Mobility edges and reentrant localization in one-dimensional dimerized non-Hermitian quasiperiodic lattice

The mobility edges and reentrant localization transitions are studied in one-dimensional dimerized lattice with nonHermitian either uniform or staggered quasiperiodic potentials.We find that the non-Hermitian uniform quasiperiodic disorder can induce an intermediate phase where the extended states coexist with the localized ones,which implies that the system has mobility edges.The localization transition is accompanied by the PT symmetry breaking transition.While if the non-Hermitian quasiperiodic disorder is staggered,we demonstrate the existence of multiple intermediate phases and multiple reentrant localization transitions based on the finite size scaling analysis.Interestingly,some already localized states will become extended states and can also be localized again for certain non-Hermitian parameters.The reentrant localization transitions are associated with the intermediate phases hosting mobility edges.Besides,we also find that the non-Hermiticity can break the reentrant localization transition where only one intermediate phase survives.More detailed information about the mobility edges and reentrant localization transitions are presented by analyzing the eigenenergy spectrum,inverse participation ratio,and normalized participation ratio.     

Stochastic webs and continuum percolation in quasiperiodic media

We report the results of an analytical and numerical study of the contour line and surface geometry in two models of continuum percolation with quasiperiodic properties. Both the fractal dimension of long isolines and the scaling coefficient nu are determined analytically for the two-dimensional percolation problem. The scaling characteristics of the isosurfaces of the three-dimensional potential function with an icosahedral symmetry are obtained using computer graphic representation.     

Quantum diffusion in semi-infinite periodic and quasiperiodic systems

This paper studies quantum diffusion in semi-infinite one-dimensional periodic lattice and quasiperiodic Fibonacci lattice. It finds that the quantum diffusion in the semi-infinite periodic lattice shows the same properties as that for the infinite periodic lattice. Different behaviour is found for the semi-infinite Fibonacci lattice. In this case, there are still C（t） - t^-δ and d（t） - t^β. However, it finds that 0 〈δ 〈 1 for smaller time, and δ = 0 for larger time due to the influence of surface localized states. Moreover, β for the semi-infinite Fibonacci lattice is much smaller than that for the infinite Fibonacci lattice. Effects of disorder on the quantum diffusion are also discussed.     

Periodic,quasiperiodic, and chaotic breathers in two-dimensional discrete β-Fermi-Pasta-Ulam lattice

Using numerical method, we investigate whether periodic, quasiperiodic, and chaotic breathers are supported by the two-dimensional discrete Fermi-Pasta-Ulam (FPU) lattice with linear dispersion term. The spatial profile and time evolution of the two-dimensional discrete β -FPU lattice are segregated by the method of separation of variables, and the numerical simulations suggest that the discrete breathers (DBs) are supported by the system. By introducing a periodic interaction into the linear interaction between the atoms, we achieve the coupling of two incommensurate frequencies for a single DB, and the numerical simulations suggest that the quasiperiodic and chaotic breathers are supported by the system, too.     

Invariable mobility edge in a quasiperiodic lattice

We analytically and numerically study a 1 D tight-binding model with tunable incommensurate potentials.We utilize the self-dual relation to obtain the critical energy,namely,the mobility edge.Interestingly,we analytically demonstrate that this critical energy is a constant independent of strength of potentials.Then we numerically verify the analytical results by analyzing the spatial distributions of wave functions,the inverse participation rate and the multifractal theory.All numerical results are in excellent agreement with the analytical results.Finally,we give a brief discussion on the possible experimental observation of the invariable mobility edge in the system of ultracold atoms in optical lattices.     

横场中非束缚类准周期伊辛链的赝临界点

本文系统地研究了有限尺寸下非束缚类准周期量子伊辛链在横场中的赝临界点的行为. 首先, 通过计算平均磁矩和协作参量, 发现这两个量的导数随着横场的变化都会出现两个清晰的峰. 这与束缚类伊辛链和无序伊辛链的结果明显不同. 其次, 研究了横场中赝临界点的概率分布情况, 发现概率分布并不是高斯型的. 这也与无序的结果不同. 最后, 分析了赝临界点产生的原因, 发现赝临界点是由非束缚类准周期伊辛链中的集团结构造成的.     

Irrational mode locking in quasiperiodic systems

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.     

Scattering of radiation by a quasiperiodic two-dimensional medium

We study the scattering of radiation by a medium presenting inhomogeneities distributed in a quasiperiodic way. We show the existence of quasiperiodic solutions of the two-dimensional stationary wave equation, under certain conditions on the index of refraction, using a technique based on Dinaburg-Sinai method for one-dimensional Schrödinger equation with a quasiperiodic potential. Moreover we show that the energy spctrum contains a nonempty absolutely continuous component, with a subset having high degeneracy, provided the inhomogeneities are small enough.     

准周期调制下自旋1/2反铁磁XY模型中的晶格畸变行为

利用严格对角化方法研究了Thue-Morse准周期调制下 自旋1/2反铁磁XY模型中的晶格畸变行为. 结果显示: 系统中每个格点的晶格畸变幅度介于均匀分布和无序分布之间. 对于较弱的准周期调制, 调制强度的增加有利于晶格畸变的形成. 但是, 对于较强的准周期调制, 调制强度的增加则阻碍晶格畸变的形成. 此外, 系统低能谱的能隙也明显受到准周期调制的影响. 关键词： 晶格畸变 Thue-Morse序列 准周期调制     

Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the ‘ant in the labyrinth’) on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.     

Localization for random and quasiperiodic potentials

A survey is made of some recent mathematical results and techniques for Schrödinger operators with random and quasiperiodic potentials. A new proof of localization for random potentials, established in collaboration with H. von Dreifus, is sketched.     

Arbitrarily slow decay of correlations in quasiperiodic systems

For diffusive motion in random media it is widely believed that the velocity autocorrelation functionc(t) exhibits power law decay as time;t. We demonstrate that the decay ofc(t) in quasiperiodic media can be arbitrarily slow within the class of integrable functions. For example, ind=1 with a potentialV(x)=cosx+coskx, there is a dense set of irrationalk's such that the decay ofc(k, t) is slower than 1/t ⁽¹⁺⁾ for any>0. The irrationals producing such a slow decay ofc(k, t) arevery well approximated by rationals.     

On quantum stability for systems under quasiperiodic perturbations

We prove that the spectrum defined in terms of the autocorrelation function of a harmonic subject to a quasiperiodic perturbation, is, at resonance, transient absolutely continuous, covering the whole line. In the nonresonant case, and under some supplementary Diophantine condition, it is pure point, coinciding with the spectrum of a special almost-periodic function.     

逾渗分立时间量子行走的传输及纠缠特性

在一维分立时间量子行走中,通过静态和动态两种方式随机地断开连接边引入无序效应,研究了静态逾渗和动态逾渗对量子行走传输特性以及位置自由度和硬币自由之间纠缠的影响.随着演化时间的增加,静态逾渗会使得量子行走从弹道传输转变为安德森局域化,而动态逾渗则会使之转变为经典扩散.理想情况下,量子纠缠在较短的时间内就达到一个常数值E_0.静态逾渗量子行走的纠缠减小,并随着时间做无规振荡,而动态逾渗量子行走的纠缠则会随着时间光滑地增加,并在某一时间超过理想情况下的常数值,表现出动态逾渗增强量子纠缠的特性.     

On the universality of crossing probabilities in two-dimensional percolation

Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratioa/b. In the limits set by the sample size and by the conventions and on the range of the ratioa/b measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.     

Bond dilution in the 3D Ising model: a Monte Carlo study

We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.Received: 24 February 2004, Published online: 28 May 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 64.60.Fr Equilibrium properties near critical points, critical exponents - 75.10.Hk Classical spin models     

Percolation, self-organized criticality, and electrical instability in carbon nanostructures

The processes of avalanche formation, percolation, and electrical instability have been investigated experimentally using multi-walled and single-walled carbon nanotubes as an example. The performed investigations are based on the comparison of electrical conductivity dynamics in classical experiments, such as the “sand heap,” the two-dimensional grid of resistances with a stochastic node blocking, and the nanosecond percolation in a mode of electrical instability in nanotube tangles/granules. The regularities of mechanisms are revealed and the general concept is formulated.     

Ionic and electronic conduction in nanostructured solids: Concepts and concerns,consensus and controversies

This paper summarizes some advances achieved in the first 10 years of studies on nanosized ionic and mixed conductors. It addresses first concepts: boundary core and space charge effects, impedance and percolation models. Then, some consensual evidence is presented: nanocrystals show no particular structural disorder and their electrical conductivity can be nicely interpreted using space charge theory, as shown for the model case of nanosized ceria. However, there are also controversial cases, such as doped zirconia, for which very different results were reported. Finally, concerns on long-time stability of nanosized solids are addressed. Recent studies of phase stability and grain growth show that these issues might be overcome.     

Fractal patterns of fluid domains for displacement processes in porous media

Percolation invasion displacement of a compressible defender is examined for two cases: when only the smallest accessible site is entered at each step and when all accessible sites less than the size given by a reducing back pressure are entered at each time step. Although the fractions of invading fluid are different, their scaling properties are equivalent. The effect of limited control of a back pressure in a real displacement and the effect of viscosity in a real time displacement are examined. In these cases the scaling properties of a percolation process at breakthrough are removed. As a result, one should expect that realistic displacement models will not have the singular properties usually attributed to percolation processes.