Anderson Localization for a Supersymmetric Sigma Model

We study a lattice sigma model which is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices in 3D. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. The existence of a diffusive phase in 3 dimensions was proved in Disertori et al. (Commun. Math. Phys., doi:, 2009) [2] for low temperatures. Here we prove localization at high temperatures for any dimension d ≥ 1. Our analysis uses Ward identities coming from internal supersymmetry.     

Anderson localization or nonlinear waves: a matter of probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.     

Coherent backscattering and localization in a self-attracting random walk model

Intensity propagation of waves in dilute 2D and 3D disordered systems is well described by a random walk path-model. In strongly scattering media, however, this model is not quite correct because of interference effects like coherent backscattering. In this letter, coherent backscattering is taken into account by a modified, self-attracting random walk. Straightforward simulations of this model essentially reproduce the results of current theories on “non-classical” transport behavior, i.e. Anderson localization in 1D and 2D for any amount of disorder and a phase transition from weak to strong localization in 3D. However, in the strongly scattering regime corrections are necessary to account for the finite number of light modes due to their non-vanishing lateral extention. Within our model this correction leads to the observation that strong localization does not take place. Received 17 September 2001     

基于声传播算子的声源定位实验模拟方法研究

声传播算子是一种高效的时域声场计算方法,它能够很方便地计算出给定系统参数下任意时刻任意位置的声场变化情况,本文采用这种方法计算所得的二维房间声场信息进行传声器阵列的声源定位仿真实验。计算结果表明,用该方法获取的阵列数据能有效地应用于阵列信号处理算法中,准确地估计出初始高斯脉冲声源的方向。声传播算子声场计算方法能为传声器阵列声源定位的实验提供方便,使得传声器阵列声源定位算法在不同混响时间的鲁棒性实验研究变得更加简捷。     

Many-body energy localization transition in periodically driven systems

According to the second law of thermodynamics the total entropy of a system is increased during almost any dynamical process. The positivity of the specific heat implies that the entropy increase is associated with heating. This is generally true both at the single particle level, like in the Fermi acceleration mechanism of charged particles reflected by magnetic mirrors, and for complex systems in everyday devices. Notable exceptions are known in noninteracting systems of particles moving in periodic potentials. Here the phenomenon of dynamical localization can prevent heating beyond certain threshold. The dynamical localization is known to occur both at classical (Fermi–Ulam model) and at quantum levels (kicked rotor). However, it was believed that driven ergodic systems will always heat without bound. Here, on the contrary, we report strong evidence of dynamical localization transition in both classical and quantum periodically driven ergodic systems in the thermodynamic limit. This phenomenon is reminiscent of many-body localization in energy space.     

Self-trapping model of desorption

Desorption induced by self-trapping (ST) of a surface excitation is studied for a model in which the localization of the surface excitation eliminates the surface bond of the desorbing particles. The adiabatic energy of the system is calculated with the use of discrete and continuum models. The free (F) state is meta- or unstable against the desorptive state for any small force to expel the adsorbate at the site of the localization. The absence of the elastic deformation energy due to bond scission causes decrease of the adiabatic energy with increase in the degree of localization, which induces a large bond dilation leading to desorption. The barrier height separating F and ST-induced desorptive states is evaluated from the adiabatic energy. The desorption probability is discussed in conjunction with its temperature and coverage dependence, where the possibility of tunneling from the metastable F state to the desorptive one is predicted.     

Coherent Motion of Excitons on a Dimerized chain

The quantum coherent motion of excitons on a dimerized chain with alternating site energies and in tersite interactions is studied analytically. Exact solutions for the propagators at any site, the mean polarization and mean-square displacement are obtained. Through these exact results and accompanying numerical analysis, it is shown that the increase of the site-energy mismatch and the hopping disturbance will tend to strengthen the localization.     

Dynamical Localization for Unitary Anderson Models

This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes.     

Spectral Analysis of the Disordered Stochastic¶1-D Ising Model

We consider the generator of the Glauber dynamics for a 1-D Ising model with random bounded potential at any temperature. We prove that for any realization of the potential the spectrum of the generator is the union of separate branches (so-called k-particles branches, k= 0,1,2,…), and with probability one it is a nonrandom set. We find the location of the spectrum and prove the localization for the one-particle branch of the spectrum. As a consequence we find a lower bound for the spectral gap for any realization of the random potential. Received: 22 September 1998 / Accepted: 12 February 1999     

Anderson localization for Bernoulli and other singular potentials

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.Partially supported by NSF grant DMS 85-03695Partially supported by NSF grant DMS 83-01889Partially supported by G.N.F.M. C.N.R.     

Localization of torsion vibrations in a discrete model of alkanes

A simple coarse-grained model of a crystal of normal paraffin (the united-atom model) is considered. By using an original semi-inverse method, it is shown that, alongside with known polymorphic transitions, the model under consideration assumes a dynamic transition, which manifests itself in the localization of vibration energy at a certain threshold value of excitation energy. The prediction of the conditions of this transition requires analytical determination of the spectrum of nonlinear normal modes with arbitrary amplitudes of vibrations because the instability of the mode with the lowest wavenumber is a necessary energy localization condition. The equations obtained make it possible to investigate the resonance interaction between the nonlinear normal modes near the low-frequency edge of the spectrum resulting in capture of the vibration energy by one of the parts of the chain. The conditions of localization of the vibration energy revealed determine the necessary initial data for computer modeling of the predicted dynamic transition in normal paraffins.     

Two interacting particles in a random potential

We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.     

Solution of a torsional Schrödinger equation with a periodic potential of general form. The probability amplitude and probability density

Analytic expressions for the probability density of states of a molecule with internal rotation and the probability of finding the state in the potential well are derived for the first time. Two methods are proposed for assigning conformers to potential wells. A quantitative measure of localization and delocalization of a state in the potential well is introduced. The rotational symmetry number is generalized to the case of asymmetric rotation. On the basis of the localization criterion, a model is developed for calculating the internal rotation contribution to thermodynamic properties of individual conformers with low rotational barriers and/or at a high temperature.     

Surface solitons in chirped photonic lattices

We study surface modes at the edge of a semi-infinite chirped photonic lattice in the framework of an effective discrete nonlinear model. We demonstrate that the lattice chirp can change dramatically the conditions for the mode localization near the surface, and we find numerically the families of discrete surface solitons in this case. Such solitons do not require any minimum power to exist provided the chirp parameter exceeds some critical value. We also analyze how the chirp modifies the interaction of a soliton with the lattice edge.     

New algorithm for footstep localization using seismic sensors in an indoor environment

In this study, we consider the use of seismic sensors for footstep localization in indoor environments. A popular strategy of localization is to use the measured differences in arrival times of source signals at multiple pairs of receivers. In the literature, most algorithms that are based on time differences of arrival (TDOA) assume that the propagation velocity is a constant as a function of the source position, which is valid for air propagation or even for narrow band signals. However a bounded medium such as a concrete slab (encountered in indoor environment) is usually dispersive and damped. In this study, we demonstrate that under such conditions, the concrete slab can be assimilated to a thin plate; considering a Kelvin–Voigt damping model, we introduce the notion of perceived propagation velocity, which decreases when the source-sensor distance increases. This peculiar behavior precludes any possibility to rely on existing localization methods in indoor environment. Therefore, a new localization algorithm that is adapted to a damped and dispersive medium is proposed, using only on the sign of the measured TDOA (SO-TDOA). A simulation and some experimental results are included, to define the performance of this SO-TDOA algorithm.     

State of stress at a crystal surface deformed by a concentrated load in the presence of a wedge twin

A procedure for the calculation of the stress fields near a wedge twin located at a crystal surface is developed using a dislocation macroscopic model. The problem is solved for the case of a concentrated load applied to the surface of a crystal with a twin. The concentrated load is found to increase the level of stresses near the wedge twin and to cause their localization at the twin boundary that is closer to the load application point.     

Localization of quasiparticles in a disordered vortex

We study the diffusive motion of low-energy normal quasiparticles along the core of a single vortex in a dirty, type-II, s-wave superconductor. The physics of this system is argued to be described by a one-dimensional supersymmetric non-linear σ model, which differs from the σ models known for disordered metallic wires. For an isolated vortex and quasiparticle energies less than the Thouless energy E_Th, we recover the spectral correlations that are predicted by random matrix theory for the universality class C. We then consider the transport problem of transmission of quasiparticles through a vortex connected to particle reservoirs at both ends. The transmittance at zero energy exhibits a weak localization correction reminiscent of quasi-one-dimensional metallic systems with symmetry index β = 1. Weak localization disappears with increasing energy over a scale set by E_Th This crossover should be observable in measurements of the longitudinal heat conductivity of an ensemble of vortices under mesoscopic conditions. In the regime of strong localization, the localization length is shown to decrease by a factor of 8 as the quasiparticle energy goes to zero.     

Localization of the Cu111 surface state by single Cu adatoms

The Cu adatom-induced localization of the two-dimensional Shockley surface state at the Cu(111) surface was identified from experimental and simulated scanning tunneling microscopy spectra. The localization gives rise to a resonance located just below the surface state band edge. The adatom-induced surface state localization is discussed in terms of the existence theorem for bound states in any attractive two-dimensional potential. We also identify adatom-induced resonance states deriving from atomic orbitals in both experimental and simulated spectra.     

Effective Rotational Potential of a Molecular Ions in a Plane Radio-Frequency Trap

Technical Physics - A model of rotational potential that represents an analog of the pseudopotential for localization of a model of diatomic particle with rigid bond in a plane quadrupole...     

Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results

The localization of the electron states and dc-conductivity of the one dimensional Anderson model are investigated with various numerical procedures. It is found that the eigenstates are always exponentially localized and that in the center of the band the localization length is proportional to the inverse square of the disorder. The dc-conductivity, as obtained by using the Kubo-Greenwood formula, obeys the central limit theorem for any finite imaginary frequency, with a variance, which is inversely proportional to the squareroot of the number of states contributing to the transport. There is no exponential length dependence of the Kubo-Greenwood conductivity within this model. The conductivity tends to zero only in the limit of vanishing imaginary frequency.