Constructing the quantum Hall system on the Grassmannians Gr2(CN)

F Ballı,A Behtash,G Ünal,S Kürkçüoğlu

doi:10.1088/1742-6596/597/1/012015

Abstract

In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(CN). We set up the Landau problem in Gr2(CN), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D-functions for charged particles on Gr2(CN) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C4) we provide explicit constructions of the single and many- particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor v = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr2(C4) and the Hall effect on the odd sphere S5, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S4 and S8, respectively. The talk is given by S. Kürkçüoğlu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S. Kürkçüoğlu, G.Ünal [1].

Highlights

A 4-dimensional generalization of the quantum Hall effect (QHE) was introduced by Hu andZhang in [2]
It is worthwhile to remark that Landau problem on two and higher dimensional spaces have close and striking connections to string physics, D-branes and stringy matrix models and to the structure fuzzy spaces such as the fuzzy sphere SF2 and fuzzy complex projective spaces CPFN
To set up and solve the Landau problem on Gr2(C4), we contemplate that SU (4) Wigner D-functions may be suitably restricted to obtain the harmonics and local sections of bundles over Gr2(C4)

Summary

Introduction

A 4-dimensional generalization of the quantum Hall effect (QHE) was introduced by Hu andZhang in [2]. The multi-particle problem in the lowest Landau level (LLL) with filling factor ν = 1 may be seen as an incompressible 4-dimensional quantum Hall liquid as demonstrated by these authors. Functions on the latter to obtain the wave functions and the energy spectrum for charged particles

Results

Conclusion