Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schro\"dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization.Comment: To appear in Commun. Math. Phys. (1999

Calculation of the Invariant Measures at Weak Disorder for the Two-Line Anderson Model

We compute the invariant measures in the weak disorder limit, for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterise the measures. It turns out that at zero energy, there is a similar anomaly as first found by Kappus and Wegner for the single chain, but that, in addition, the measures take a different form on different regions of the spectrum

Invariant Measures for One-Dimensional Anderson Localisation

We compute the invariant measures for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterise the measures. It turns out that there is a similar anomaly as first found by Kappus and Wegner for the single chain, but that, in addition, the measures take a different form on different regions of the spectrum

Lowest Lyapunov Exponents for the Armchair Nanotube

We compute sum of the two the lowest Lyapunov exponents γ_(2N−1) + γ_2N of a tight-binding model for an single-wall armchair carbon nanotube with point impurities to lowest (second) order in the disorder parameter λ. The result is that γ_(2N−1) + γ_2N ∼ (λ^2)(N^−1) , where N is the number of hexagons around the perimeter. This is similar to the result of Schulz-Baldes [20] for the standard Anderson model on a strip, but because there are only two conducting channels near the Fermi level (centre of the spectral band), this implies that the scattering length is proportional to the diameter of the tube as predicted by Todorov and White [10]

Coding Theorem for a Class of Quantum Channels with Long-Term Memory

In this paper we consider the transmission of classical information through a class of quantum channels with long-term memory, which are given by convex combinations of product channels. Hence, the memory of such channels is given by a Markov chain which is aperiodic but not irreducible. We prove the coding theorem and weak converse for this class of channels. The main techniques that we employ, are a quantum version of Feinstein's Fundamental Lemma and a generalization of Helstrom's Theorem.Comment: Some typos correcte

The invalidity of a strong capacity for a quantum channel with memory

The strong capacity of a particular channel can be interpreted as a sharp limit on the amount of information which can be transmitted reliably over that channel. To evaluate the strong capacity of a particular channel one must prove both the direct part of the channel coding theorem and the strong converse for the channel. Here we consider the strong converse theorem for the periodic quantum channel and show some rather surprising results. We first show that the strong converse does not hold in general for this channel and therefore the channel does not have a strong capacity. Instead, we find that there is a scale of capacities corresponding to error probabilities between integer multiples of the inverse of the periodicity of the channel. A similar scale also exists for the random channel.Comment: 7 pages, double column. Comments welcome. Repeated equation removed and one reference adde

Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems

We study the spectrum of a random Schroedinger operator for an electron submitted to a magnetic field in a finite but macroscopic two dimensional system of linear dimensions equal to L. The y direction is periodic and in the x direction the electron is confined by two smooth increasing boundary potentials. The eigenvalues of the Hamiltonian are classified according to their associated quantum mechanical current in the y direction. Here we look at an interval of energies inside the first Landau band of the random operator for the infinite plane. In this energy interval, with large probability, there exist O(L) eigenvalues with positive or negative currents of O(1). Between each of these there exist O(L^2) eigenvalues with infinitesimal current O(exp(-cB(log L)^2)). We explain what is the relevance of this analysis to the integer quantum Hall effect.Comment: 29 pages, no figure

Perfect Transfer of Arbitrary States in Quantum Spin Networks

We propose a class of qubit networks that admit perfect state transfer of any two-dimensional quantum state in a fixed period of time. We further show that such networks can distribute arbitrary entangled states between two distant parties, and can, by using such systems in parallel, transmit the higher dimensional systems states across the network. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to $N$-qubit spin networks of identical qubit couplings, we show that $2\log_3 N$ is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits then perfect state transfer can be achieved over arbitrarily long distances in a linear chain. This paper expands and extends the work done in PRL 92, 187902.Comment: 12 pages, 3 figures with updated reference

The Phase Diagram of a Spin Glass on a Tree with Ferromagnetic Interactions

A spin glass problem on a Cayley tree with ferromagnetic interactions is solved rigorously. Using a level-I large deviation argument together with the martingale approach used by Bullet, Patrick and Pulé [1], explicit expressions for the free energy are derived in different regions of the phase diagram. It is found that there are four phases: a paramagnetic phase, a spin-glass phase, a ferromagnetic phase and a mixed phase. The nature of the phase diagram depends on the power with which the ferromagnetic term occurs in the Hamiltonian

Large Deviations and the Random Energy Model

We present a simple proof of the formula for the free energy of the random-energy model using a large deviation property which holds almost surely with respect to the randomness. This proof is extended to the case with external magnetic field leading to the solution of a model with higher-order ferromagnetic term. It is shown that this model is useful for Sourlas' application to error-correcting codes as was already pointed out in a recent letter by the authors