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

On the regularity of the Hausdorff distance between spectra of perturbed magnetic Hamiltonians

We study the regularity properties of the Hausdorff distance between spectra of continuous Harper-like operators. As a special case we obtain H\"{o}lder continuity of this Hausdorff distance with respect to the intensity of the magnetic field for a large class of magnetic elliptic (pseudo)differential operators with long range magnetic fields.Comment: to appear in the Proceedings of the 'Spectral Days' conference, Santiago de Chile 201

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

Piezoelectricity: Quantized Charge Transport Driven by Adiabatic Deformations

We study the (zero temperature) quantum piezoelectric response of Harper-like models with broken inversion symmetry. The charge transport in these models is related to topological invariants (Chern numbers). We show that there are arbitrarily small periodic modulations of the atomic positions that lead to nonzero charge transport for the electrons.Comment: Latex, letter. Replaced version with minor change in style. 1 fi

The Spectral Structure of the Electronic Black Box Hamiltonian

We give results on the absence of singular continuous spectrum of the one-particle Hamiltonian underlying the electronic black box model.Comment: 11 page

Quantum response of dephasing open systems

We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect.Comment: Article rewritten, two appendices added. 16 pages, 2 figure

Topological Invariants in Fermi Systems with Time-Reversal Invariance

We discuss topological invariants for Fermi systems that have time-reversal invariance. The TKN^2 integers (first Chern numbers) are replaced by second Chern numbers, and Berry's phase becomes a unit quaternion, or equivalently an element of SU(2). The canonical example playing much the same role as spin ½ in a magnetic field is spin ½ in a quadrupole electric field. In particular, the associated bundles are nontrivial and have ± 1 second Chern number. The connection that governs the adiabatic evolution coincides with the symmetric SU(2) Yang-Mills instanton

Effects of interaction on an adiabatic quantum electron pump

We study the effects of inter-electron interactions on the charge pumped through an adiabatic quantum electron pump. The pumping is through a system of barriers, whose heights are deformed adiabatically. (Weak) interaction effects are introduced through a renormalisation group flow of the scattering matrices and the pumped charge is shown to {\it always} approach a quantised value at low temperatures or long length scales. The maximum value of the pumped charge is set by the number of barriers and is given by $Q_{\rm max} = n_b -1$. The correlation between the transmission and the charge pumped is studied by seeing how much of the transmission is enclosed by the pumping contour. The (integer) value of the pumped charge at low temperatures is determined by the number of transmission maxima enclosed by the pumping contour. The dissipation at finite temperatures leading to the non-quantised values of the pumped charge scales as a power law with the temperature ($Q-Q_{\rm int} \propto T^{2\alpha}$), or with the system size ($Q-Q_{\rm int} \propto L_s^{-2\alpha}$), where $\alpha$ is a measure of the interactions and vanishes at $T=0 ~(L_s=\infty)$. For a double barrier system, our result agrees with the quantisation of pumped charge seen in Luttinger liquids.Comment: 9 pages, 9 figures, better quality figures available on request from author