Landau Models and Matrix Geometry
National Institute of Technology, Sendai College, Ayashi, Sendai, 989-3128, Japan
July 30, 2019
We develop an in-depth analysis of the Landau models on in the monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the monopole are introduced to provide a concrete coordinate representation of operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the covariance of the eigenfunctions. With the spin connection of , we construct an invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too. With the developed technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac-Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them.
The Landau models are physical models that manifest the non-commutative geometry in a most obvious way. It is well known [Hasebe-2015, Hatusda-Iso-Umetsu-2003] that the fuzzy two-sphere geometry [berezin1975, Hoppe1982, madore1992] is realized in the Landau model [Dirac-1931, Wu-Yang-1976] that provides a set-up of the 2D quantum Hall effect [Haldane-1983]. Similarly the set-up of the Landau model [Yang-1978-I, Yang-1978-II] is used for the construction of the 4D quantum Hall effect [Zhang-Hu-2001] whose underlying geometry is the fuzzy four-sphere [Grosse-Klimcik-Presnajder-1996, Castelino-Lee-Taylor-1997, Kimura2002]. The correspondence was further explored on [Hasebe-2014-1, Hasebe-Kimura-2003] and the Landau model was shown to realize the geometry of fuzzy -sphere [Ho-Ramgoolam-2002, Kimura2003]. Besides spheres, there are many manifolds that incorporate non-commutative geometry, and Landau models have been constructed on various manifolds, , supermanifolds, hyperboloids, etc. [Karabali-Nair-2002, Bernevig-Hu-Toumbas-Zhang-2003, Hasebe-2005, Jellal-2005, Hasebe-2008, Daoud-Jellal-2008, Hasebe-2010, Balli-Behtash-Kurkcuoglu-Unal-2014, Karabali-Nair-2016, Lapa-Jian-Ye-Hughes-2016, Heckman-Tizzano-2017]. The works have brought deeper understanding of the Landau physics and the associated fuzzy geometry as well. The magnetic field is the vital for the realization of the non-commutative geometry in the Landau model, and for spheres, the magnetic field is brought by the monopole at the center of the spheres. Since the monopole charge mathematically corresponds to the Chern number that is defined on even dimensional manifold, all of the manifolds used in the above works are even dimensional. Also in the viewpoint of the non-commutative geometry, adoption of the even dimensional manifolds is quite reasonable, because the geometric quantization is performed by replacing the Poisson bracket with the commutator, and even dimensional symplectic manifold generally accommodates non-commutative structure by such a quantization procedure.
|Monopole gauge group|
|Landau model||Landau model||Landau model||Landau model|
|Quantum Hall effect||2D QHE||3D QHE||4D QHE|