Extended states in a Lifshitz tail regime for random Schrödinger operators on trees

We resolve an existing question concerning the location of the mobility edge for operators with a hopping term and a random potential on the Bethe lattice. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies which have been suggested with localization issues for many particle systems. We find that extended states appear through disorder enabled resonances well beyond the energy band of the operator's hopping term. For weak disorder this includes a Lifshitz tail regime of very low density of states.     

DNA分子链电子结构特性研究

在单电子紧束缚近似下，建立了一维无序二元DNA分子链模型，计算了链长为2×10⁴个碱基对的DNA分子链的电子态密度、局域化特性，并探讨了碱基对的不同组分、格点能量无序度对电子局域态的影响.结果表明：由于DNA分子链中格点能量无序及碱基对的不同组分的存在，其电子波函数呈现出局域化的特性，而局域长度作为衡量电子局域化程度的一个尺度，受碱基对的组分及格点能量无序度的影响. 关键词： DNA分子链 电子结构 电子局域态 局域长度     

Diagram technique for finding of vertex functions in the Landau theory of heteropolymer liquids

The problem on finding the coefficients of the Landau free energy expansion into the power series of parameter of order has been considered for solutions and melts of linear heteropolymers whose molecules comprise several types monomeric units arranged stochastically. The presence of such a quenched structural disorder places this problem outside the framework of the traditional statistical physics inviting for its solution special approaches. One of them, based on the replica concept and actively engaged in theoretical physics of disordered systems, has been invoked in this paper to derive expressions for the vertex functions in the Landau theory of heteropolymer liquids. An algorithm has been formulated which permits one resorting to the simple diagram technique to write down expressions for these functions of any order in terms of the statistical characteristics of chemical quenched structure of polymer molecules. Explicit expressions for the contributions to the Landau free energy up to the fourth degree of order parameters for polymer systems with an arbitrary structural disorder have been presented to illustrate this general algorithm. Its potentialities have been also exemplified for the melt of random m-component copolymer where exact analytical formulas for these contributions up to n=6 at an arbitrary m have been derived for the first time.     

Pair scattering of electrons in edge channels of opposite chiralities in the presence of a disorder potential

The nonequilibrium transfer of the energy between electrons of counter-propagating quasi-one-dimensional systems has been perturbatively calculated for edge channels in a two-dimensional system in the integer quantum Hall effect. The processes involving two electrons that are allowed only in the system with disorder have been taken into account. Expressions for the cases of Coulomb scattering and transfer of nonequilibrium phonons have been obtained. The energy transferred per unit time has a quasi-threshold dependence on the degree of nonequilibrium of the hot channel. According to numerical estimates for electrons in GaAs, Coulomb scattering processes dominate in the energy transfer and the expected effect can be experimentally observed.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.     

Nonextensive statistical analysis of the data on the high-speed impact fracture of solids

The high-speed detection of impact-induced cracks in samples of materials that differ strongly in their degree of homogeneity (fused quartz and granite) has been performed by the acoustic emission method. The experimental energy distributions of the acoustic emission signals that correspond to the energy distribution in the events of the nucleation, growth, and coalescence of the microcracks have been interpreted in terms of the Tsallis statistics, which has been developed to generalize classical thermodynamics over the case of nonequilibrium systems. This allowed us to estimate the degree of correlation in the process of crack formation and to compare the energy release densities in various materials and at various stages of the impact fracture.     

Disorder and pseudogap in strongly correlated systems: Phase diagram in the DMFT + Σ approach

The influence of disorder and pseudogap fluctuations on the Mott insulator-metal transition in strongly correlated systems has been studied in the framework of the generalized dynamic mean field theory (DMFT + Σ approach). Using the results of investigations of the density of states (DOS) and optical conductivity, a phase diagram (disorder-Hubbard interaction-temperature) is constructed for the paramagnetic Anderson-Hubbard model, which allows both the effects of strong electron correlations and the influence of strong disorder to be considered. Strong correlations are described using the DMFT, while a strong disorder is described using a generalized self-consistent theory of localization. The DOS and optical conductivity of the paramagnetic Hubbard model have been studied in a pseudogap state caused by antiferromagnetic spin (or charge) short-range order fluctuations with a finite correlation length, which have been modeled by a static Gaussian random field. The effect of a pseudogap on the Mott insulator-metal transition has been studied. It is established that, in both cases, the static Gaussian random field (related to the disorder or pseudogap fluctuations) leads to suppression of the Mott transition, broadening of the coexistence region of the insulator and metal phases, and an increase in the critical temperature at which the coexistence region disappears.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

多孔硅材料光学Anderson局域的研究

本文基于Anderson局域效应和有限时域差分(FDTD)计算方法,对具有一定无序性的多孔硅材料内光的横向局域进行了理论分析,考察了不同无序度、孔洞排列的不同方式对硅材料光学性质的影响。计算结果表明,多孔硅内光的行为和多孔硅的结构参数有很大关系。当结构参数合适时,光束可以呈现比较明显的横向局域效果。适当改变材料中孔道的排列方式,可以有效的调节光在材料中的传播方式与局域效果。设计了具有"边墙"效果的孔洞排列方式,不仅能够固定光斑的位置,而且还可实现光束的局域增强效果。     

Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals

In this paper, the localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. Numerical results of the localization lengths of SH-wave are presented and discussed in ordered and disordered piezoelectric phononic crystals to identify the different effect degrees for the decay of electrical potential in the polymers and the randomness on the localization level. For the disordered case, disorder in the thickness of the polymers and disorder in the elastic constant of the piezoelectric ceramics are all considered. The results show that some parameters such as the incident angle of elastic wave, the randomness degree and the piezoelectricity of piezoelectric ceramics and so on have pronounced effects on the frequency-dependent localization length.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

Wave localization in two-dimensional periodic systems with randomly disordered size

The expression of the localization factor in the two-dimensional periodic systems is derived based on the plane-wave expansion, transfer matrix and matrix eigenvalue methods. A comprehensive study is performed for the wave localization in the phononic crystal which is composed of steel cylinders embedded in epoxy matrix with the randomly disordered rod size. From the results, it can be observed that with the increase of the disorder degree, the localization phenomenon is strengthened. Furthermore, the filling fraction has significant effects on the wave localization characteristics.     

Localization length of wave functions in one-dimensional disordered systems with periodic modulation

The localization length of wave functions in one-dimensional disordered systems with periodic modulation is studied. The role of spatial inhomogeneity in the problem is considered. We calculate the localization length varying with energy for the compositionally modulated systems with either the disorder of Anderson distributions or the randomness of periods. The results show that the non-uniformity of disordered systems leads to some different properties of the localized wave functions.     

无序度对一维长程关联无序系统中局域化-退局域化转变的影响

对长程幂律关联能量序列进行了修正,使其能体现出无序度在一维长程关联无序系统中的影响,并利用重正化群方法,计算了能反映该系统局域化-退局域化转变的Lyapunov指数.结果表明,在由于关联指数p的影响而在系统中出现的局域化向退局域化的转变中,无序度起着相反的作用.当关联指数p一定而无序度W增大时,系统中心能区范围内由于长程关联而引起的扩展态逐渐向局域态转变.当无序度W增大到某一临界值W_c时,系统中所有本征态均转变为局 关键词： 长程关联 Lyapunov指数 无序度 局域化-退局域化转变     

Electron spectrum and wave functions of icosahedral quasicrystals

Electron spectra and wave functions of icosahedral quasicrystals have been investigated in the tight-binding approximation using the two-fragment structural model (the Amman-MacKay network) with “central” decoration. A quasicrystal has been considered as a limiting structure in a set of optimal cubic approximants with increasing lattice constants. The method of level statistics indicates that the energy spectrum of an icosahedral quasicrystal contains a singular (nonsmooth) component. The density of electron states has been calculated for the first four optimal cubic approximants of the icosahedral quasicrystal, and the respective Lebesgue measures of energy spectra of these approximants have been obtained. Unlike the case of a one-dimensional quasiperiodic structure, the energy spectrum of an icosahedral quasicrystal does not contain a hierarchical gap structure typical of the Cantor set of measure zero in a one-dimensional quasicrystal. Localization of wave functions in an icosahedral quasicrystal has been studied, and their “critical” behavior has been detected. The effect of disorder due to substitutional impurities on electron properties of icosahedral quasicrystals has been investigated. This disorder makes the electron spectrum “smoother” and leads to a tendency to localization of wave functions. Zh. éksp. Teor. Fiz. 113, 1009–1025 (March 1998)     

Electronic spectrum and localization of electronic states in aperiodic quantum dot chains

The electronic energy spectra of aperiodic Thue-Morse, Rudin-Shapiro, and double-periodic quantum dot chains are investigated in the tight-binding approximation. The dependence of the spectrum on all parameters of a “mixed” aperiodic chain model is studied: the electronic energy at quantum dots and the hopping integrals. The electronic degree of localization in the chains under consideration is determined by analyzing the inverse participation ratio. Its spectral distribution and the dependence of the band-averaged degree of localization on these model parameters have been calculated. It is shown that a transition of the system’s sites to a resonant state in which the degree of electron localization decreases, while an overlap between the subbands occurs in the spectrum is possible when the parameters are varied.     

Localization of an electron on a one-dimensional chain of periodically arrayed random δ potentials

A new method for exact averaging of the resistivity of a chain of periodically arrayed random short-range potentials has been developed. It is shown that the dependence of the average resistivity of the system on the length of a system having an arbitrary disorder at its centers is a sum of three exponential functions. The nature of the localization of the states as a function of the system’s disorder parameters and the energy of the single-electron states have been investigated. Fiz. Tverd. Tela (St. Petersburg) 41, 1851–1855 (October 1999)     

Many-body localization in disorder-free systems: The importance of finite-size constraints

Recently it has been suggested that many-body localization (MBL) can occur in translation-invariant systems, and candidate 1D models have been proposed. We find that such models, in contrast to MBL systems with quenched disorder, typically exhibit much more severe finite-size effects due to the presence of two or more vastly different energy scales. In a finite system, this can artificially split the density of states (DOS) into bands separated by large gaps. We argue for such models to faithfully represent the thermodynamic limit behavior, the ratio of relevant coupling must exceed a certain system-size depedent cutoff, chosen such that various bands in the DOS overlap one another. Setting the parameters this way to minimize finite-size effects, we study several translation-invariant MBL candidate models using exact diagonalization. Based on diagnostics including entanglement and local observables, we observe thermal (ergodic), rather than MBL-like behavior. Our results suggest that MBL in translation-invariant systems with two or more very different energy scales is less robust than perturbative arguments suggest, possibly pointing to the importance of non-perturbative effects which induce delocalization in the thermodynamic limit.     

Numerical studies on the anderson localization problem

The electron localization is studied for Anderson's tight-binding model with diagonal and off-diagonal disorder for a very large square lattice (10,000 sites) and diamond lattice (27,000 sites). The numerical investigations are based on the Lanczos recursion method. The convergence of the recursion coefficientsa _n ,b _n is discussed with regard to the electron localization.From Anderson's criterion and an exact real space renormalization method the energy of the localization edge is found as a function of the degree of disorder. Also the dependence of the spatial decay rate of localized wave functions on the energy and the degree of disorder is evaluated. Near the Anderson transition, where all states become localized, we get two critical exponentsv _E andv _W , which lead us to the tentative suggestion of multicritical scaling laws for this transition.