Effect
• Quantum Hall Effect in Higher Dimensions
• The Eight Dimensional Quantum Hall Effect and the Octonions
• On spherical harmonics for fuzzy spheres in diverse dimensions
• Effective field theory description of the higher dimensional quantum Hall liquid
• The effective action for edge states in higher dimensional quantum Hall systems
• How Bob Laughlin Tamed the Giant Graviton from Taub-NUT space
• Higher-Dimensional Quantum Hall Effect in String Theory
• Edge States for quantum Hall droplets in higher dimensions and a generalized WZW model
• The twistor geometry of three-qubit entanglement
• The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields
• Quantum Hall effect on $S^3$, edge states and fuzzy $S^3/{\mathbf{Z}}_2$
• Dimensional Hierarchy in Quantum Hall Effects on Fuzzy Spheres
• The Gromov width of complex Grassmannians
• Fuzzy Complex Grassmannian Spaces and their Star Products
• Fuzzy spaces, the M(atrix) model and the quantum Hall effect
• Geometric phase for SU(N) through fiber bundles and unitary integration
• Quantum Hall Effect on the Flag Manifold F_2

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• $SO(4)$ Landau Models and Matrix Geometry
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• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background