Hyperbolic Supersymmetric Quantum Hall Effect

Developing a non-compact version of the SUSY Hopf map, we formulate the quantum Hall effect on a super-hyperboloid. Based on $OSp(1|2)$ group theoretical methods, we first analyze the one-particle Landau problem, and successively explore the many-body problem where Laughlin wavefunction, hard-core pseudo-potential Hamiltonian and topological excitations are derived. It is also shown that the fuzzy super-hyperboloid emerges in the lowest Landau level.Comment: 14 pages, two columns, no figures, published version, typos correcte

Unification of Laughlin and Moore-Read States in SUSY Quantum Hall Effect

Based on the recently proposed SUSY quantum Hall effect, we show that Laughlin and Moore-Read states are related by a hidden SUSY transformation. Regarding the SUSY Laughlin wavefunction as a master wavefunction, Laughlin and Moore-Read states appear as two extreme limits of component wavefunctions. Realizations of topological excitations on Laughlin and Moore-Read states are also discussed in the SUSY formalism. We develop a streographically projected formulation of the SUSY quantum Hall effect. With appropriate interpretation of Grassmann odd coordinates, we illustrate striking analogies between SUSY quantum Hall effect and superfluidity.Comment: 5 pages, 1 figure, typos fixe

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.

SO(4)SO(4) Landau Models and Matrix Geometry

We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct an $SO(4)$ 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 $SO(4)$ 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 $S^3$ 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

Free infinite divisibility for beta distributions and related ones

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1.Comment: 37 pages, 6 figures, slightly different from the published versio

Conditionally monotone independence I: Independence, additive convolutions and related convolutions

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.Comment: 41 pages; small mistakes revised; to appear in Infin. Dimens. Anal. Quantum Probab. Relat. To

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