One-dimensional Quasiperiodic Systems Research Articles

Entanglement entropy scaling in critical phases of one-dimensional quasiperiodic systems

Entanglement entropy scaling in critical phases of one-dimensional quasiperiodic systems

Two-dimensional vertex-decorated Lieb lattice with exact mobility edges and robust flat bands

The mobility edge (ME) that marks the energy separating extended and localized states is a most important concept in understanding the metal-insulator transition induced by disordered or quasiperiodic potentials. MEs have been extensively studied in three-dimensional disorder systems and one-dimensional quasiperiodic systems. However, the studies of MEs in two-dimensional (2D) systems are rare. Here, we propose a class of 2D vertex-decorated Lieb lattice models with quasiperiodic potentials only acting on the vertices of the Lieb lattice or extended Lieb lattices. By mapping these models to the 2D Aubry-Andr\'e model, we obtain exact expressions of MEs and the localization lengths of localized states, and further demonstrate that the flat bands remain unaffected by the quasiperiodic potentials. Finally, we propose a highly feasible scheme to experimentally realize our model in a quantum dot array. Our results open the door to studying and realizing exact MEs and robust flat bands in 2D systems.

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.

Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals

The Majorana edge states of the Kitaev chain model have attracted extensive attention on their stability and experimental realization. One of the interesting aspects is finding the exotic proximity effect, which guarantees the presence of the Majorana modes, further enables efficient braidings between them. In this paper, we explore the superconducting proximity effect for quasiperiodic quantum wires and discuss how quasiperiodic patterns affect the stability of the Majorana modes. Considering the Kitaev chain model of the one-dimensional quasiperiodic system, we discuss the pattern-dependent proximity effects. First, we argue that the presence of quasiperiodic hoppings energetically induces the $p$-wave pairing also to be quasiperiodic rather than uniform pairing. More interestingly, when the normal metallic wire is adjacent to the quasiperiodic superconducting wire, we have found that the Majorana edge modes are transferred to the edge of the normal metallic side with enhanced stability. Finally, we discover the proximity effect on the strengths of the quasiperiodicities with a general power-law relationship, whose power depends on the tiling pattern. Our results show how quasiperiodic patterns play a crucial role in the Kitaev chain and the stabilization of the Majorana mode.

Exact mobility edges and topological phase transition in two-dimensional non-Hermitian quasicrystals

The emergence of the mobility edge (ME) has been recognized as an important characteristic of Anderson localization. The difficulty in understanding the physics of the MEs in three-dimensional (3D) systems from a microscopic image encourages the development of models in lower-dimensional systems that have exact MEs. While most of the previous studies are concerned with one-dimensional (1D) quasiperiodic systems, the analytic results that allow for an accurate understanding of two-dimensional (2D) cases are rare. In this work, we disclose an exactly solvable 2D quasicrystal model with parity-time ( $$\cal{P}\cal{T}$$ ) symmetry displaying exact MEs. In the thermodynamic limit, we unveil that the extended-localized transition point, observed at the $$\cal{P}\cal{T}$$ symmetry breaking point, is topologically characterized by a hidden winding number defined in the dual space. The coupling waveguide platform can be used to realize the 2D non-Hermitian quasicrystal model, and the excitation dynamics can be used to detect the localization features.

Topological charge pumping in quasiperiodic systems characterized by the Bott index

We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, regardless of the detail of quasiperiodic models. We apply this formalism to the Fibonacci-Rice-Mele model which we made from Fibonacci lattice, a well-known quasiperiodic system, and Rice-Mele model. We find that these quasiperiodic systems show topological charge pumping with a multi-level behavior due to the fractal nature of the Fibonacci lattice. Such multi-level pumping behaviors can be understood by a real space renormalization group analysis.

Duality between two generalized Aubry-André models with exact mobility edges

A mobility edge (ME) in energy separating extended from localized states is a central concept in understanding various fundamental phenomena, such as the metal-insulator transition in disordered systems. In one-dimensional quasiperiodic systems, there exist a few models with exact MEs, and these models are beneficial to provide exact understanding of ME physics. Here we investigate two widely studied models including exact MEs, one with an exponential hopping and one with a special form of incommensurate on-site potential. We analytically prove that the two models are mutually dual and further give the numerical verification by calculating the inverse participation ratio and Husimi function. Our result may provide insight into realizing and observing exact MEs in both theory and experiment.

Readout of quasiperiodic systems using qubits

We develop a theoretical scheme to perform a readout of the properties of a quasiperiodic system by coupling it to one or two qubits. We show that the decoherence dynamics of a single qubit coupled via a pure dephasing type term to a one-dimensional quasiperiodic system with a potential given by the Andr\'e-Aubry-Harper (AAH) model and its generalized versions (GAAH model) is sensitive to the nature of the single-particle eigenstates (SPEs). More specifically, we can use the non-Markovianity of the qubit dynamics as quantified by the backflow of information to clearly distinguish the localized, delocalized, and mixed regimes with a mobility edge of the AAH and GAAH models and evidence the transition between them. By attaching two qubits at distinct sites of the system, we demonstrate that the transport property of the quasiperiodic system is encoded in the scaling of the threshold time to develop correlations between the qubits with the distance between the qubits. This scaling can also be used to distinguish and infer different regimes of transport such as ballistic, diffusive, and no transport engendered by SPEs that are delocalized, critical, and localized respectively. In addition, the localization length of the SPEs can also be gleaned from the exponential decay of correlations at long times as a function of distance between qubits. When there is a mobility edge allowing the coexistence of different kinds of SPEs in the spectrum, such as the coexistence of localized and delocalized states in the GAAH models, we find that the transport behavior and the scaling of the threshold time with qubit separation are governed by the fastest spreading states.

Bandwidth convertible mid-infrared absorption of one-dimensional quasi-periodic system containing Dirac semimetals

The absorption properties of one-dimensional quasi-periodic (Fibonacci and Thue Morse sequences) systems composed of Dirac semimetal are investigated. Numerical results show that nearly perfect broadband or narrowband absorption in mid-infrared range can be obtained using the same constituents with different sequences. For the Fibonacci sequence, a 90% absorption band with a relative bandwidth over 38% can be obtained, which results from the superimposing multiple resonances, and the average absorption over the absorption band can reach 97.2%. A nearly perfect narrowband absorption can be realized by the Thue Morse sequence due to the strong field localization effect. The absorption can be tunable by changing the Fermi energy of the Dirac semimetal, the geometry parameters and the permittivity of the constituents. In addition, the effects of the polarization and the incident angles of the incident wave on the absorption performance are also calculated. Our demonstrated structures have promising absorption properties and the results are valuable in designing tunable optoelectronic devices such as mid-infrared detectors and filters.

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.

One-Dimensional Quasicrystals with Power-Law Hopping.

One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

Polaron in a non-Abelian Aubry-André-Harper model with p -wave superfluidity

We theoretically investigate the behavior of a mobile impurity immersed in a one-dimensional quasi-periodic Fermi system with topological $p$-wave superfluidity. This polaron problem is solved by using a standard variational approach, the so-called Chevy ansatz. The polaron states are found to be strongly affected by the strength of the quasi-disorder and the amplitude of the $p$-wave pairing. We analyze the phase diagram of the polaron ground state and find four phases: two extended phases, a weakly-localized phase and a strongly-localized phase. It is remarkable that these polaron phases are directly corresponding to the four distinct phases experienced by the underlying background Fermi system. In particular, the weakly-localized polaron phase corresponds to an intriguing critical phase of the Fermi system. Therefore, the different phases of the background system can be unambiguously probed by measuring the polaron properties via radio-frequency spectroscopy. We also investigate the high-lying excited polaron states at an infinite temperature and address the possibility of studying many-body localization (MBL) of these states. We find that the introduction of $p$-wave pairing may delocalize the many-body localized states and make the system easier to thermalize. Our results could be observed in current state-of-the-art cold-atom experiments.

5a-F-14 1次元準周期系のシュタルクラダーV

5a-F-14 1次元準周期系のシュタルクラダーV

Tamm-plasmon polaritons in one-dimensional photonic quasi-crystals.

We present an investigation to ascertain the existence of Tamm-plasmon-polariton-like modes in one-dimensional (1D) quasi-periodic photonic systems. Photonic bandgap formation in quasi-crystals is essentially a consequence of long-range periodicity exhibited by multilayers and, thus, it can be explained using the dispersion relation in the Brillouin zone. Defining a "Zak"-like topological phase in 1D quasi-crystals, we propose a recipe to ascertain the existence of Tamm-like photonic surface modes in a metal-terminated quasi-crystal lattice. Additionally, we also explore the conditions of efficient excitation of such surface modes along with their dispersion characteristics.

Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems.

In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.

Nonequilibrium phase diagram of a one-dimensional quasiperiodic system with a single-particle mobility edge

We investigate and map out the non-equilibrium phase diagram of a generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of AAH model. By calculating the imbalance, we get hints of a rich phase diagram. We find a fascinating connection between single particle wavefunctions near the mobility edge of GAAH model and the wavefunctions of the critical AAH model. By placing this model far-from-equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of non-equilibrium steady state (NESS) current. Current is calculated by fully exact non-equilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a 'critical' line separating regions of ballistic and localized transport. Like critical point of AAH model, current scales sub-diffusively with system size on the 'critical' line ($I\sim N^{-2\pm0.1}$). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where $I\sim N^{-1.4\pm0.05}$). All these results can be understood from the above-mentioned connection between states near mobility edge of GAAH model and those of critical AAH model. A very interesting high temperature non-equilibrium phase diagram of the GAAH model emerges from our calculations.

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.

Phase diagram of a non-Abelian Aubry-André-Harper model withp-wave superfluidity

We theoretically study a one-dimensional quasi-periodic Fermi system with topological $p$-wave superfluidity, which can be deduced from a topologically non-trivial tight-binding model on the square lattice in a uniform magnetic field and subject to a non-Abelian gauge field. The system may be regarded a non-Abelian generalization of the well-known Aubry-Andr\'e-Harper model. We investigate its phase diagram as functions of the strength of the quasi-disorder and the amplitude of the $p$-wave order parameter, through a number of numerical investigations, including a multifractal analysis. There are four distinct phases separated by three critical lines, i.e., two phases with all extended wave-functions (I and IV), a topologically trivial phase (II) with all localized wave-functions and a critical phase (III) with all multifractal wave-functions. The phase I is related to the phase IV by duality. It also seems to be related to the phase II by duality. Our proposed phase diagram may be observable in current cold-atom experiments, in view of simulating non-Abelian gauge fields and topological insulators/superfluids with ultracold atoms.

Topologically universal spectral hierarchies of quasiperiodic systems

Topological properties of energy spectra of general one-dimensional quasiperiodic systems, describing also Bloch electrons in magnetic fields, are studied for an infinity of irrational modulation frequencies corresponding to irrational numbers of flux quanta per unit cell. These frequencies include well-known ones considered in works on Fibonacci quasicrystals. It is shown that the spectrum for any such frequency exhibits a self-similar hierarchy of clusters characterized by universal (system-independent) values of Chern topological integers which are exactly determined. The cluster hierarchy provides a simple and systematic organization of all the spectral gaps, labeled by universal topological numbers which are exactly determinable, thus avoiding their numerical evaluation using rational approximants of the irrational frequency. These numbers give both the quantum Hall conductance of the system and the winding number of the edge-state energy traversing a gap as a Bloch quasimomentum is varied.

Localization in One-dimensional Quasi-periodic Nonlinear Systems

To investigate localization in one-dimensional quasi-periodic nonlinear systems, we consider the Schrodinger equation $${\rm i}\dot{q}_n+\epsilon(q_{n+1}+q_{n-1})+V(n\tilde{\alpha}+x)q_n+ |q_n|^2q_n=0,\quad n\in\mathbb{Z},$$ as a model, with V a nonconstant real-analytic function on $${\mathbb{R}/\mathbb{Z}}$$ , and $${\tilde{\alpha}}$$ satisfying a certain Diophantine condition. It is shown that, if $${\epsilon}$$ is sufficiently small, then for a.e. $${x\in\mathbb{R}/\mathbb{Z}}$$ , dynamical localization is maintained for “typical” solutions in a quasi-periodic time-dependent way.