In the present work, we investigated the correlation-induced localization-delocalization transition in the one-dimensional tight-binding model with fractal disorder. We obtained a phase transition diagram from localized to extended states based on the normalized localization length by controlling the correlation and the disorder strength of the potential. In addition, the transition of the diffusive property of wavepacket dynamics is shown around the critical point.
In the previous work, we investigated the correlation-induced localization-delocalization transition (LDT) of the wavefunction at band center ($E=0$) in the one-dimensional tight-binding model with fractal disorder [Yamada, EPJB (2015) 88, 264]. In the present work, we study the energy ($E \neq 0$) dependence of the normalized localization length and the delocalization of the wavefunction at the different energy in the same system. The mobility edges in the LDT arise when the fractal dimension of the potential landscape is larger than the critical value depending on the disorder strength, whic…
We use a numerical implementation of the strong disorder renormalization group (RG) method to study the low-energy fixed points of random Heisenberg and tight-binding models on different types of fractal lattices. For the Heisenberg model new types of infinite disorder and strong disorder fixed points are found. For the tight-binding model we add an orbital magnetic field and use both diagonal and off-diagonal disorder. For this model besides the gap spectra we study also the fraction of frozen sites, the correlation function, the persistent current and the two-terminal current. The lattices w…
We investigated numerically the distribution of participation numbers in the 3d Anderson tight-binding model at the localization-delocalization threshold. These numbers in {\em one} disordered system experience strong level-to-level fluctuations in a wide energy range. The fluctuations grow substantially with increasing size of the system. We argue that the fluctuations of the correlation dimension, $D_2$ of the wave functions are the main reason for this. The distribution of these correlation dimensions at the transition is calculated. In the thermodynamic limit ($L\to \infty$) it does not de…
Waves propagating in certain one-dimensional quasiperiodic lattices are known to exhibit a sharp localization transition. We theoretically predict and experimentally observe that the localization of light in one-dimensional photonic quasicrystals may be followed by a second delocalization transition for some states on increasing quasiperiodic modulation strength - an example of a reentrant transition. We further propose that this phenomenon can be qualitatively captured by a dimerized tight-binding model with long-range couplings.
A weakly disordered quasi-one-dimensional tight-binding hopping model with $N$ rows is considered. The probability distribution of the Landauer conductance is calculated exactly in the middle of the band, $ε=0$, and it is shown that a delocalization transition at this energy takes place if and only if $N$ is odd. This even-odd effect is explained by level repulsion of the transmission eigenvalues.
We investigate the effect of a two-level jump process or random telegraph noise on a square wave driven tight-binding lattice. In the absence of the noise, the system is known to exhibit dynamical localization for specific ratios of the amplitude and the frequency of the drive. We obtain an exact expression for the probability propagator to study the stability of dynamical localization against telegraph noise. Our analysis shows that in the presence of noise, a proper tuning of the noise parameters destroys dynamical localization of the clean limit in one case, while it induces dynamical local…
Power-law random banded unitary matrices (PRBUM), whose matrix elements decay in a power-law fashion, were recently proposed to model the critical statistics of the Floquet eigenstates of periodically driven quantum systems. In this work, we numerically study in detail the statistical properties of PRBUM ensembles in the delocalization-localization transition regime. In particular, implications of the delocalization-localization transition for the fractal dimension of the eigenvectors, for the distribution function of the eigenvector components, and for the nearest neighbor spacing statistics …
Stationary states of non-interacting electrons on a Koch fractal are investigated within a tight binding approach. It is observed that if a hierarchically long range hopping is allowed, a suitable correlation between the parameters defining the Hamiltonian leads to spectacular changes in the transport properties of finite, but arbitrarily large fractals. Topologically identical structures, that are found to support the same distribution of the amplitudes of eigenstates, are conducting in some cases and insulating in the others, depending on the choice of the hierarchy parameter. The values of …
Motivated by the constrained many-body dynamics, the stability of the localization-delocalization properties to the inclusion of the soft constraints is addressed in random matrix models. These constraints are modeled by correlations in long-ranged hopping with Pearson correlation coefficient different from zero or unity. Counterintuitive robustness of delocalized phases, both ergodic and (multi)fractal, in these models is numerically observed and confirmed by the analytical calculations. First, matrix inversion trick is used to uncover the origin of such robustness. Next, to characterize delo…
In disordered two dimensional Chern insulators, a single bulk extended mode is predicted to exist per band, up to a critical disorder strength; all the other bulk modes are localized. This behavior contrasts strongly with topologically trivial two-dimensional phases, whose modes all become localized in the presence of disorder. Using a tight-binding model of a realistic photonic Chern insulator, we show that delocalized bulk eigenstates can be observed in an experimentally realistic setting. This requires the selective use of resonator losses to suppress topological edge states, and acquiring …
We use supersymmetric (SUSY) methods to study the delocalization transition at zero energy in a one-dimensional tight-binding model of spinless fermions with particle-hole symmetric disorder. Like the McCoy-Wu random transverse-field Ising model to which it is related, the fermionic problem displays two different correlation lengths for typical and mean correlations. Using the SUSY technique, mean correlators are obtained as quantum mechanical expectation values for an U(2|1,1) ``superspin''. In the scaling limit, this quantum mechanics is closely related to a 0+1-dimensional Liouville theory,…
We consider d dimensional systems which are localized in the absence of interactions, but whose single particle (SP) localization length diverges near a discrete set of (single-particle) energies, with critical exponent ν. This class includes disordered systems with intrinsic- or symmetry-protected- topological bands, such as disordered integer quantum Hall insulators. In the absence of interactions, such marginally localized systems exhibit anomalous properties intermediate between localized and extended including: vanishing DC conductivity but sub-diffusive dynamics, and fractal entanglement…
We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson's model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emerge…
Finite strips, composed of a periodic stacking of infinite quasiperiodic Fibonacci chains, have been investigated in terms of their electronic properties. The system is described by a tight binding Hamiltonian. The eigenvalue spectrum of such a multi-strand quasiperiodic network is found to be sensitive on the mutual values of the intra-strand and inter-strand tunnel hoppings, whose distribution displays a unique three-subband self-similar pattern in a parameter subspace. In addition, it is observed that special numerical correlations between the nearest and the next-nearest neighbor hopping i…
In this paper, we explore the localization transition and the scaling properties of both quasi-one-dimensional and two-dimensional quasiperiodic systems, which are constituted from coupling several Aubry-André (AA) chains along the transverse direction, in the presence of next-nearest-neighbor (NNN) hopping. The localization length, two-terminal conductance, and participation ratio are calculated within the tight-binding Hamiltonian. Our results reveal that a metal-insulator transition could be driven in these systems not only by changing the NNN hopping integral but also by the dimensionality…
We perform both analytical and numerical studies of the one-dimensional tight-binding Hamiltonian with stochastic uncorrelated on-site energies and non-fluctuating long-range hopping integrals . It was argued recently [A. Rodriguez at al., J. Phys. A: Math. Gen. 33, L161 (2000)] that this model reveals a localization-delocalization transition with respect to the disorder magnitude provided . The transition occurs at one of the band edges (the upper one for and the lower one for). The states at the other band edge are always localized, which hints on the existence of a single mobility edge. We …
We consider the one-dimensional $XX$-model in a quasi-periodic transverse-field described by the Harper potential, which is equivalent to a tight-binding model of spinless fermions with a quasi-periodic chemical potential. For weak transverse field (chemical potential), $h<h_c$, the excitations (fermions) are delocalized, but become localized for $h>h_c$. We study the non-equilibrium relaxation of the system by applying two protocols: a sudden change of $h$ (quench dynamics) and a slow change of $h$ in time (adiabatic dynamics). For a quench into the delocalized (localized) phase, the entangle…
We study the hopping transport of a quantum particle through randomly diluted percolation clusters in two dimensions realized both on the square and triangular lattices. We investigate the nature of localization of the particle by calculating the transmission coefficient as a function of energy (-2 < E < 2 in units of the hopping integral in the tight-binding Hamiltonian) and disorder, q (probability that a given site of the lattice is not available to the particle). Our study based on finite size scaling suggests the existence of delocalized states that depends on energy and amount of disorde…
Quantum particles in a disordered potential, photons or classical waves in a random medium, or the universe expansion in a fluctuating cosmic field, all share Anderson localization as a communality. In general, localization is enhanced for strong disorder and low dimensions. In one dimension and for discrete uncorrelated random potentials, such as tight binding models, all states are localized for any disorder strength. This is in contrast to continuous random potentials, where we show here that regardless of the strength of the random potential, we have delocalization in the limit where the r…
Localization--non-ergodic transition in controllable-dimension fractal networks from diffusion-limited aggregation
Institute for Basic Science（基础科学研究院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Korea University of Science and Technology（韩国科学技术大学）；Polish Academy of Sciences（波兰科学院）；Russian Quantum Center, Skolkovo（俄罗斯量子中心（斯科尔科沃））
We demonstrate, by explicit construction, that a single band tight binding Hamiltonian defined on a class of deterministic fractals of the b = 3N Sierpinski type can give rise to an infinity of dispersionless, flat-band like states which can be worked out analytically using the scale invariance of the underlying lattice. The states are localized over clusters of increasing sizes, displaying the existence of a multitude of localization areas. The onset of localization can, in principle, be delayed in space by an appropriate choice of the energy of the electron. A uniform magnetic field threadin…
If a localized quantum state in a tight-binding model with structural aperiodicity is subject to noisy evolution, then it is generally expected to result in diffusion and delocalization. In this work, it is shown that the localized phase of the kicked Aubry-André-Harper (AAH) model is robust to the effects of noisy evolution, for long times, provided that some kick is delivered once every time period. However, if strong noisy perturbations are applied by randomly missing kicks, a sharp dynamical transition from a ballistic growth phase at initial times to a diffusive growth phase for longer ti…
A Sierpinski gasket fractal network model is studied in respect of its electronic spectrum and magneto-transport when each arm of the gasket is replaced by a diamond shaped Aharonov-Bohm interferometer, threaded by a uniform magnetic flux. Within the framework of a tight binding model for non-interacting, spinless electrons and a real space renormalization group method we unravel a class of extended and localized electronic states. In particular, we demonstrate the existence of extreme localization of electronic states at a special finite set of energy eigenvalues, and an infinite set of energ…
Effective delocalization in the one-dimensional Anderson model with stealthy disorder
Princeton University（普林斯顿大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；The Pennsylvania State University（宾夕法尼亚州立大学）；Columbia University（哥伦比亚大学）
We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasi-periodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at th…
It is reported a combined numerical approach to study the localization properties of the one-dimensional tight-binding model with potential modulated along the prime numbers. A localization-delocalization transition was found as function of the potential intensity; it is also argued that there are delocalized states for any value of the potential intensity.
A particle that is immersed in a uniform temperature bath performs Brownian diffusion, as discussed by Einstein. But Sinai has realized that in a "random environment" the diffusion is suppressed. Follow-up works have pointed out that in the presence of bias $f$ there are delocalization and sliding transitions, with threshold value $f_c$ that depends on the disorder strength. We discuss in a critical way the emergence of Sinai physics for both passive and active Brownian particles. Tight-binding and Fokker-Planck versions of the model are addressed on equal footing. We assume that the transitio…
The nature of extended states in disordered tight binding models with a constant imaginary vector potential is explored. Such models, relevant to vortex physics in superconductors and to population biology, exhibit a delocalization transition and a band of extended states even for a one dimensional ring. Using an analysis of eigenvalue trajectories in the complex plane, we demonstrate that each delocalized state is characterized by an (integer) winding number, and evaluate the associated complex current. Winding numbers in higher dimensions are also discussed.
The diffusion of electronic wave packets in one-dimensional systems with on-site, binary disorder is numerically investigated within the framework of a single-band tight-binding model. Fractal properties are incorporated by assuming that the distribution of distances $\ell$ between consecutive impurities obeys a power law, $P(\ell) \sim \ell^{-α}$. For suitable ranges of $α$, one finds system-wide anomalous diffusion. Asymmetric diffusion effects are introduced through the application of an external electric field, leading to results similar to those observed in the case of photogenerated elec…
Eigenstate transitions and critical transport in 3D Aubry-André model
J. Stefan Institute（约热·斯蒂芬研究所）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Theoretical Physics, Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学理论物理研究所）
We clarify universal critical properties of delocalization-localization transitions in three-dimensional (3D) unitary and orthogonal classes with particle-hole and/or chiral symmetries (classes AIII, BDI, D, C and CI). We first introduce tight-binding models on cubic lattice that belong to these five nonstandard symmetry classes respectively. Unlike the Bogoliubov-de Gennes Hamiltonian for superconductors, all the five models have finite areas of Fermi surfaces in the momentum space in the clean limit. Thereby, the scaling theory of the Anderson transition guarantees the presence of the deloca…
We study the time evolution of the survival probability $P(t)$ in open one-dimensional quasiperiodic tight-binding samples of size $L$, at critical conditions. We show that it decays algebraically as $P(t)\sim t^{-α}$ up to times $t^*\sim L^γ$, where $α= 1-D_0^E$, $γ=1/D_0^E$ and $D_0^E$ is the fractal dimension of the spectrum of the closed system. We verified these results for the Harper model at the metal-insulator transition and for Fibonacci lattices. Our predictions should be observable in propagation experiments with electrons or classical waves in quasiperiodic superlattices or dielect…
We analyze the out-of-equilibrium dynamics of a quantum particle coupled to local magnetic degrees of freedom that undergo a classical phase transition. Specifically, we consider a two-dimensional tight-binding model that interacts with a background of classical spins in thermal equilibrium, which are subject to Ising interactions and act as emergent, correlated disorder for the quantum particle. Particular attention is devoted to temperatures close to the ferromagnet-to-paramagnet transition. To capture the salient features of the classical transition, namely the effects of long-range correla…
Scaling analysis and renormalization group on the mobility edge in the quantum random energy model
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；SISSA（的里雅斯特国际高等研究学校）；INFN, Sezione di Trieste（意大利国家核物理研究院的里雅斯特分部）；The Abdus Salam ICTP（阿卜杜斯·萨拉姆理论物理中心）；Department of Physics, Princeton University（普林斯顿大学物理系）
We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with {\it correlated} nearest-neighbor $XY$ interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of ${\mathbb Z}_3$ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that, despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp con…
