18 research outputs found
Dynamics of localized waves in 1D random potentials: statistical theory of the coherent forward scattering peak
As recently discovered [PRL 190601(2012)], Anderson localization
in a bulk disordered system triggers the emergence of a coherent forward
scattering (CFS) peak in momentum space, which twins the well-known coherent
backscattering (CBS) peak observed in weak localization experiments. Going
beyond the perturbative regime, we address here the long-time dynamics of the
CFS peak in a 1D random system and we relate this novel interference effect to
the statistical properties of the eigenfunctions and eigenspectrum of the
corresponding random Hamiltonian. Our numerical results show that the dynamics
of the CFS peak is governed by the logarithmic level repulsion between
localized states, with a time scale that is, with good accuracy, twice the
Heisenberg time. This is in perfect agreement with recent findings based on the
nonlinear -model. In the stationary regime, the width of the CFS peak
in momentum space is inversely proportional to the localization length,
reflecting the exponential decay of the eigenfunctions in real space, while its
height is exactly twice the background, reflecting the Poisson statistical
properties of the eigenfunctions. Our results should be easily extended to
higher dimensional systems and other symmetry classes.Comment: See the published article for the updated versio
Momentum-space dynamics of Dirac quasiparticles in correlated random potentials: Interplay between dynamical and Berry phases
We consider Dirac quasi-particles, as realized with cold atoms loaded in a
honeycomb lattice or in a -flux square lattice, in the presence of a weak
correlated disorder such that the disorder fluctuations do not couple the two
Dirac points of the lattices. We numerically and theoretically investigate the
time evolution of the momentum distribution of such quasi-particles when they
are initially prepared in a quasi-monochromatic wave packet with a given mean
momentum. The parallel transport of the pseudo-spin degree of freedom along
scattering paths in momentum space generates a geometrical phase which alters
the interference associated with reciprocal scattering paths. In the massless
case, a well-known dip in the momentum distribution develops at backscattering
(respective to the Dirac point considered) around the transport mean free time.
This dip later vanishes in the honeycomb case because of trigonal warping. In
the massive case, the dynamical phase of the scattering paths becomes crucial.
Its interplay with the geometrical phase induces an additional transient broken
reflection symmetry in the momentum distribution. The direction of this
asymmetry is a property of the Dirac point considered, independent of the
energy of the wave packet. These Berry phase effects could be observed in
current cold atom lattice experiments.Comment: Additional data and explanations compared to version 1. See published
article for the latest versio
Analytical and numerical study of uncorrelated disorder on a honeycomb lattice
We consider a tight-binding model on the regular honeycomb lattice with
uncorrelated on-site disorder. We use two independent methods (recursive
Green's function and self-consistent Born approximation) to extract the
scattering mean free path, the scattering mean free time, the density of states
and the localization length as a function of the disorder strength. The two
methods give excellent quantitative agreement for these single-particle
properties. Furthermore, a finite-size scaling analysis reveals that all
localization lengths for different lattice sizes and different energies
(including the energy at the Dirac points) collapse onto a single curve, in
agreement with the one-parameter scaling theory of localization. The
predictions of the self-consistent theory of localization however fail to
quantitatively reproduce these numerically-extracted localization lengths.Comment: 19 pages, 25 figure
Periodic and discrete Zak bases
Weyl's displacement operators for position and momentum commute if the
product of the elementary displacements equals Planck's constant. Then, their
common eigenstates constitute the Zak basis, each state specified by two phase
parameters. Upon enforcing a periodic dependence on the phases, one gets a
one-to-one mapping of the Hilbert space on the line onto the Hilbert space on
the torus. The Fourier coefficients of the periodic Zak bases make up the
discrete Zak bases. The two bases are mutually unbiased. We study these bases
in detail, including a brief discussion of their relation to Aharonov's modular
operators, and mention how they can be used to associate with the single degree
of freedom of the line a pair of genuine qubits.Comment: 15 pages, 3 figures; displayed abstract is shortened, see the paper
for the complete abstrac