Competition of disorder and interchain hopping in a two-chain Hubbard model

We study the interplay of Anderson localization and interaction in a two chain Hubbard ladder allowing for arbitrary ratio of disorder strength to interchain coupling. We obtain three different types of spin gapped localized phases depending on the strength of disorder: a pinned 4k _F Charge Density Wave (CDW) for weak disorder, a pinned 2k _F CDW^π for intermediate disorder and two independently pinned single chain 2k _F CDW for strong disorder. Confinement of electrons can be obtained as a result of strong disorder or strong attraction. We give the full phase diagram as a function of disorder, interaction strength and interchain hopping. We also study the influence of interchain hopping on localization length and show that localization is enhanced by a small interchain hopping but suppressed by a large interchain hopping. Received 6 April 2001     

Anderson localization for multi-dimensional systems at large disorder or large energy

We prove that discrete Schrödinger operators on ^d with a random-potential have almost-surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy. This is the first proof of localization for multi-dimensional Anderson models.Groupe de recherche 048 du CNRS     

Thermal conductivity of nonlinear waves in disordered chains

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity κ(T). We find indications for an asymptotic low-temperature κ ∼ T ⁴ and intermediate temperature κ ∼ T ² laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (Laptyeva et al, Europhys. Lett. 91, 30001 (2010)).     

一维长程关联无序系统中的电子态

利用傅里叶滤波法在一维Anderson无序系统中产生了具有幂律谱密度公式s(q)∝q^-p形式的长程关联随机能量序列，并利用传输矩阵方法计算了系统中引入了长程关联后的局域长度，同时应用负本征值理论对系统中的电子态密度进行了分析，并分别把计算结果与系统中不具有长程关联时的局域长度与电子态密度进行了比较.结果表明，长程幂律关联的引入对电子态的性质产生了很大的影响，当关联指数p≥2.0时，在系统能带中心范围内发生了部分局域态向退局域态的转变，而同时电子态密度也发生了很大的变化，出现了六个范霍夫奇点，系统的能带范围也相应地得到展宽. 关键词： 无序系统 长程关联 局域长度 电子态密度     

Finite-size scaling of the level compressibility at the Anderson transition

We compute the number level variance Σ ₂ and the level compressibility χ from high precision data for the Anderson model of localization and show that they can be used in order to estimate the critical properties at the metal-insulator transition by means of finite-size scaling. With N, W, and L denoting, respectively, linear system size, disorder strength, and the average number of levels in units of the mean level spacing, we find that both χ(N, W) and the integrated Σ ₂ obey finite-size scaling. The high precision data was obtained for an anisotropic three-dimensional Anderson model with disorder given by a box distribution of width W/2. We compute the critical exponent as ν≈ 1.45±0.12 and the critical disorder as W _c≈ 8.59±0.05 in agreement with previous transfer-matrix studies in the anisotropic model. Furthermore, we find χ≈ 0.28±0.06 at the metal-insulator transition in very close agreement with previous results. Received 1st November 2001 and Received in final form 8 March 2002 Published online 6 June 2002     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

The Weak Localization for the Alloy-Type Anderson Model on a Cubic Lattice

We consider alloy type random Schrödinger operators on a cubic lattice whose randomness is generated by the sign-indefinite single-site potential. We derive Anderson localization for this class of models in the Lifshitz tails regime, i.e. when the coupling parameter λ is small, for the energies E≤?Cλ ².     

Series expansion study of quantum percolation on the square lattice

We study the site and bond quantum percolation model on the two-dimensional square lattice using series expansion in the low concentration limit. We calculate series for the averages of , where T _ij (E) is the transmission coefficient between sites i and j, for k=0, 1, , 5 and for several values of the energy E near the center of the band. In the bond case the series are of order p¹⁴ in the concentration p(some of those have been formerly available to order p¹⁰) and in the site case of order p¹⁶. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, shows clear evidence for a delocalization transition (from exponentially localized to extended or power-law-decaying states) at an energy-dependent threshold p _q(E) in the range , confirming previous results (e.g. and for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different kis characterized by a constant gap exponent, which is identified as the localization length exponent from a general scaling assumption. We obtain estimates of . These values violate the bound of Chayes et al. Received 28 February 2000     

Localization of classical waves I: Acoustic waves

We consider classical acoustic waves in a medium described by a position dependent mass density (x). We assume that (x) is a reandom perturbation of a periodic function ₀(x) and that the periodic acoustic operator has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators onL ²( ^d ). We prove that, in the random medium described by (x), the random operatorA exhibits Anderson localization inside the gap in the spectrum ofA ₀. This is shown even in situations when the gap is totally filled by the spectrum of the random opertor; we can prescribe random environments that ensure localization in almost the whole gap.This author was supported by the U.S. Air Force Grant F49620-94-1-0172.This author was supported in part by the NSF Grants DMS-9208029 and DMS-9500720.     

The metal-insulator transition in extrinsic semiconductors

The main purpose of this article is to describe the electrical and magnetic properties of extrinsic semiconductors when the concentration of donors varies, particularly for concentrations near that for which a metal-insulator transition takes place. Since the donor centres are distributed at random in space, the combined effects of correlation and Anderson localization have to be considered. As an introduction, in § 1 we give an outline of our present understanding of the transition in some crystalline materials, particularly V₂O₃. For this the metallic phase, a highly correlated electron gas, is discussed, as is also the question of a discontinuous change in the number of current carriers. In § 2 we discuss Si : P and similar materials. For uncompensated samples, the material near the transition is to be described by overlapping Hubbard bands, and the metal-insulator transition occurs when the states at the Fermi energy show Anderson localization. Just on the insulator side of the transition conduction is by variable-range hopping for uncompensated as for compensated materials, and a variation of log (resistivity) as T ^?1/4 is observed. The existence or otherwise of moments in the metallic phase is discussed, as is also the magnetoresistance, Hall effect and Knight shift. In § 3 the transition is described for narrow-band semiconductors, particularly La_1?x Sr _x VO₃. In § 4 some compounds of rare earths are treated, including Sm_1?x Nd _x Se.     

Electron Metal-Dielectric Phase Transitions in Semiconductors under Pressure

The effect of uniform pressure on the character of metal-dielectric transitions in compensated weakly- and highly doped semiconductors (WHDCS) is considered. The influence of hybridization of resonance quasilocalized impurity states with band continuum states on the transition is shown. Minimum metallic conductivities in p-CdSnAs₂ are determined for Mott and Anderson transitions. Special features of the metal-dielectric transformation in weakly-doped narrow- and high energy-gap semiconductors are discussed for the case of hydrogen-like impurities. Anderson localization in WHDCS is also considered. Phase diagrams are presented.     

Spectral Theory of Pseudo-Ergodic Operators

We define a class of pseudo-ergodic non-self-adjoint Schr?dinger operators acting in spaces l ²(X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting onl ²(Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators. Received: 8 August 2000 / Accepted: 3 October 2000     

Localization in the CPA frame for alloys

To describe electron localization in substitutionally random alloysA _c B _1–c the coherent potential approximation (CPA) is incorporated into the self-consistent theory of Anderson localization in the form developed by Vollhardt and Wölfle. Modifications of the localization theory arise from the tight-binding model with bimodal diagonal disorder of arbitrary strength. The mean-free path, correlation and localization lengths, and the zero-temperature conductivity are calculated at dimensionalityd=3. The metal-insulator transition is studied numerically for a CPA-induced band structure under semielliptical model assumptions.     

Disorder-induced Anderson localization in GaAs1−xPx

We report on low-temperature photoluminescence studies of undoped GaAs_1?xP_x alloys. Arguments are presented for the identification of the strongly asymmetrically shaped peak, called M^X_O, with the recombination of electrons and holes bound to different regions of the alloy. It is proposed that at low temperatures, the carriers are subject to Anderson localization below a mobility edge in the potential wells which are induced by the fluctuating alloy potential. Variations in its properties with varying alloy composition suggest that the localization effects are primarily related to electrons with X-conduction band character.     

Energy level statistics in disordered metals with an anderson transition

We present numerical scaling results for the energy level statistics in orthogonal and symplectic tight-binding Hamiltonian random matrix ensembles defined on disordered two and three-dimensional electronic systems with and without spinorbit coupling (SOC), respectively. In the metallic phase for weak disorder the nearest level spacing distribution functionP(S), the number variance <(N)²>, and the two-point correlation functionK ₂(), are shown to be described by the Gaussian random matrix theories. In the insulating phase, for strong disorder, the correlations vanish for large scales and the ordinary Poisson statistics is asymptotically recovered, which is consistent with localization of the corrosponding eigenstates. At the Anderson metal-insulator transition we obtain new universal scale-invariant distribution functions describing the critical spectral density fluctuations.     

Systematic treatment of the anderson localization on the basis of the projection operator formalism

A derivation within the projection operator technique is given for the density and current response functions of a system of independent particles moving in a random potential. The essential point is the derivation of kinetic equations for the current relaxation kernel instead of for the density propagator as in a previous treatment on the basis of the projection operator formalism. In these equations the divergent contributions from the 2k _F -scattering mechanism can be systematically separated from those of diffusional scattering. Especially, both the self-consistent current relaxation theory, developed by Götze and the self-consistent treatment of Vollhardt and Wölfle are rederived from simple approximations of the kinetic equations. The outlined method represents a systematic approach to the Anderson localization. It may be applicable also to more realistic models for the Anderson localization as well as extended to the evaluation of general transport coefficients on a level far beyond the usual perturbation theories.     

Anderson Localization for Time Quasi-Periodic Random Schrödinger and Wave Equations

We prove that at large disorder, with large probability and for a corresponding set of Diophantine frequencies of large measure, Anderson localization in ^d is stable under localized time quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. This paper paves the way for the construction of time quasi-periodic KAM type of solutions of non linear random Schrödinger equations in [BW].Wei-Min Wang thanks A. Soffer and T. Spencer for many useful conversations and for initiations to the subject. She also thanks M. Combescure and J. Sjöstrand for helpful discussions on the quasi-energy operator formulation of time dependent Schrödinger equations. The support of NSF grant DMS 9729992 is gratefully acknowleged.     

One-dimensional localization in porous a-Si1−c Mnc

The temperature dependence of the conductance of porous silicon doped with manganese up to densities corresponding to the metallic side of the Anderson transition is investigated. It is found that in the temperature range below T=40–60 K the conductance decreases with T as G(T)∝T ^−1/3. This behavior corresponds to one-dimensional electron localization in silicon wires under conditions of inelastic electron-electron collisions with a small energy transfer. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 4, 265–269 (25 February 1998)     

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 impurity model: Vertex corrections within the finite U non-crossing approximation

We revise the simplest possible approximations to solve numerically the vertex equations for the single impurity Anderson model (SIAM) within the finite U non-crossing approximation (UNCA), considering the self-energies at lowest order in the 1/N diagrammatic expansion. We introduce an approximation to the vertex corrections that includes the double energy dependence and compare it with an approximation (NCAf²v) that neglects a second energy argument. Finally, we analyse the influence of the different approximations on the estimated Kondo scale for simple electronic models.