158 research outputs found
Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model
One-dimensional quasiperiodic systems, such as the Harper model and the
Fibonacci quasicrystal, have long been the focus of extensive theoretical and
experimental research. Recently, the Harper model was found to be topologically
nontrivial. Here, we derive a general model that embodies a continuous
deformation between these seemingly unrelated models. We show that this
deformation does not close any bulk gaps, and thus prove that these models are
in fact topologically equivalent. Remarkably, they are equivalent regardless of
whether the quasiperiodicity appears as an on-site or hopping modulation. This
proves that these different models share the same boundary phenomena and
explains past measurements. We generalize this equivalence to any
Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.Comment: 7 pages, 2 figures, minor change
Universality in quantum chaos and the one parameter scaling theory
We adapt the one parameter scaling theory (OPT) to the context of quantum
chaos. As a result we propose a more precise characterization of the
universality classes associated to Wigner-Dyson and Poisson statistics which
takes into account Anderson localization effects. Based also on the OPT we
predict a new universality class in quantum chaos related to the
metal-insulator transition and provide several examples. In low dimensions it
is characterized by classical superdiffusion or a fractal spectrum, in higher
dimensions it can also have a purely quantum origin as in the case of
disordered systems. Our findings open the possibility of studying the metal
insulator transition experimentally in a much broader type of systems.Comment: 4 pages, 2 figures, acknowledgment added, typos correcte
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
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
Spectral Properties of Three Dimensional Layered Quantum Hall Systems
We investigate the spectral statistics of a network model for a three
dimensional layered quantum Hall system numerically. The scaling of the
quantity is used to determine the critical exponent for
several interlayer coupling strengths. Furthermore, we determine the level
spacing distribution as well as the spectral compressibility at
criticality. We show that the tail of decays as with
and also numerically verify the equation
, where is the correlation dimension and the
spatial dimension.Comment: 4 pages, 5 figures submitted to J. Phys. Soc. Jp
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
Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States
We investigate extrinsic wormhole-like twist defects that effectively increase the genus of space in lattice versions of multi-component fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a “proof of concept” demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects
Universality of the Wigner time delay distribution for one-dimensional random potentials
We show that the distribution of the time delay for one-dimensional random
potentials is universal in the high energy or weak disorder limit. Our
analytical results are in excellent agreement with extensive numerical
simulations carried out on samples whose sizes are large compared to the
localisation length (localised regime). The case of small samples is also
discussed (ballistic regime). We provide a physical argument which explains in
a quantitative way the origin of the exponential divergence of the moments. The
occurence of a log-normal tail for finite size systems is analysed. Finally, we
present exact results in the low energy limit which clearly show a departure
from the universal behaviour.Comment: 4 pages, 3 PostScript figure
Conductivity of 2D lattice electrons in an incommensurate magnetic field
We consider conductivities of two-dimensional lattice electrons in a magnetic
field. We focus on systems where the flux per plaquette is irrational
(incommensurate flux). To realize the system with the incommensurate flux, we
consider a series of systems with commensurate fluxes which converge to the
irrational value. We have calculated a real part of the longitudinal
conductivity . Using a scaling analysis, we have found
behaves as \,
when and the Fermi energy is near
zero. This behavior is closely related to the known scaling behavior of the
spectrum.Comment: 16 pages, postscript files are available on reques
Bethe ansatz for the Harper equation: Solution for a small commensurability parameter
The Harper equation describes an electron on a 2D lattice in magnetic field
and a particle on a 1D lattice in a periodic potential, in general,
incommensurate with the lattice potential. We find the distribution of the
roots of Bethe ansatz equations associated with the Harper equation in the
limit as alpha=1/Q tends to 0, where alpha is the commensurability parameter (Q
is integer). Using the knowledge of this distribution we calculate the higher
and lower boundaries of the spectrum of the Harper equation for small alpha.
The result is in agreement with the semiclassical argument, which can be used
for small alpha.Comment: 17 pages including 5 postscript figures, Latex, minor changes, to
appear in Phys.Rev.
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