Braiding fluxes in Pauli Hamiltonian

Aharonov and Casher showed that Pauli Hamiltonians in two dimensions have gapless zero modes. We study the adiabatic evolution of these modes under the slow motion of $N$ fluxons with fluxes $\Phi_a\in\mathbb{R}$. The positions, $\mathbf{r}_a\in\mathbb{R}^2$, of the fluxons are viewed as controls. We are interested in the holonomies associated with closed paths in the space of controls. The holonomies can sometimes be abelian, but in general are not. They can sometimes be topological, but in general are not. We analyze some of the special cases and some of the general ones. Our most interesting results concern the cases where holonomy turns out to be topological which is the case when all the fluxons are subcritical, $\Phi_a<1$, and the number of zero modes is $D=N-1$. If $N\ge3$ it is also non-abelian. In the special case that the fluxons carry identical fluxes the resulting anyons satisfy the Burau representations of the braid group

Large coupling behaviour of the Lyapunov exponent for tight binding one-dimensional random systems

Studies the Lyapunov exponent gamma lambda (E) of (hu)(n)=u(n+1)+u(n-1)+ lambda V(n)u(n) in the limit as lambda to infinity where V is a suitable random potential. The authors prove that gamma lambda (E) approximately ln lambda as lambda to infinity uniformly as E/ lambda runs through compact sets. They also describe a formal expansion (to order lambda -2) for random and almost periodic potentials

Visualizing Two Qubits

The notions of entanglement witnesses, separable and entangled states for two qubits system can be visualized in three dimensions using the SLOCC equivalence classes. This visualization preserves the duality relations between the various sets and allows us to give ``proof by inspection'' of a non-elementary result of the Horodeckies that for two qubits, Peres separability test is iff. We then show that the CHSH Bell inequalities can be visualized as circles and cylinders in the same diagram. This allows us to give a geometric proof of yet another result of the Horodeckies, which optimizes the violation of the CHSH Bell inequality. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use three rather than two measurements each, does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class.Comment: 22 pages, 5 figures. Added several reference