One-dimensional Quasiperiodic Lattices Research Articles

Super-ballistic diffusion induced by nonlinear interactions in a one-dimensional quasiperiodic lattice

We numerically investigate the spreading dynamics of wave packets in a one-dimensional lattice with quasiperiodic disorders and nonlinear interactions. Under moderate disorder strength, we reveal an intriguing super-ballistic diffusion driven by increasing the interaction strength from the ballistic diffusion in the non- and weakly interacting cases. We obtain a wide parameter region for such a faster-than-ballistic spreading of wave packets in the quasiperiodic lattice, although the nonlinear self-trapping is dominated under strong interactions. The super-ballistic diffusion is further analyzed based on the point source model. We also show the interaction-induced sub-diffusive spreading in the Anderson localized regime. The super-ballistic and sub-diffusive dynamics are exhibited in the spreading that starts from both the single-site and Gaussian wave packets. Our results showcase the interaction-enabled super-ballistic diffusion, which is scarce and experimentally realizable in artificial atomic or photonic lattices.

Renormalization group theory of one-dimensional quasiperiodic lattice models with commensurate approximants

Renormalization group theory of one-dimensional quasiperiodic lattice models with commensurate approximants

Quench-induced chaotic dynamics of Anderson-localized interacting Bose-Einstein condensates in one dimension

We study the effect of atomic interaction on the localization and the associated dynamics of Bose-Einstein condensates in a one-dimensional quasiperiodic optical lattice and random disordered potentials. When the interactions are absent, the condensates exhibit localization, which weakens as we increase the interaction strength beyond a threshold value for both potential types. We inspect the localized and delocalized states by perturbing the system via quenching the interaction strength instantaneously to zero and studying the dynamics of the condensate, which we further corroborate using the out-of-time-order correlator. The temporal behavior of the time correlator displays regular dynamics for the localized state, while it shows temporal chaos for the delocalized state. We confirm this dynamical behavior by analyzing the power spectral density of the time correlator. We further identify that the condensate admits a quasiperiodic route to chaotic dynamics for both the potentials. Finally, we present the variation of the maximal Lyapunov exponents for different nonlinearity and disorder strengths that have a positive value in the regime where the time-correlator function shows chaotic behavior. Through this, we establish the strong connection between the spatially delocalized state of the condensate and its temporal chaos.

Eigenstates in gain–loss systems of metallic-mean quasiperiodic chains

The eigenvalue spectrum of a gain–loss one-dimensional quasiperiodic lattice was examined. An eigenvalue analysis was performed for golden-, silver-, and bronze–mean systems. The eigenvalues of these systems tended to coalesce in discrete small areas of a complex plane, and the eigenvalues exhibited a hierarchical spatial distribution on that plane. In addition, the effect of positional disorder in this system was investigated. The scaling property of the eigenvectors in the golden-mean system was clarified. Multifractal properties of the system were observed except for the outermost edge of the eigenvalue spectrum. For more advanced control of wave phenomena, clarification of physical properties of gain–loss systems in quasi-periodic systems is expected to lead to new technical applications in the future.

Critical analysis of the reentrant localization transition in a one-dimensional dimerized quasiperiodic lattice

A re-entrant localization transition has been predicted recently in a one-dimensional quasiperiodic lattice with dimerized hopping between the nearest-neighbour sites (Phys. Rev. Lett. {\bf 126} 106803 (2021)) \cite{PhysRevLett.126.106803}. It has been shown that the interplay between the hopping dimerization and a staggered quasi-periodic disorder manifests two localization transitions through two intermediate phases resulting in four critical points as a function of the quasiperiodic potential. In this paper, we study the phenomenon of the re-entrant localization transition by examining the spectral properties of the states. By performing a systematic finite-size scaling analysis for a fixed value of the hopping dimerization, we obtain accurate critical disorder strengths for different transitions and the associated critical exponents. Moreover, through a multifractal analysis, we study the critical nature of the states across the localization transitions by computing the mass exponents and the corresponding fractal dimensions of the states. Further, we complement the critical nature of the states by computing the Hausdorff dimensions.

Emergence of multiple localization transitions in a one-dimensional quasiperiodic lattice

Low dimensional quasiperiodic systems exhibit localization transitions by turning all quantum states localized after a critical quasidisorder. While certain systems with modified or constrained quasiperiodic potential undergo multiple localization transitions in one dimension, we predict an emergence of multiple localization transitions without directly imposing any constraints on the quasiperiodic potential. By considering a one-dimensional system described by the Aubry-And\'{r}e (AA) model, we show that an additional staggered onsite potential can drive the system through a series of localization transitions as a function of the staggered potential. Interestingly, we find that the number of localization transitions strongly depends on the strength of the quasiperiodic potential. Moreover, we obtain the signatures of these localization transitions in the expansion dynamics and propose an experimental scheme for their detection in the quantum gas experiment.

Quantum transport in a one-dimensional quasicrystal with mobility edges

Quantum transport in a one-dimensional (1D) quasiperiodic lattice with mobility edges is explored. We first investigate the adiabatic pumping between left and right edge modes by resorting to two edge-bulk-edge channels and demonstrate that the success or failure of the adiabatic pumping depends on whether the corresponding bulk subchannel undergoes a localization-delocalization transition. Compared with the paradigmatic Aubry-Andr\'e (AA) model, the introduction of mobility edges triggers an opposite outcome for successful pumping in the two channels, showing a discrepancy of critical condition, and facilitates the robustness of the adiabatic pumping against quasidisorder. We also consider the transfer between excitations at both boundaries of the lattice and an anomalous phenomenon characterized by the enhanced quasidisorder contributing to the excitation transfer is found. Furthermore, there exists a parametric regime where a nonreciprocal effect emerges in the presence of mobility edges, which leads to a unidirectional transport for the excitation transfer and enables potential applications in the engineering of quantum diodes.

Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice

We analytically determine the non-Hermitian mobility edges of a one-dimensional quasiperiodic lattice model with exponential decaying hopping and complex potentials as well as its dual model, which is just a non-Hermitian generalization of the Ganeshan-Pixley-Das Sarma model with nonreciprocal nearest-neighboring hopping. The presence of non-Hermitian term destroys the self-duality symmetry and thus prevents us exploring the localization-delocalization point through looking for self-dual points. Nevertheless, by applying Avila's global theory, the Lyapunov exponent of the Ganeshan-Pixley-Das Sarma model can be exactly derived, which enables us to get an analytical expression of mobility edge of the non-Hermitian dual model. Consequently, the mobility edge of the original model is obtained by using the dual transformation, which creates exact mappings between the spectra and wavefunctions of these two models.

Learning single-particle mobility edges by a neural network based on data compression

The breaking of the well-known Aubry-Andr\'e model with a single-particle mobility edge (SPME) is a intriguing subject that has not yet been fully understood. In particular, how to accurately and efficiently recognize a SPME in an optical lattice is currently under active debate. In this work, we develop a data compression-based neural network (DCNN) approach to identify SPMEs in one-dimensional quasiperiodic optical lattice using eigenstates as the sole diagnostic. We find that such method can successfully identify SPMEs of a large system only using a small network trained by the small system data, without onerously and repetitively training a new and large-scale network by massive data of a large system. Furthermore, we show that this method is also applicable to recognize more complex phase transitions, such as many-body localization. Our DCNN approach first paves the way for the development of a generic tool for identifying unexplored phase transitions in large systems.

One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges.

The mobility edges (MEs) in energy that separate extended and localized states are a central concept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems, while MEs may exist for certain cases, the analytic results that allow for an exact understanding are rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, where quasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analytical solutions provide the exact results not only for the MEs, but also for the localization and extended features of all states in the spectra, as derived through computing the Lyapunov exponents from Avila's global theory and also numerically verified by calculating the fractal dimension. We further propose a novel scheme with experimental feasibility to realize our model based on an optical Raman lattice, which paves the way for experimental exploration of the predicted exact ME physics.

Localization Driven Superradiant Instability.

The prominent Dicke superradiant phase arises from coupling an ensemble of atoms to a cavity optical field when an external optical pumping exceeds a threshold strength. Here we report a prediction of the superradiant instability driven by Anderson localization, realized with a hybrid system of the Dicke and Aubry-André (DAA) model for bosons trapped in a one-dimensional (1D) quasiperiodic optical lattice and coupled to a cavity. Our central finding is that for bosons condensed in a localized phase given by the DAA model, the resonant superradiant scattering is induced, for which the critical optical pumping of the superradiant phase transition approaches zero, giving an instability driven by the Anderson localization. The superradiant phase for the DAA model with or without a mobility edge is investigated, showing that the localization driven superradiant instability is in sharp contrast to the superradiance as widely observed for a Bose-Einstein condensate in extended states, and should be insensitive to the temperature of the system. This study unveils a novel effect of localization on the Dicke superradiance, and is well accessible based on the current experiments.

Mobility edge and intermediate phase in one-dimensional incommensurate lattice potentials

We study theoretically the localization properties of two distinct one-dimensional quasiperiodic lattice models with a single-particle mobility edge (SPME) separating extended and localized states in the energy spectrum. The first one is the familiar Soukoulis-Economou trichromatic potential model with two incommensurate potentials, and the second is a system consisting of two coupled 1D Aubry-Andre chains each containing one incommensurate potential. We show that as a function of the Hamiltonian model parameters, both models have a wide single-particle intermediate phase (SPIP), defined as the regime where localized and extended single-particle states coexist in the spectrum, leading to a behavior intermediate between purely extended or purely localized when the system is dynamically quenched from a generic initial state. Our results thus suggest that both systems could serve as interesting experimental platforms for studying the interplay between localized and extended states, and may provide insight into the role of the coupling of small baths to localized systems. We also calculate the Lyapunov (or localization) exponent for several incommensurate 1D models exhibiting SPME, finding that such localization critical exponents for quasiperiodic potential induced localization are nonuniversal and depend on the microscopic details of the Hamiltonian.

Phononic topological states in 1D quasicrystals

In this work, we address the study of phonons propagating on a one-dimensional quasiperiodic lattice, where the atoms are considered bounded by springs whose strength are modulated by equivalent Aubry–André hoppings. As an example, from the equations of motion, we obtained the equivalent phonon spectrum of the well known Hofstadter butterfly. We have also obtained extended, critical, and localized regimes in this spectrum. By introducing the equivalent Aubry–André model through the variation of the initial phase , we have shown that border states for phonons are allowed to exist. These states can be classified as topologically protected states (topological states). By calculating the inverse participation rate, we describe the localization of phonons and verify a phase transition, characterized by the critical value of , where the states of the system change from extended to localized, precisely like in a metal-insulator phase transition.

Properties and applications of one dimensional quasiperiodic lattices

Quasiperiodic lattices have been widely studied in cold atoms, which make one study extended-Anderson localization transition in one-dimensional (1D) or two-dimensional (2D) systems. In 2008, Inguscio's group prepared one-dimensional quasiperiodic lattice in cold atomic system and observed Anderson localization, which promoted the theoretical and experimental studies of quasiperiodic systems. Later, Bloch's group observed the many body localization in their prepared 1D and 2D quasiperiodic lattices. Recently, they also successfully observed the mobility edge and many body localization in a system with mobility edge in quasiperiodic lattices. These cold atom experiments have promoted the researches of many body localization and mobility edge. Quasiperiodic lattices have become a platform and its effects on many physical phenomena are being extensively studied, which can be expectantly observed in cold atomic experiments. This paper is based on the authors' some related work and briefly review some recent studies on the 1D quasiperiodic lattices, including some important cold atom experiments, some important properties of the quasiperiodic lattices and their effects on some physical phenomena, such as topological states.

Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice.

A single-particle mobility edge (SPME) marks a critical energy separating extended from localized states in a quantum system. In one-dimensional systems with uncorrelated disorder, a SPME cannot exist, since all single-particle states localize for arbitrarily weak disorder strengths. However, in a quasiperiodic system, the localization transition can occur at a finite detuning strength and SPMEs become possible. In this Letter, we find experimental evidence for the existence of such a SPME in a one-dimensional quasiperiodic optical lattice. Specifically, we find a regime where extended and localized single-particle states coexist, in good agreement with theoretical simulations, which predict a SPME in this regime.

腔诱导的冷原子在一维准周期性光晶格中的布洛赫振荡

腔诱导的冷原子在一维准周期性光晶格中的布洛赫振荡

Almost mobility edges and the existence of critical regions in one-dimensional quasiperiodic lattices

We study a one-dimensional quasiperiodic system described by the Aubry-Andr\'e model in the small wave vector limit and demonstrate the existence of almost mobility edges and critical regions in the system. It is well known that the eigenstates of the Aubry-Andr\'e model are either extended or localized depending on the strength of incommensurate potential $V$ being less or bigger than a critical value $V_c$, and thus no mobility edge exists. However, it was shown in a recent work that this conclusion does not hold true when the wave vector $\alpha$ of the incommensurate potential is small, and for the system with $V<V_c$, there exist almost mobility edges at the energy $E_{c_{\pm}}$, which separate the robustly delocalized states from "almost localized" states. We find that, besides $E_{c_{\pm}}$, there exist additionally another energy edges $E_{c'_{\pm}}$, at which abrupt change of inverse participation ratio occurs. By using the inverse participation ratio and carrying out multifractal analyses, we identify the existence of critical regions among $|E_{c_{\pm}}| \leq |E| \leq |E_{c'_{\pm}}|$ with the almost mobility edges $E_{c_{\pm}}$ and $E_{c'_{\pm}}$ separating the critical region from the extended and localized regions, respectively. We also study the system with $V>V_c$, for which all eigenstates are localized states, but can be divided into extended, critical and localized states in their dual space by utilizing the self-duality property of the Aubry-Andr\'e model.

Anderson localization in one-dimensional quasiperiodic lattice models with nearest- and next-nearest-neighbor hopping

We explore the reduced relative Shannon information entropies SR for a quasiperiodic lattice model with nearest- and next-nearest-neighbor hopping, where an irrational number is in the mathematical expression of incommensurate on-site potentials. Based on SR, we respectively unveil the phase diagrams for two irrationalities, i.e., the inverse bronze mean and the inverse golden mean. The corresponding phase diagrams include regions of purely localized phase, purely delocalized phase, pure critical phase, and regions with mobility edges. The boundaries of different regions depend on the values of irrational number. These studies present a more complete picture than existing works.

Non-additive model for specific heat of electrons

By using non-additive Tsallis entropy we demonstrate numerically that one-dimensional quasicrystals, whose energy spectra are multifractal Cantor sets, are characterized by an entropic parameter, and calculate the electronic specific heat, where we consider a non-additive entropy Sq. In our method we consider an energy spectra calculated using the one-dimensional tight binding Schrödinger equation, and their bands (or levels) are scaled onto the [0,1] interval. The Tsallis' formalism is applied to the energy spectra of Fibonacci and double-period one-dimensional quasiperiodic lattices. We analytically obtain an expression for the specific heat that we consider to be more appropriate to calculate this quantity in those quasiperiodic structures.

Freed by interaction kinetic states in the Harper model

We study the problem of two interacting particles in a one-dimensional quasiperiodic lattice of the Harper model. We show that a short or long range interaction between particles leads to emergence of delocalized pairs in the non-interacting localized phase. The properties of these freed by interaction kinetic states (FIKS) are analyzed numerically including the advanced Arnoldi method. We find that the number of sites populated by FIKS pairs grows algebraically with the system size with the maximal exponent b = 1, up to a largest lattice size N = 10 946 reached in our numerical simulations, thus corresponding to a complete delocalization of pairs. For delocalized FIKS pairs the spectral properties of such quasiperiodic operators represent a deep mathematical problem. We argue that FIKS pairs can be detected in the framework of recent cold atom experiments [M. Schreiber et al., Science 349, 842 (2015)] by a simple setup modification. We also discuss possible implications of FIKS pairs for electron transport in the regime of charge-density wave and high T c superconductivity.