57 research outputs found
Non-Commutative Geometry in Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
We clarify relations between the higher dimensional quantum Hall effect and
A-class topological insulator. In particular, we elucidate physical
implications of the higher dimensional non-commutative geometry in the context
of A-class topological insulator. This presentation is based on
arXiv:1403.5066.Comment: 5 pages, 1 table; contribution to the proceedings of the Workshop on
Noncommutative Field Theory and Gravity, Corfu, Greece, September 8-15, 2013,
Fortschritte der Physik 201
Quantum Hall Liquid on a Noncommutative Superplane
Supersymmetric quantum Hall liquids are constructed on a noncommutative
superplane.
We explore a supersymmetric formalism of the Landau problem. In the lowest
Landau level, there appear spin-less bosonic states and spin-1/2 down fermionic
states, which exhibit a super-chiral property. It is shown the Laughlin
wavefunction and topological excitations have their superpartners. Similarities
between supersymmetric quantum Hall systems and bilayer quantum Hall systems
are discussed.Comment: 11 pages, 3 figures, 1 table, minor corrections, published in
Phys.Rev.
Landau Models and Matrix Geometry
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.Comment: 1+59 pages, 8 figures, 1 table, minor corrections, published versio
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and
their mutual relations. The Hopf maps of division algebras provide a prototype
relation between monopoles and fuzzy spheres. Generalization of complex numbers
to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres
to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an
interesting hierarchical structure made of "compounds" of lower dimensional
spheres. We give a physical interpretation for such particular structure of
fuzzy spheres by utilizing Landau models in generic even dimensions. With
Grassmann algebra, we also introduce a graded version of the Hopf map, and
discuss its relation to fuzzy supersphere in context of supersymmetric Landau
model.Comment: v2: note and references added; v3: references adde
SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry
We review the recent developments of the SUSY quantum Hall effect
[hep-th/0409230, hep-th/0411137, hep-th/0503162, hep-th/0606007,
arXiv:0705.4527]. We introduce a SUSY formulation of the quantum Hall effect on
supermanifolds. On each of supersphere and superplane, we investigate SUSY
Landau problem and explicitly construct SUSY extensions of Laughlin
wavefunction and topological excitations. The non-anti-commutative geometry
naturally emerges in the lowest Landau level and brings particular physics to
the SUSY quantum Hall effect. It is shown that SUSY provides a unified picture
of the original Laughlin and Moore-Read states. Based on the charge-flux
duality, we also develop a Chern-Simons effective field theory for the SUSY
quantum Hall effect.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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