Electron correlation effects and two-photon absorption in diamond shaped graphene quantum dots

In quasi-1D $\pi$-conjugated polymers such as \emph{trans}-polyacetylene and polyenes, electron correlation effects determine the "reversed" excited state ordering in which the lowest two-photon $2A_{g}$ state lies below the lowest one-photon $1B_{u}$ state. In this work, we present conclusive theoretical evidence of reversed excited state ordering in fairly 2D $\pi$-conjugated systems, namely, diamond-shaped graphene quantum dots (DQDs). Our electron correlated calculations show that DQDs begin to exhibit reversed excited ordering with increasing size, in disagreement with independent-particle picture. This signals the onset of strong correlation effects which renders them nonluminescent. Further, we calculate and analyze the two-photon absorption (TPA) spectra as well as photoinduced absorption (PA) spectra of these systems and find excellent agreement with the available experimental results. Our investigations demonstrate that unlike a strictly 1D system like\emph{trans}-polyacetylene, the non-linear and excited state absorptions in DQDs are highly intricate, with several even parity states responsible for strong absorptions. Our results could play an important role in the design of graphene-based non-linear optical devices.Comment: 21 pages + 6 figures (main text), 16 pages + 5 figures (supporting information

The complex Lorentzian Leech lattice and the bimonster

We find 26 reflections in the automorphism group of the the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.Comment: 24 pages, 3 figures, revised and proof corrected. Some small results added. to appear in the Journal of Algebr

Reflection group of the quaternionic Lorentzian Leech lattice

In this article we study a second example of the phenomenon studied in "Complex Lorentzian Leech lattice and bimonster".(Arxiv. math.GR/0508228). The results and methods of proof are similar. We find 14 roots in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram D given by the incidence graph of projective plane over finite field of order 2. We prove that the reflections in these 14 roots generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the reflection group. Much of the work follows the analogy that D is like the Coxeter-Dynkin diagram for this hyperbolic reflection group.Comment: 12 pages, 1 figure, Revised and proof corrected. To appear in the Journal of Algebr