Quasiperiodic Systems Research Articles

Rational approximations of quasiperiodicity via projected Green's functions

We introduce the projected Green's function technique to study quasiperiodic systems such as the Aubry-Andr\'e-Harper (AAH) model and beyond. In particular, we use projected Green's functions to construct a rational approximate sequence of transfer matrix equations consistent with quasiperiodic topology, where convergence of these sequences corresponds to the existence of extended eigenfunctions. We motivate this framework by applying it to a few well-studied cases such as the almost-Mathieu operator (AAH model), as well as more generic no-dual models that challenge standard routines. The technique is flexible and can be used to extract both analytic and numerical results, e.g., we analytically extract a modified phase diagram for Liouville irrationals. As a numerical tool, it does not require the fixing of boundary conditions and circumvents a primary failing of numerical techniques in quasiperiodic systems---extrapolation from finite size. Instead, it uses finite-size scaling to define convergence bounds on the full irrational limit.

Exponential size scaling of the Liouvillian gap in boundary-dissipated systems with Anderson localization

We carry out a systematical study of the size scaling of Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling $\Delta_{g}\propto L^{- 3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation $\Delta_{g}\propto e^{- \kappa L}$ in the localized phase, where $\kappa$ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-Andr\'{e}-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent $\kappa$ coincides pretty well with the analytical result of Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.

Optimal design for vibration energy harvesters based on quasi-periodic structures

In this paper, the design of large-scale quasi-periodic Vibration Energy Harvesters (VEH) is optimized to enhance the harvested power of an electromagnetic mode localized structure. This work aims to optimize the output power by employing the energy localization phenomenon in a large-scale periodic configuration by introducing the minimum number of perturbations. The harvested power, number and location of perturbations are among the objectives that need to be optimized. A genetic-based mixed-integer optimization algorithm is used to meet the objective functions within a constraint on the system kinetic energy. Numerical simulations for quasi-periodic systems with 20 and 100 Degrees of Freedom (DOF) are performed. It is shown that the ratio of harvested power increases as the number of perturbations rises and it exceeds 80% of the total output power by perturbing almost one-third of the total DOFs. The proposed methodology is a decision-making aid to provide an optimal design in a generalized quasi-periodic VEH in order to reduce the number of harvesting transducers while providing a significantly high amount of harvested power.

Quantum transport in quasiperiodic lattice systems in the presence of Büttiker probes

Quasi-periodic lattice systems offer diverse transport properties. In this work, we investigate the environment induced effects on transport properties for quasi-periodic systems, namely the one-dimensional Aubry-Andr\'e-Harper (AAH) lattice chain and its generalized version (GAAH) by considering the B\"uttiker probe approach. We first consider voltage probe situation and study the electrical conductance properties in the linear response regime. At zero temperature, we observe enhancement in conductance at all the no-transport regimes, located both inside and outside of the band of the original system, for small probe coupling strength $\gamma$ with a power-law scaling $\gamma^4$. Whereas, for large probe coupling strengths, the conductance at all Fermi energies is the same and decays as a power-law with scaling $1/\gamma^4$. This particular scaling survives even in the finite-temperature limit. Interestingly, this scaling result is different from the one recently predicted following the local Lindblad master equation approach. The transport eventually becomes diffusive for all energy ranges and in all regimes of the original model for a sufficiently strong coupling with the probes. We further extend our study and consider voltage-temperature probes to analyze the thermoelectric performance of the chain in terms of the figure of merit. We also demonstrate the validity of two recently obtained bounds on thermoelectric efficiency that are tighter than the seminal Carnot bound and express the same in terms of the Onsager's transport coefficients.

A Direct Approach to Compute the Lyapunov–Perron Transformation for Linear Quasi-periodic Systems

A Direct Approach to Compute the Lyapunov–Perron Transformation for Linear Quasi-periodic Systems

Challenges to observation of many-body localization

We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.

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.

1D quasicrystals and topological markers

Local topological markers are effective tools for determining the topological properties of both homogeneous and inhomogeneous systems. The Chern marker is an established topological marker that has previously been shown to effectively reveal the topological properties of 2D systems. In an earlier work, the present authors have developed a marker that can be applied to 1D time-dependent systems which can be used to explore their topological properties, like charge pumping under the presence of disorder. In this paper, we show how to alter the 1D marker so that it can be applied to quasiperiodic and aperiodic systems. We then verify its effectiveness against different quasicrystal Hamiltonians, some which have been addressed in previous studies using existing methods, and others which possess topological structures that have been largely unexplored. We also demonstrate that the altered 1D marker can be productively applied to systems that are fully aperiodic.

Efficient algorithm for simulating particles in true quasiperiodic environments

We introduce an algorithm based on generalized dual method (GDM) to efficiently study the dynamics of a particle in quasiperiodic environments without the need to use periodic approximations or to save the information of the vertices that make up the quasiperiodic lattice. We show that the computation time and the memory required to find the tile in which a particle is located as a function of the distance R to the center of symmetry remains constant in our algorithm, while using the GDM directly both quantities go like R 2. This allows us to perform realistic simulations with low consumption of computational resources. The algorithm can be used to study any quasiperiodic lattice that can be produced by the cut-and-project method. Using this algorithm, we have calculated the free path length distribution in quasiperiodic Lorentz gases reproducing previous results and for systems with high symmetries at the Boltzmann–Grad limit. We have found for the Boltzmann–Grad limit, that the distribution of free paths depends on the rank r of the quasiperiodic system and not on its symmetry. The distribution as a function of the free path length l appears to be a combination of exponential decay and a power-law behavior. The latter seems to become important only for probabilities less than , showing an exponential decaying free-path length distribution for r → ∞, similar to what is observed in disordered systems.

Stability and Robustness Analysis of Quasi-Periodic System Subjected to Uncertain Parametric Excitations and Nonlinear Perturbations

Abstract In this work, the asymptotic stability bounds are identified for a class of linear quasi-periodic dynamical systems with stochastic parametric excitations and nonlinear perturbations. The application of a Lyapunov–Perron (L-P) transformation converts the linear part of such systems to a linear time-invariant form. In the past, using the Infante’s approach for linear time-invariant systems, stability theorem and corollary were derived and demonstrated for time periodic systems with variation in stochastic parameters. In this study, the same approach is extended toward linear quasi-periodic with stochastic parameter variations. Furthermore, the Lyapunov’s direct approach is employed to formulate the stability conditions a for quasi-periodic system with nonlinear perturbations. If the nonlinearities satisfy a bounding condition, sufficient conditions for asymptotic stability can be derived for such systems. The applications of stability theorems are demonstrated with practical examples of commutative and noncommutative quasi-periodic systems.

Order Reduction of Nonlinear Quasi-Periodic Systems Using Lyapunov–Perron Transformation

Abstract In this paper, multiple order reduction techniques for parametrically excited nonlinear quasi-periodic systems are presented. The linear time-varying part of the quasi-periodic system is transformed into a linear time-invariant form via the Lyapunov–Perron (L–P) transformation. The analytical computation of such a transformation is performed using an intuitive state augmentation and the normal forms technique. This L–P transformation is further utilized in analyzing the nonlinear part of the original quasi-periodic system. Using the L–P transformation, three-order reduction techniques are detailed in this work. First, a Guyan linear reduction method is applied to reduce the order. The second method is to determine a nonlinear projection based on the singular perturbation method. In the third technique, the method of Invariant Manifold is applied to identify a relationship between the dominant and nondominant system states. Furthermore, in this work, all three order reduction techniques are demonstrated on the class of commutative and noncommutative/Hills-type nonlinear quasi-periodic systems. The behavior of the reduced system states of the resulting solution is compared with the numerical integration results and their performance is studied using the error plots for each technique.

Positive piecewise continuous quasi-periodic solutions to logistic impulsive differential equations

ABSTRACT To prove the existence of piecewise continuous solutions to a logistic quasi-periodic differential system with impulses (whose coefficients have rationally independent periods), this system is divided into a differential equation and a difference equation. The quasi-periodicity of a function is proved by showing that this function is the uniform limit of a series of trigonometric polynomials with a finite total number of frequencies. The asymptotically stable quasi-periodic positive and piecewise continuous solution is proved to exist and to be unique. Quasi-periodic variation of the environment leads to a quasi-periodic growth of the population size in the sense that the rationally independent frequencies of the system are also frequencies of the quasi-periodic solution. The positive solutions have a repeated behavior similar to that of the quasi-periodic solution for a sufficiently long time due to asymptotical stability. The separation of the continuous-discrete system into a differential equation and a difference equation is a method of proving the existence of a quasi-periodic solution with perturbed coefficients of the impulsive system.

Lieb–Robinson bound in one-dimensional inhomogeneous quantum systems

Lieb–Robinson bound (LRB) in one-dimensional noninteracting many-electron systems with the disordered and quasiperiodic on-site potentials is studied numerically. For the short-range hopping system, a logarithmic light cone (i.e. |x|=βlogt+x0) is found in the system with the disordered on-site potential for small time. The coefficient β decreases with the increasing strength of disordered. When time is large, the bound does not change with time (i.e. |x|=C). For the generalized Fibonacci quasiperiodic (GFQ) system, the on-site potential is taken as V or −V according to two kinds of GFQ sequences. It is found that the system has a power-law light cone (i.e. |x|∝tγ, with 0<γ<1). The exponent γ decreases with the increasing V. We also find that γ for the first class of GFQ system is larger than that for the second class of GFQ system with the same V. Finally, the effects of the long-range hopping which decays like r−α with the distance r on LRB are discussed.

Design of tunable hierarchical waveguides based on Fibonacci-like microstructure

A novel class of Fibonacci-based high performance tunable hierarchical waveguides is here proposed and special focus is devoted to the design of adaptive-passive control systems for the Bloch wave propagation. By exploiting the features of periodic approximants, the metamaterial is conceived by the proper repetition of an elementary cell along a fixed direction. This elementary cell is made up of two building blocks repeated according to the Fibonacci sequence, to form a quasi-periodic finite microstructured system The former building block is made of a homogeneous elastic material, while the latter is a microstructured two-phase laminate. One of these phases is piezoelectric shunted by a suitably conceived electrical circuit, such that the constitutive properties of the piezoelectric phase are tuned by adjusting its equivalent impedance/admittance. Moreover, the overall constitutive properties of the microstructured layer are determined, exploiting the scale separation, via an asymptotic homogenization scheme. Then, it is possible to determine the frequency band structure of the tunable waveguide by exploiting the transfer matrix approach. With the aim of providing broad design directions, attention is paid to characterizing the dispersion waves properties of the tunable Fibonacci-like superlattices described by several generations of the Fibonacci sequence and for different values of the orientation angle of the two-phase laminate.

Unfolded band structures of photonic quasicrystals and moiré superlattices

We apply the band-unfolding approach to explore the dispersions of different types of photonic quasiperiodic structures, including photonic quasicrystals that have high-rotational symmetry but lack translational symmetry, and moir\'e superlattices, which are a twist of two primary lattices. The band-unfolding approach provides a simple picture to get the band structure of such quasiperiodic structures, which can be directly compared with experimental results. We envision this approach will be a basic method for introducing the rich concepts of photonics in condensed-matter physics to photonic quasiperiodic systems.

Fermionic Many-Body Localization for Random and Quasiperiodic Systems in the Presence of Short- and Long-Range Interactions.

We study many-body localization (MBL) for interacting one-dimensional lattice fermions in random (Anderson) and quasiperiodic (Aubry-Andre) models, focusing on the role of interaction range. We obtain the MBL quantum phase diagrams by calculating the experimentally relevant inverse participation ratio (IPR) at half-filling using exact diagonalization methods and extrapolating to the infinite system size. For short-range interactions, our results produce in the phase diagram a qualitative symmetry between weak and strong interaction limits. For long-range interactions, no such symmetry exists as the strongly interacting system is always many-body localized, independent of the effective disorder strength, and the system is analogous to a pinned Wigner crystal. We obtain various scaling exponents for the IPR, suggesting conditions for different MBL regimes arising from interaction effects.

On the reducibility of two-dimensional quasi-periodic systems with Liouvillean basic frequencies and without non-degeneracy condition

In this paper we consider the linear quasi-periodic systemx˙=(A+P(t,ϵ))x,x∈R2, where A is a 2×2 constant matrix with different eigenvalues, P(t,ϵ) is Cm(m=0,1)-smooth in ϵ and analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), with α being irrational. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, and 0≤β(α)<r, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, r is the initial radius of analytic domain, it is proved that for many sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x,x∈R2, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

Order reduction of nonlinear quasi-periodic systems subjected to external excitations

This paper presents order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov–Perron (L–P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P Transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L–P Transformation. This approach is similar to the Lyapunov–Floquet (L–F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems’ essential dynamics. This work presents reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations. The methods proposed here use the L–P Transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The ‘quasi-periodic invariant manifold’ based technique yields ‘reducibility conditions’. These conditions (referred to in the perturbation literature as resonance conditions) help us understand the system’s various types of resonant interactions. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one must preserve all these ‘resonant’ states in the reduced-order model. Thus, if the ‘reducibility conditions’ are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.

Localization phenomena and electronic transport in irradiated Aubry–André–Harper systems

The role of light irradiation on electronic localization is critically investigated for the first time in a tight-binding lattice where site energies are modulated in the cosine form following the Aubry–André–Harper (AAH) model. The critical point of transition from delocalized-to-localized phase can be monitored selectively by regulating the light parameters that is extremely useful to have controlled electron transmission across the system. Starting with a strictly one-dimensional (1D) AAH chain, we extend our analysis considering a two-stranded ladder model which brings peculiar signatures in presence of irradiation. Unlike 1D system, AAH ladder exhibits a mixed phase (MP) zone where both extended and localized energy eigenstates co-exist. This is the fundamental requirement to have mobility edge in energy band spectrum. A mathematical description is given for decoupling the irradiated ladder into two effective 1D AAH chains. The underlying mechanism of getting a MP zone relies on the availability of two distinct critical points (CPs) of the decoupled chains, in presence of second-neighbor hopping between the two strands. Using a minimal coupling scheme the effect of light irradiation is incorporated following the Floquet–Bloch ansatz. The localization behaviors of different energy eigenstates are studied by calculating inverse participation ratio, and, are further explained in a more compact way by calculating two-terminal transmission probabilities together with average density of states. Finally, the decoupling procedure is extended for a more general multi-stranded AAH ladders where multiple CPs and thus multiple mobility edges are found. Our analysis may provide a new route of engineering localization properties in similar kind of other fascinating quasiperiodic systems.

Editorial: Localization of Light in Randomly Disordered and Quasiperiodic Systems

Editorial: Localization of Light in Randomly Disordered and Quasiperiodic Systems