29 research outputs found
Nonlinear Lattice Waves in Random Potentials
Localization of waves by disorder is a fundamental physical problem
encompassing a diverse spectrum of theoretical, experimental and numerical
studies in the context of metal-insulator transition, quantum Hall effect,
light propagation in photonic crystals, and dynamics of ultra-cold atoms in
optical arrays. Large intensity light can induce nonlinear response, ultracold
atomic gases can be tuned into an interacting regime, which leads again to
nonlinear wave equations on a mean field level. The interplay between disorder
and nonlinearity, their localizing and delocalizing effects is currently an
intriguing and challenging issue in the field. We will discuss recent advances
in the dynamics of nonlinear lattice waves in random potentials. In the absence
of nonlinear terms in the wave equations, Anderson localization is leading to a
halt of wave packet spreading.
Nonlinearity couples localized eigenstates and, potentially, enables
spreading and destruction of Anderson localization due to nonintegrability,
chaos and decoherence. The spreading process is characterized by universal
subdiffusive laws due to nonlinear diffusion. We review extensive computational
studies for one- and two-dimensional systems with tunable nonlinearity power.
We also briefly discuss extensions to other cases where the linear wave
equation features localization: Aubry-Andre localization with quasiperiodic
potentials, Wannier-Stark localization with dc fields, and dynamical
localization in momentum space with kicked rotors.Comment: 45 pages, 19 figure
Re-localization due to finite response times in a nonlinear Anderson chain
We study a disordered nonlinear Schr\"odinger equation with an additional
relaxation process having a finite response time . Without the relaxation
term, , this model has been widely studied in the past and numerical
simulations showed subdiffusive spreading of initially localized excitations.
However, recently Caetano et al.\ (EPJ. B \textbf{80}, 2011) found that by
introducing a response time , spreading is suppressed and any
initially localized excitation will remain localized. Here, we explain the lack
of subdiffusive spreading for by numerically analyzing the energy
evolution. We find that in the presence of a relaxation process the energy
drifts towards the band edge, which enforces the population of fewer and fewer
localized modes and hence leads to re-localization. The explanation presented
here is based on previous findings by the authors et al.\ (PRE \textbf{80},
2009) on the energy dependence of thermalized states.Comment: 3 pages, 4 figure
Quantum correction to the Kubo formula in closed mesoscopic systems
We study the energy dissipation rate in a mesoscopic system described by the
parametrically-driven random-matrix Hamiltonian H[\phi(t)] for the case of
linear bias \phi=vt. Evolution of the field \phi(t) causes interlevel
transitions leading to energy pumping, and also smears the discrete spectrum of
the Hamiltonian. For sufficiently fast perturbation this smearing exceeds the
mean level spacing and the dissipation rate is given by the Kubo formula. We
calculate the quantum correction to the Kubo result that reveals the original
discreteness of the energy spectrum. The first correction to the system
viscosity scales proportional to v^{-2/3} in the orthogonal case and vanishes
in the unitary case.Comment: 4 pages, 3 eps figures, REVTeX
Localization Bounds for Multiparticle Systems
We consider the spectral and dynamical properties of quantum systems of
particles on the lattice , of arbitrary dimension, with a Hamiltonian
which in addition to the kinetic term includes a random potential with iid
values at the lattice sites and a finite-range interaction. Two basic
parameters of the model are the strength of the disorder and the strength of
the interparticle interaction. It is established here that for all there
are regimes of high disorder, and/or weak enough interactions, for which the
system exhibits spectral and dynamical localization. The localization is
expressed through bounds on the transition amplitudes, which are uniform in
time and decay exponentially in the Hausdorff distance in the configuration
space. The results are derived through the analysis of fractional moments of
the -particle Green function, and related bounds on the eigenfunction
correlators
Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states
We consider low-temperature behavior of weakly interacting electrons in
disordered conductors in the regime when all single-particle eigenstates are
localized by the quenched disorder. We prove that in the absence of coupling of
the electrons to any external bath dc electrical conductivity exactly vanishes
as long as the temperatute does not exceed some finite value . At the
same time, it can be also proven that at high enough the conductivity is
finite. These two statements imply that the system undergoes a finite
temperature Metal-to-Insulator transition, which can be viewed as Anderson-like
localization of many-body wave functions in the Fock space. Metallic and
insulating states are not different from each other by any spatial or discrete
symmetries. We formulate the effective Hamiltonian description of the system at
low energies (of the order of the level spacing in the single-particle
localization volume). In the metallic phase quantum Boltzmann equation is
valid, allowing to find the kinetic coefficients. In the insulating phase,
, we use Feynmann diagram technique to determine the probability
distribution function for quantum-mechanical transition rates. The probability
of an escape rate from a given quantum state to be finite turns out to vanish
in every order of the perturbation theory in electron-electron interaction.
Thus, electron-electron interaction alone is unable to cause the relaxation and
establish the thermal equilibrium. As soon as some weak coupling to a bath is
turned on, conductivity becomes finite even in the insulating phase
Two-level system with a thermally fluctuating transfer matrix element: Application to the problem of DNA charge transfer
Charge transfer along the base-pair stack in DNA is modeled in terms of
thermally-assisted tunneling between adjacent base pairs. Central to our
approach is the notion that tunneling between fluctuating pairs is rate-limited
by the requirement of their optimal alignment. We focus on this aspect of the
process by modeling two adjacent base pairs in terms of a classical damped
oscillator subject to thermal fluctuations as described by a Fokker-Planck
equation. We find that the process is characterized by two time scales, a
result that is in accord with experimental findings.Comment: original file is revtex4, 10 pages, three eps figure
Effect of Holstein phonons on the optical conductivity of gapped graphene
We study the optical conductivity of a doped graphene when a sublattice
symmetry breaking is occurred in the presence of the electron-phonon
interaction. Our study is based on the Kubo formula that is established upon
the retarded self-energy. We report new features of both the real and imaginary
parts of the quasiparticle self-energy in the presence of a gap opening. We
find an analytical expression for the renormalized Fermi velocity of massive
Dirac Fermions over broad ranges of electron densities, gap values and the
electron-phonon coupling constants. Finally we conclude that the inclusion of
the renormalized Fermi energy and the band gap effects are indeed crucial to
get reasonable feature for the optical conductivity.Comment: 12 pages, 4 figures. To appear in Eur. Phys. J.
Thermal conductivity in harmonic lattices with random collisions
We review recent rigorous mathematical results about the macroscopic
behaviour of harmonic chains with the dynamics perturbed by a random exchange
of velocities between nearest neighbor particles. The random exchange models
the effects of nonlinearities of anharmonic chains and the resulting dynamics
have similar macroscopic behaviour. In particular there is a superdiffusion of
energy for unpinned acoustic chains. The corresponding evolution of the
temperature profile is governed by a fractional heat equation. In non-acoustic
chains we have normal diffusivity, even if momentum is conserved.Comment: Review paper, to appear in the Springer Lecture Notes in Physics
volume "Thermal transport in low dimensions: from statistical physics to
nanoscale heat transfer" (S. Lepri ed.
Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems
We study properties of chaos in generic one-dimensional nonlinear Hamiltonian
lattices comprised of weakly coupled nonlinear oscillators, by numerical
simulations of continuous-time systems and symplectic maps. For small coupling,
the measure of chaos is found to be proportional to the coupling strength and
lattice length, with the typical maximal Lyapunov exponent being proportional
to the square root of coupling. This strong chaos appears as a result of
triplet resonances between nearby modes. In addition to strong chaos we observe
a weakly chaotic component having much smaller Lyapunov exponent, the measure
of which drops approximately as a square of the coupling strength down to
smallest couplings we were able to reach. We argue that this weak chaos is
linked to the regime of fast Arnold diffusion discussed by Chirikov and
Vecheslavov. In disordered lattices of large size we find a subdiffusive
spreading of initially localized wave packets over larger and larger number of
modes. The relations between the exponent of this spreading and the exponent in
the dependence of the fast Arnold diffusion on coupling strength are analyzed.
We also trace parallels between the slow spreading of chaos and deterministic
rheology.Comment: 15 pages, 14 figure
Inhomogeneous Josephson junction chains: a superconducting meta-material for superinductance optimization
We report a theoretical study of the low-frequency impedance of a Josephson junction chain whose parameters vary in space. Our goal is to find the optimal spatial profile which maximizes the total inductance of the chain without shrinking the low-frequency window where the chain behaves as an inductor. If the spatial modulation is introduced by varying the junction areas, we find that the best result is obtained for a spatially homogeneous chain, reported earlier in the literature. An improvement over the homogeneous result can be obtained by representing the junctions by SQUIDs with different loop areas, so the inductances can be varied by applying a magnetic field. Still, we find that this improvement becomes less important for longer chains