71 research outputs found
Localization and Fluctuations in Quantum Kicked Rotors
We address the issue of fluctuations, about an exponential lineshape, in a
pair of one-dimensional kicked quantum systems exhibiting dynamical
localization. An exact renormalization scheme establishes the fractal character
of the fluctuations and provides a new method to compute the localization
length in terms of the fluctuations. In the case of a linear rotor, the
fluctuations are independent of the kicking parameter and exhibit
self-similarity for certain values of the quasienergy. For given , the
asymptotic localization length is a good characteristic of the localized
lineshapes for all quasienergies. This is in stark contrast to the quadratic
rotor, where the fluctuations depend upon the strength of the kicking and
exhibit local "resonances". These resonances result in strong deviations of the
localization length from the asymptotic value. The consequences are
particularly pronounced when considering the time evolution of a packet made up
of several quasienergy states.Comment: REVTEV Document. 9 pages, 4 figures submitted to PR
``Critical'' phonons of the supercritical Frenkel-Kontorova model: renormalization bifurcation diagrams
The phonon modes of the Frenkel-Kontorova model are studied both at the
pinning transition as well as in the pinned (cantorus) phase. We focus on the
minimal frequency of the phonon spectrum and the corresponding generalized
eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown
to have nontrivial scaling properties not only at the pinning transition point
but also in the cantorus regime. Therefore the phonons defy localization and
remain critical even where the associated area-preserving map has a positive
Lyapunov exponent. In this region, the critical scaling properties vary
continuously and are described by a line of renormalization limit cycles.
Interesting renormalization bifurcation diagrams are obtained by monitoring the
cycles as the parameters of the system are varied from an integrable case to
the anti-integrable limit. Both of these limits are described by a trivial
decimation fixed point. Very surprisingly we find additional special parameter
values in the cantorus regime where the renormalization limit cycle degenerates
into the above trivial fixed point. At these ``degeneracy points'' the phonon
hull is represented by an infinite series of step functions. This novel
behavior persists in the extended version of the model containing two
harmonics. Additional richnesses of this extended model are the one to two-hole
transition line, characterized by a divergence in the renormalization cycles,
nonexponentially localized phonons, and the preservation of critical behavior
all the way upto the anti-integrable limit.Comment: 10 pages, RevTeX, 9 Postscript figure
Self-Similarity and Localization
The localized eigenstates of the Harper equation exhibit universal
self-similar fluctuations once the exponentially decaying part of a wave
function is factorized out. For a fixed quantum state, we show that the whole
localized phase is characterized by a single strong coupling fixed point of the
renormalization equations. This fixed point also describes the generalized
Harper model with next nearest neighbor interaction below a certain threshold.
Above the threshold, the fluctuations in the generalized Harper model are
described by a strange invariant set of the renormalization equations.Comment: 4 pages, RevTeX, 2 figures include
Binary Tree Approach to Scaling in Unimodal Maps
Ge, Rusjan, and Zweifel (J. Stat. Phys. 59, 1265 (1990)) introduced a binary
tree which represents all the periodic windows in the chaotic regime of
iterated one-dimensional unimodal maps. We consider the scaling behavior in a
modified tree which takes into account the self-similarity of the window
structure. A non-universal geometric convergence of the associated superstable
parameter values towards a Misiurewicz point is observed for almost all binary
sequences with periodic tails. There are an infinite number of exceptional
sequences, however, which lead to superexponential scaling. The origin of such
sequences is explained.Comment: 25 pages, plain Te
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