We study out of time ordered correlators (OTOC) in a free fermionic model with a quasi-periodic potential. This model is equivalent to the Aubry-André model and features a phase transition from an extended phase to a localized phase at a non-zero value of the strength of the quasi-periodic potential. We investigate five different time-regimes of interest for out of time ordered correlators; early, wavefront, $x=v_Bt$, late time equilibration and infinite time. For the early time regime we observe a power law for all potential strengths. For the time regime preceding the wavefront we confirm a recently proposed universal form and use it to extract the characteristic velocity of the wavefront for the present model. A Gaussian waveform is observed to work well in the time regime surrounding $x=v_Bt$. Our main result is for the late time equilibration regime where we derive a finite time equilibration bound for the OTOC, bounding the correlator's distance from its late time value. The bound impose strict limits on equilibration of the OTOC in the extended regime ans is valid not only for the Aubry-André model but for any quadratic model model. Finally, momentum out of time ordered correlators for the Aubry-André model are studied where large values of the OTOC are observed at late times at the critical point.
Non-Hermitian extensions of the Anderson and Aubry-André-Harper models are attracting a considerable interest as platforms to study localization phenomena, metal-insulator and topological phase transitions in disordered non-Hermitian systems. Most of available studies, however, resort to numerical results, while few analytical and rigorous results are available owing to the extraordinary complexity of the underlying problem. Here we consider a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-André-Harper model, undergoing a topological metal-insulator phase transition, and provide…
We study the high temperature transport behavior of the Aubry-André-Harper (AAH) model, both in the isolated thermodynamic limit and in the open system. At the critical point of the AAH model, we find hints of super-diffusive behavior from the scaling of spread of an initially localized wavepacket. On the other hand, when connected to two baths with different chemical potentials at the two ends, we find that the critical point shows clear sub-diffusive scaling of current with system size. We provide an explanation of this by showing that the current scaling with system-size is entirely governe…
As opposed to random disorder, which localizes single-particle wave-functions in 1D at arbitrarily small disorder strengths, there is a localization-delocalization transition for quasi-periodic disorder in the 1D Aubry-André model at a finite disorder strength. On the single-particle level, many properties of the ground-state critical behavior have been revealed by applying a real-space renormalization-group scheme; the critical properties are determined solely by the continued fraction expansion of the incommensurate frequency of the disorder. Here, we investigate the many-particle localizati…
A recent experiment by P. Bordia et al. (Periodically Driving a Many Body Localized Quantum System, Nat Phys, Jan 2017) has demonstrated that periodically modulating the potential of a localised many-body quantum system described by the Aubry-André Hamiltonian with on-site interactions can lead to a many-body localisation-delocalisation transition, provided the modulation amplitude is big enough. Here, we consider the noninteracting counterpart of that model in order to explore its phase diagram as a function of the strength of the disordered potential, the driving frequency and its amplitude.…
In this paper, we explore quantum criticality in the disordered Aubry-André (AA) model. For the pure AA model, it is well-known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $Δ$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $ξ$ scales with $Δ$ as $ξ\proptoΔ^{-ν_Δ}$ with $ν_Δ$ a new critical exponent, which is estimated to be $ν_Δ\approx0.46$. This value is r…
We numerically study the evolution of spectral correlations in the entanglement Hamiltonian (EH) of non-interacting fermions in the Aubry-André-Harper (AAH) model. We analyze the time evolution of the EH spectrum in a nonequilibrium setting by studying several quantities: spectral distribution, level statistics, entanglement entropy, and spectral form factor (SFF) in the context of the delocalization-localization transition in the AAH model. It is observed that the SFF of the entanglement spectrum in the delocalized phase and at the phase-transition point evolves in three-time intervals. We ma…
We study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-André potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-André model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown…
Here we study the phase diagram of the Aubry-Andre-Harper model in the presence of strong interactions as the strength of the quasiperiodic potential is varied. Previous work has established the existence of many-body localized phase at large potential strength; here, we find a rich phase diagram in the delocalized regime characterized by spin transport and unusual correlations. We calculate the non-equilibrium steady states of a boundary-driven strongly interacting Aubry-Andre-Harper model by employing the time-evolving block decimation algorithm on matrix product density operators. From thes…
The Aubry-André 1D lattice model describes a particle hopping in a pseudo-random potential. Depending on its strength $λ$, all eigenstates are either localized ($λ>1$) or delocalized ($λ<1$). Near the transition, the localization length diverges like $ξ\sim(λ-1)^{-ν}$ with $ν=1$. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then -- in analogy with the Kibble-Zurek mechanism -- it enters the delocalized phase having finite localization length $\hatξ\simτ_Q^{ν/(1+zν)}$. Here $τ_Q$ is ramp/…
We discuss and compare two recently proposed toy models for anomalous transport and Griffiths effects in random systems near the Many-Body Localization transitions: the random dephasing model, which adds thermal inclusions in an Anderson Insulator as local Markovian dephasing channels that heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach which models them in terms of ensembles of random regular graphs. For these two settings we discuss and compare transport and dissipative properties and their statistics. We show that both types of dissipation lead to similar Grif…
Fibonacci number systems and the localization criterion in the many-body Aubry-André model
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；MTA-BME Lendület ”Momentum” Open Quantum Systems Research Group, Institute of Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学物理研究所 MTA-BME 杠杆力矩开放量子系统研究组）；Institute for Solid State Physics and Optics, HUN-REN Wigner Research Centre for Physics（匈牙利研究与教育网络维格纳物理研究中心凝聚态物理与光学研究所）
We uncover a systematic structure in the single particle phase-diagram of the quasiperiodic Aubry-André-Harper(AAH) model with power-law hoppings ($\sim \frac{1}{r^σ}$) when the quasiperiodicity parameter is chosen to be a member of the `metallic mean family' of irrational Diophantine numbers. In addition to the fully delocalized and localized phases we find a co-existence of multifractal (localized) states with the delocalized states for $σ<1$ ($σ>1$). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical pr…
We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-André potential and a time-periodic hopping amplitude as a function of the drive frequency $ω_D$ using exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate $ω_D$, the Floquet eigenstates exhibit non-trivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a g…
Motivated by the recent experiment by Bordia et al [Nat. Phys. 13, 460 (2017)], we study single particle delocalization phenomena of Aubry-André (AA) model subjected to periodic drives. In two distinct cases we construct an equivalent classical description to illustrate that the drive induced delocalization phenomena stems from an instability and onset of chaos in the underlying dynamics. In the first case we analyze the delocalization and the thermalization in a time modulated AA potential with respect to driving frequency and demonstrate that there exists a threshold value of the amplitude o…
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 have investigated scaling properties of the Aubry-André model and related one-dimensional quasiperiodic Hamiltonians near their localisation transitions. We find numerically that the scaling of characteristic energies near the ground state, usually captured by a single dynamical exponent, does not obey a power law relation. Instead, the scaling behaviour depends strongly on the correlation length in a manner governed by the continued fraction expansion of the irrational number $β$ describing incommensurability in the system. This dependence is, however, found to be universal between a range…
Equilibrium and nonequlibrium scaling behaviors of localization transition in a non-Hermitian Aubry-André model with onsite gain and loss
The School of Mathematics and Physics, Jiangsu University of Technology（江苏理工学院数学与物理学院）；The Jiangsu Key Laboratory of Clean Energy Storage and Conversion, Jiangsu University of Technology（江苏理工学院清洁能源存储与转化江苏省重点实验室）；Jiangsu Physical Science Research Center（江苏物理科学研究中心）
