Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums

For Anderson tight-binding models in dimension $d$ with random on-site energies $\epsilon_{\vec r}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality regime corresponding to small $V$ can be studied via the standard perturbation theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios $Y_q(L)$, which are the order parameters of Anderson transitions, can be written in terms of weighted L\'evy sums of broadly distributed variables (as a consequence of the presence of on-site random energies in the denominators of the perturbation theory). We compute at leading order the typical and disorder-averaged multifractal spectra $\tau_{typ}(q)$ and $\tau_{av}(q)$ as a function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum $\tau_{typ}(q)=\tau_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method yields the same disorder-averaged spectrum $\tau_{av}(q)$ of order $O(V)$ as obtained previously via the Levitov renormalization method by Mirlin and Evers [Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly the typical spectrum, also of order $O(V)$, but with a different $q$-dependence $\tau_{typ}(q) \ne \tau_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find that the corresponding singularity spectra $f_{typ}(\alpha)$ and $f_{av}(\alpha)$ differ even in the positive region $f>0$, and vanish at different values $\alpha_+^{typ} > \alpha_+^{av}$, in contrast to the standard picture. We also obtain that the saddle value $\alpha_{typ}(q)$ of the Legendre transform reaches the termination point $\alpha_+^{typ}$ where $f_{typ}(\alpha_+^{typ})=0$ only in the limit $q \to +\infty$.Comment: 13 pages, 2 figures, v2=final versio

Shear modulated fluid amplifier Patent

Shear modulated fluid amplifier of high pressure hydraulic vortex amplifier typ

Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization

We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance $\overline{J^2(r)} \propto r^{-2\sigma}$ and a uniform transverse field $h$. The ground state is studied via real-space renormalization to characterize the spinglass-paramagnetic zero temperature quantum phase transition as a function of the control parameter $h$. In the spinglass phase $h<h_c$, the typical renormalized coupling grows with the length scale $L$ as the power-law $J_L^{typ}(h) \propto \Upsilon(h) L^{\theta}$ with the classical droplet exponent $\theta=1-\sigma$, where the stiffness modulus vanishes at criticality $\Upsilon(h) \propto (h_c-h)^{\mu}$, whereas the typical renormalized transverse field decays exponentially $h^{typ}_L(h) \propto e^{- \frac{L}{\xi}}$ where the correlation length diverges at the transition $\xi \propto (h_c-h)^{-\nu}$. At the critical point $h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c)$ and the typical renormalized transverse field $h^{typ}_L(h_c)$ display the same power-law behavior $L^{-z}$ with a finite dynamical exponent $z$. The RG rules are applied numerically to chains containing $L=2^{12}=4096$ spins in order to measure these critical exponents for various values of $\sigma$ in the region $1/2<\sigma<1$.Comment: 9 pages, 7 figure

Scattering at the Anderson transition: Power--law banded random matrix model

We analyze the scattering properties of a periodic one-dimensional system at criticality represented by the so-called power-law banded random matrix model at the metal insulator transition. We focus on the scaling of Wigner delay times $\tau$ and resonance widths $\Gamma$. We found that the typical values of $\tau$ and $\Gamma$ (calculated as the geometric mean) scale with the system size $L$ as $\tau^{\tiny typ}\propto L^{D_1}$ and $\Gamma^{\tiny typ} \propto L^{-(2-D_2)}$, where $D_1$ is the information dimension and $D_2$ is the correlation dimension of eigenfunctions of the corresponding closed system.Comment: 6 pages, 8 figure

The Posthumous Depiction of Youths in Late Hellenistic and Early Imperial Gymnasia

Dieser Beitrag untersucht posthume Ehrungen und Darstellungen von jungen Männern im Gymnasion, die in der Forschung bislang nicht umfassend untersucht worden sind. Nach einem Überblick über das bekannte Skulpturenrepertoire in Gymnasia werden die epigraphischen Quellen posthumer Ehrungen von jungen Männern diskutiert, die im Gymnasion trainierten und vorzeitig verstarben. Der Fokus liegt dann auf der Identifizierung von Skulpturen die, dem Kontext und der Ikonographie zufolge, als posthume Ehrungen von Jugendlichen gedient haben könnten. Darunter ist z.B. die Statue des Kleoneikos von Eretria. Es wird dargelegt, dass drei verschiedene ikonographische Typen für diese Ehrungen verwendet wurden: der nackte ‚heroische‘ Typ, der Himation-Typ und die Herme