Quantum Hall effect on odd spheres

Abstract

We solve the Landau problem for charged particles on odd-dimensional spheres S 2k−1 in the background of constant SO(2k − 1) gauge fields carrying the irreducible representation I 2 , I 2 , · · · , I 2 . We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner D-functions, and for odd values of I the explicit local form of the wave functions in the lowest Landau level (LLL). Spectrum of the Dirac operator on S 2k−1 in the same gauge field background together with its degeneracies is also determined and in particular the number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial S 2k−2 is captured by constructing the relevant projective modules. For the Landau problem on S 5 , we demonstrate an exact correspondence between the union of Hilbert spaces of LLL's with I ranging from 0 to I max = 2K or I max = 2K + 1 to the Hilbert spaces of the fuzzy CP 3 or that of winding number ±1 line bundles over CP 3 at level K, respectively.

DFT investigations of the unusual reactivity of 2‐pyridinecarboxaldehyde in base‐catalyzed aldol reactions with acetophenone

“…* In a related investigation, 2-and 3-fluoro benzaldehydes were also found to give rise to diketones in base-catalyzed aldol reactions with 2-acetylbenzofuran. [8] However, the yields in these reactions were somewhat lower than similar reactions involving 2-pyridinecarboxaldehyde. When we compared the LUMO energies of fluoro-substituted chalcones ( Figure 10), calculations reveal that the LUMO energies of trans-1-phenyl-3-(2 0 -fluoro or 3 0 -fluorophenyl)-2-propenone are indeed lower than the LUMO energy of trans-chalcone but significantly higher than that of the trans isomer of 1-phenyl-3-(2-pyridinyl)-2-propen-1-one.…”

“…Moreover, Coskun reported similar results in base-catalyzed aldol reactions of 2-pyridinecarboxaldehyde, 2-fluoro-, and 3-fluorobenzaldehyde with 2-acetylbenzofuran. [8] We had previously hypothesized that, in the reactions of 2-pyridinecarboxaldehyde with acetophenone, sodium ion complexation by the heteroaromatic ring may contribute to the stereoselectivity of the reaction and the ensuing reactivity of the aldol condensation product. [7] Coordination of the metal ion by the aromatic rings nitrogen and both oxygens of the aldolate may favor a conformation where the pyridinyl ring is gauche rather than anti to the benzoyl group.…”

“…In base-promoted aldol condensation reactions, loss of hydroxide from enolate 7 is considered to be the slowest step in the elimination and, hence, is rate determining. [8] Numerous theoretical studies of elimination reactions have been published supporting the importance of an antiperiplanar transition state in which the dihedral angle between the proton being removed and the nucleofuge (in this case, the hydroxyl group) is 180 . [26,29,30] As described by Mohrig, [31] anti stereospecificity results from the coordinated with a single sodium ion.…”

“…Recently, similar results were reported by Coskun in the reactions of 2-pyridinecarboxaldehyde, 2-fluoro-, and 3-fluorobenzaldehyde with 2-acetylbenzofuran. [8] In each of these reactions, the aldol condensation product rapidly undergoes attack at the β-carbon of the α,β-unsaturated ketone by a second equivalent of the enolate resulting in formation of a symmetrically substituted diketone.…”

SO(4) Landau models and matrix geometry

“…We also demonstrated that similar relation holds for arbitrary odd and even dimensional Landau models [37] and the dimensional relation has its origin in differential topology; the dimensional ladder of anomaly or the spectral flow of Atiyah-Patodi-Singer. Though the foundation of the odd dimensional Landau models was thus established, there are merely a handful of works about them up to the present [30,36,37,38]. (See also [39,40,41] for odd dimensional topological insulator Landau models based on the Dirac oscillator.…”