Quasiperiodic Modulated-Spring Model

We study the classical vibration problem of a chain with spring constants which are modulated in a quasiperiodic manner, {\it i. e.}, a model in which the elastic energy is $\sum_j k_j( u_{j-1}-{u_j})^2$, where $k_j=1+\Delta cos[2\pi\sigma(j-1/2)+\theta]$ and $\sigma$ is an irrational number. For $\Delta < 1$, it is shown analytically that the spectrum is absolutely continuous, {\it i.e.}, all the eigen modes are extended. For $\Delta=1$, numerical scaling analysis shows that the spectrum is purely singular continuous, {\it i.e.}, all the modes are critical.Comment: REV TeX fil

Novel Heterocyclic Arylidene Derivatives as Anticancer Agents Against U87 Human Glioblastoma

The primary objectives of this interdisciplinary study were the synthesis of novel heterocyclic arylidenes and the investigation of their anticancer activity against U87 glioblastoma cell viability. Recently, novel hybrid derivatives have been considered as potential candidates for treating glioblastoma, demonstrating a synergistic anticancer effect in previous studies. 12 heterocyclic arylidenes with various functional groups, including halogens and boronic acid, were produced via a Knoevenagel condensation. These compounds and their starting reagents were then administered to U87 glioblastoma cancer cells at graded concentrations within a 12-well cell viability assay to determine each compound’s lethal concentration 50 (LC50). The LC50 of each compound was then compared to determine the effects of substituent type and position on anticancer activity. Although these arylidenes displayed some anticancer effects, their high LC50 suggest they have no significant effect on U87 glioblastoma cell viability and proliferation

Non-equivalence between Heisenberg XXZ spin chain and Thirring model

The Bethe ansatz equations for the spin 1/2 Heisenberg XXZ spin chain are numerically solved, and the energy eigenvalues are determined for the anti-ferromagnetic case. We examine the relation between the XXZ spin chain and the Thirring model, and show that the spectrum of the XXZ spin chain is different from that of the regularized Thirring model.Comment: 10 pages. 2figure

Density Matrix Renormalization Group Study of the S=1/2 Anisotropic Antiferromagnetic Heisenberg Chains with Quasiperiodic Exchange Modulation

The low energy behavior of the S=1/2 antiferromagnetic XY-like XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap of the chain with length N scales as $\exp (-cN^{\omega})$ with nonuniversal exponent $\omega$ if the Ising component of the exhange coupling is antiferromagnetic. This behavior is expected to be the characteristic feature of the quantum spin chains with relevant aperiodicity. This is in contrast to the XY chain for which the precious mean exchange modulation is marginal and the gap scales as $N^{-z}$. On the contrary, it is also verified that the energy gap scales as $N^{-1}$ if the Ising component of the exhange coupling is ferromagnetic. Our results are not only consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling regime which is inaccesssible by the bosonization approach.Comment: 8 pages, 15 figures, 1 table; Proceedings of the workshop 'Frontiers in Magnetism', Kyoto, Oct. 199

Mode entanglement of electrons in the one-dimensional Frenkel-Kontorova model

We study the mode entanglement in the one-dimensional Frenkel-Kontorova model, and found that behaviors of quantum entanglement are distinct before and after the transition by breaking of analyticity. We show that the more extended the electron is, the more entangled the corresponding state. Finally, a quantitative relation is given between the average square of the concurrence quantifying the degree of entanglement and the participation ratio characterizing the degree of localization.Comment: 4 pages, 4 figures. V

Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field

The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives the Hamiltonian. We obtain analytical and numerical solutions for the wave functions by solving the Bethe Ansatz equations, proposed by Wiegmann and Zabrodin on the basis of above observation. The semi-classical case with the flux per plaquette $\phi=1/Q$ is analyzed in detail, by exploring a structure of the Bethe Ansatz equations. We also reveal the multifractal structure of the Bethe Ansatz solutions and corresponding wave functions when $\phi$ is irrational, such as the golden or silver mean.Comment: 30 pages, 11 GIF figures(use xv, or WWW browser

Self-similarity under inflation and level statistics: a study in two dimensions

Energy level spacing statistics are discussed for a two dimensional quasiperiodic tiling. The property of self-similarity under inflation is used to write a recursion relation for the level spacing distributions defined on square approximants to the perfect quasiperiodic structure. New distribution functions are defined and determined by a combination of numerical and analytical calculations.Comment: Latex, 13 pages including 6 EPS figures, paper submitted to PR