Resonances in 1D disordered systems: localization of energy and resonant transmission

Localized states in one-dimensional open disordered systems and their connection to the internal structure of random samples have been studied. It is shown that the localization of energy and anomalously high transmission associated with these states are due to the existence inside the sample of a transparent (for a given resonant frequency) segment with the minimal size of order of the localization length. A mapping of the stochastic scattering problem in hand onto a deterministic quantum problem is developed. It is shown that there is no one-to-one correspondence between the localization and high transparency: only small part of localized modes provides the transmission coefficient close to one. The maximal transmission is provided by the modes that are localized in the center, while the highest energy concentration takes place in cavities shifted towards the input. An algorithm is proposed to estimate the position of an effective resonant cavity and its pumping rate by measuring the resonant transmission coefficient. The validity of the analytical results have been checked by extensive numerical simulations and wavelet analysis

Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution

We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2 \pi p/q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows to consider them further in the limit $p, q \to \infty$ by the technique of Thermodynamic Bethe Ansatz and analyze Hofstadter problem for the irrational flux.Comment: 7 pages, 2 figures, Revte

The longitudinal conductance of mesoscopic Hall samples with arbitrary disorder and periodic modulations

We use the Kubo-Landauer formalism to compute the longitudinal (two-terminal) conductance of a two dimensional electron system placed in a strong perpendicular magnetic field, and subjected to periodic modulations and/or disorder potentials. The scattering problem is recast as a set of inhomogeneous, coupled linear equations, allowing us to find the transmission probabilities from a finite-size system computation; the results are exact for non-interacting electrons. Our method fully accounts for the effects of the disorder and the periodic modulation, irrespective of their relative strength, as long as Landau level mixing is negligible. In particular, we focus on the interplay between the effects of the periodic modulation and those of the disorder. This appears to be the relevant regime to understand recent experiments [S. Melinte {\em et al}, Phys. Rev. Lett. {\bf 92}, 036802 (2004)], and our numerical results are in qualitative agreement with these experimental results. The numerical techniques we develop can be generalized straightforwardly to many-terminal geometries, as well as other multi-channel scattering problems.Comment: 13 pages, 11 figure

Quenching across quantum critical points in periodic systems: dependence of scaling laws on periodicity

We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-1/2 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the z direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a natural number. For a linear quench of the magnetic field strength (or potential strength) at rate 1/\tau across a quantum critical point, we find that the density of defects thereby produced scales as 1/\tau^{q/(q+1)}, deviating from the 1/\sqrt{\tau} scaling that is ubiquitous to a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum k undergoing a non-linear quench as well as by performing numerical simulations. We also find that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of \tau although it may exhibit a cross-over at intermediate values of \tau. Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and interaction strength.Comment: 13 pages including 11 figure

On semiclassical dispersion relations of Harper-like operators

We describe some semiclassical spectral properties of Harper-like operators, i.e. of one-dimensional quantum Hamiltonians periodic in both momentum and position. The spectral region corresponding to the separatrices of the classical Hamiltonian is studied for the case of integer flux. We derive asymptotic formula for the dispersion relations, the width of bands and gaps, and show how geometric characteristics and the absence of symmetries of the Hamiltonian influence the form of the energy bands.Comment: 13 pages, 8 figures; final version, to appear in J. Phys. A (2004

Band-Contact Lines in Electron Energy Spectrum of Graphite

We discuss the known experimental data on the phase of the de Haas -van Alphen oscillations in graphite. These data can be understood if one takes into account that four band-contact lines exist near the HKH edge of the Brillouin zone of graphite.Comment: 5 pages, 2 fifures. To appear in Physical Review B (B15

Hofstadter butterfly as Quantum phase diagram

The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labelled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.Comment: 4 prl pages 1 colored figure typos corrected, reference [26] added, "Ten Martini" assumption adde

Extended states in 1D lattices: application to quasiperiodic copper-mean chain

The question of the conditions under which 1D systems support extended electronic eigenstates is addressed in a very general context. Using real space renormalisation group arguments we discuss the precise criteria for determining the entire spertrum of extended eigenstates and the corresponding eigenfunctions in disordered as well as quasiperiodic systems. For purposes of illustration we calculate a few selected eigenvalues and the corresponding extended eigenfunctions for the quasiperiodic copper-mean chain. So far, for the infinite copper-mean chain, only a single energy has been numerically shown to support an extended eigenstate [ You et al. (1991)] : we show analytically that there is in fact an infinite number of extended eigenstates in this lattice which form fragmented minibands.Comment: 10 pages + 2 figures available on request; LaTeX version 2.0

Bloch electron in a magnetic field and the Ising model

The spectral determinant det(H-\epsilon I) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux \Phi=2\pi P/Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.Comment: 4 pages, 1 figure, REVTE