Spectral flow and level spacing of edge states for quantum Hall hamiltonians

We consider a non relativistic particle on the surface of a semi-infinite cylinder of circumference $L$ submitted to a perpendicular magnetic field of strength $B$ and to the potential of impurities of maximal amplitude $w$. This model is of importance in the context of the integer quantum Hall effect. In the regime of strong magnetic field or weak disorder $B>>w$ it is known that there are chiral edge states, which are localised within a few magnetic lengths close to, and extended along the boundary of the cylinder, and whose energy levels lie in the gaps of the bulk system. These energy levels have a spectral flow, uniform in $L$, as a function of a magnetic flux which threads the cylinder along its axis. Through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by $2\pi\alpha L^{-1}$ with $\alpha>0$, independent of $L$, and of the configuration of impurities. This implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual Anderson model and their statistics cannot obey one of the Gaussian ensembles. Our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum Hall systems.Comment: 22 pages, no figure

Statistical Mechanics for Unstable States in Gel'fand Triplets and Investigations of Parabolic Potential Barriers

Free energies and other thermodynamical quantities are investigated in canonical and grand canonical ensembles of statistical mechanics involving unstable states which are described by the generalized eigenstates with complex energy eigenvalues in the conjugate space of Gel'fand triplet. The theory is applied to the systems containing parabolic potential barriers (PPB's). The entropy and energy productions from PPB systems are studied. An equilibrium for a chemical process described by reactions $A+CB\rightleftarrows AC+B$ is also discussed.Comment: 14 pages, AmS-LaTeX, no figur

On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schroedinger operators

We show for a large class of discrete Harper-like and continuous magnetic Schrodinger operators that their band edges are Lipschitz continuous with respect to the intensity of the external constant magnetic field. We generalize a result obtained by J. Bellissard in 1994, and give examples in favor of a recent conjecture of G. Nenciu.Comment: 15 pages, accepted for publication in Annales Henri Poincar

Device-independent quantum key distribution secure against collective attacks

Device-independent quantum key distribution (DIQKD) represents a relaxation of the security assumptions made in usual quantum key distribution (QKD). As in usual QKD, the security of DIQKD follows from the laws of quantum physics, but contrary to usual QKD, it does not rely on any assumptions about the internal working of the quantum devices used in the protocol. We present here in detail the security proof for a DIQKD protocol introduced in [Phys. Rev. Lett. 98, 230501 (2008)]. This proof exploits the full structure of quantum theory (as opposed to other proofs that exploit the no-signalling principle only), but only holds again collective attacks, where the eavesdropper is assumed to act on the quantum systems of the honest parties independently and identically at each round of the protocol (although she can act coherently on her systems at any time). The security of any DIQKD protocol necessarily relies on the violation of a Bell inequality. We discuss the issue of loopholes in Bell experiments in this context.Comment: 25 pages, 3 figure