In the presence of quasiperiodic potentials, the celebrated Kitaev chain presents an intriguing phase diagram with ergodic, localized and and multifractal states. In this work, we generalize these results by studying the localization properties of the Aubry-André-Harper model in the presence of long-range hopping and superconducting pairing amplitudes. These amplitudes decay with power-law exponents $ξ$ and $α$ respectively. To this end, we review and compare a toolbox of global and local characterization methods in order to investigate different types of transitions between ergodic, localized and multifractal states. We report energy-dependent transitions from ergodic to multifractal states for pairing terms with $α<1$ and energy-dependent transitions from ergodic to localized states with an intermediate multifractal region for $α>1$. The size of the intermediate multifractal region depends not only on the value of the superconducting pairing term $Δ$, but also on the energy band. The transitions are not described by a mobility edge, but instead we report hybridization of bands with different types of localization properties. This leads to coexisting multifractal regimes where fractal dimensions follow different distributions.
Higher-order topological phases of matter have been extensively studied in various areas of physics. While the Aubry-André-Harper model provides a paradigmatic example to study topological phases, it has not been explored whether a generalized Aubry-André-Harper model can exhibit a higher-order topological phenomenon. Here, we construct a two-dimensional higher-order topological insulator with chiral symmetry based on the Aubry-André-Harper model. We find the coexistence of zero-energy and nonzero energy corner-localized modes. The former is protected by the quantized quadrupole moment, while …
Mobility edge (ME), a critical energy separating localized and extended states in spectrum, is a central concept in understanding the localization physics. However, there are few models with exact MEs. In the paper, we generalize the Aubry-André-Harper model proposed in [Phys. Rev. Lett. 114, 146601 (2015)] and recently realized in [Phys. Rev. Lett. 126, 040603 (2021)], by introducing a relative phase in the quasiperiodic potential. Applying Avila's global theory we analytically compute localization lengths of all single-particle states and determine the exact expression of ME, which both sign…
We show the localization transition and its effect on two dynamical processes for an extended Aubry-André-Harper model with incommensurate on-site and hopping potentials. After specifying an extended Aubry-André-Harper model, we check the localization transition for all the eigenstates and eigenenergy band splitting behavior versus a system parameter. To examine the effect of localization transition on dynamical processes, firstly, the slowly pumping of the edge states are examined. In the dynamical processes, the system acts as conductor for the excitation in the nonlocal region and insulator…
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…
Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize phases of a generalized Aubry-André-Harper model. The persistent entropy and mean squared lifetime of features obtained using persistent homology behave similarly to conventional measures (Shannon entropy and inverse participation ratio) and can distinguish localized, extended, and crticial phases. However, we find that the persistent entropy also clearly disting…
We study non-Hermitian Aubry-André-Harper models with p-wave pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. By analysing the $\mathcal{PT}$ symmetry breaking, winding numbers of energy spectra, localization and fractal dimensions of states, and fate of Majorana fermions, a complete phase diagram on Anderson localization and topological phase transitions is obtained. In particular, the non-Hermitian topological nature of Anderson localization phase transitions from extended to critical and then to localized phases is identified, using both analytic…
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…
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…
We investigate the Anderson localization in non-Hermitian Aubry-André-Harper (AAH) models with imaginary potentials added to lattice sites to represent the physical gain and loss during the interacting processes between the system and environment. By checking the mean inverse participation ratio (MIPR) of the system, we find that different configurations of physical gain and loss have very different impacts on the localization phase transition in the system. In the case with balanced physical gain and loss added in an alternate way to the lattice sites, the critical region (in the case with p-…
We investigate the quantum criticality and universality in Aubry-André-Harper (AAH) model with $p$-wave superconducting pairing $Δ$ in terms of the generalized fidelity susceptibility (GFS). We show that the higher-order GFS is more efficient in spotlighting the critical points than lower-order ones, and thus the enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic systems. The GFS obeys power-law scaling for localization transitions and thus scaling properties of the GFS provide compelling values of critical expon…
The Aubry-André-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization phase transition at a finite critical value $V_c$ of the quasiperiodic potential amplitude $V$. In terms of dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent $δ$ of wave packet spreading, with $δ=1$ in the delocalized phase $V<V_c$ (ballistic transport), $δ\simeq 1/2$ at the critical point $V=V_c$ (diffusive transport), and $δ=0$ in the localized phase $V>V_c$ (dynamical local…
The Anderson localization phase transition in the Aubry-André-Harper (AAH) model with \textit{p}-wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolutions of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies …
We investigate the nonequilibrium dynamics of the one-dimension Aubry-André-Harper model with $p$-wave superconductivity by changing the potential strength with slow and sudden quench. Firstly, we study the slow quench dynamics from localized phase to critical phase by linearly decreasing the potential strength $V$. The localization length is finite and its scaling obeys the Kibble-Zurek mechanism. The results show that the second-order phase transition line shares the same critical exponent $zν$, giving the correlation length $ν=0.997$ and dynamical exponent $z=1.373$, which are different fro…
Topological phonon polaritons (TPhPs) are promising optical modes relevant in long-range radiative heat transfer, information processing and infrared sensing, whose topological protection is expected to enable their robust existence and transport. In this work we show that TPhPs can be supported in one-dimensional (1D) bichromatic silicon carbide nanoparticle (NP) chains, and demonstrate that they can considerably enhance radiative heat transfer for an array much longer than the wavelength of radiation. By introducing incommensurate or commensurate modulations on the interparticle distances, t…
Imaginary Gauge-steerable Edge Modes In Non-Hermitian Aubry-André-Harper Model
School of Science, Qingdao University of Technology（青岛理工大学理学院）；Center for Quantum Sciences and School of Physics, Northeast Normal University（东北师范大学量子科学中心与物理学院）
We have investigated scaling properties near the quantum critical point between the extended phase and the critical phase in the Aubry-André-Harper model with p-wave pairing, which have rarely been exploited as most investigations focus on the localization transition from the critical phase to the localized phase. We find that the spectrum averaged entanglement entropy and the generalized fidelity susceptibility act as eminent universal order parameters of the corresponding critical point without gap closing. We introduce a Widom scaling ansatz for these criticality probes to develop a unified…
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 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…
We investigate the nonequilbrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is dif…
We study an extended Aubry-Andr{é}-Harper model with simultaneous modulation of hopping, on-site potential, and $p$-wave superconducting pairing. For the case of commensurate modulation of $β= 1/2$ it is shown that the model hosts four different types of topological states: adiabatic cycles can be defined which pump particles, two types of Majorana fermions, or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multi-fractal analysis. In addition, we …
Spread complexity as a probe in generalized and long-range Aubry-Andre-Harper models
Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Homi Bhabha National Institute（霍米·巴哈国立研究所）；Instytut Fizyki Teoretycznej, Wydział Fizyki, Astronomii i Informatyki Stosowanej, Uniwersytet Jagielloński（雅盖隆大学物理、天文学与应用信息学院理论物理研究所）
We study the localization transitions for coupled one-dimensional lattices with quasiperiodic potential. Besides the localized and extended phases there is an intermediate mixed phase which can be easily explained decoupling the system so as to deal with effective uncoupled Aubry-André chains with different transition points. We clarify, therefore, the origin of such an intermediate phase finding the conditions for getting a uniquely defined mobility edge for such coupled systems. Finally we consider many coupled chains with an energy shift which compose an extension of the Aubry-André model i…
Generalized Aubry-André-Harper model with power-law quasiperiodic potentials
College of Science, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院）；Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT（工业和信息化部航空航天信息材料与物理重点实验室（南航））；Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University（山西大学理论物理研究所及量子光学与量子光学器件国家重点实验室）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）
We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbour coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-André and the Fibonacci models. We find that stronger many-body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation. As a result, the positi…
The single particle eigenstates of the Aubry-André-Harper model are known to show a delocalization-localization transition at a finite strength of the quasi-periodic disorder. In this work, we point out that an intimate relationship exists between the sub-band structure of the spectrum and transport properties of the model. To capture the transport properties we have not only used a variety of single-particle measures like inverse participation ratio, and von Neumann entropy, but also many-particle measures such as persistent current and its variance, and many body entanglement entropy. The ma…
We demonstrate the existence of generalized Aubry-André self-duality in a class of non-Hermitian quasi-periodic lattices with complex potentials. From the self-duality relations, the analytical expression of mobility edges is derived. Compared to Hermitian systems, mobility edges in non-Hermitian ones not only separate localized from extended states, but also indicate the coexistence of complex and real eigenenergies, making it possible a topological characterization of mobility edges. An experimental scheme, based on optical pulse propagation in synthetic photonic mesh lattices, is suggested …
The nearest-neighbor Aubry-André quasiperiodic localization model is generalized to include power-law translation-invariant hoppings $T_l\propto t/l^a$ or power-law Fourier coefficients $W_m \propto w/m^b$ in the quasi-periodic potential. The Aubry-André duality between $T_l$ and $W_m$ is manifest when the Hamiltonian is written in the real-space basis and in the Fourier basis on a finite ring. The perturbative analysis in the amplitude $t$ of the hoppings yields that the eigenstates remain power-law localized in real space for $a>1$ and are critical for $a_c=1$ where they follow the Strong Mu…
We study a class of off-diagonal quasiperiodic hopping models described by one-dimensional Su-Schrieffer-Heeger chain with quasiperiodic modulations. We unveil a general dual-mapping relation in parameter space of the dimerization strength $λ$ and the quasiperiodic modulation strength $V$, regardless of the specific details of the quasiperiodic modulation. Moreover, we demonstrated semi-analytically and numerically that under the specific quasiperiodic modulation, quantum criticality can emerge and persist in a wide parameter space. These unusual properties provides a distinctive paradigm comp…
We study a generalized Aubry-Andre model that obeys $\mathcal{PT}$-symmetry. We observe a robust $\mathcal{PT}$-symmetric phase with respect to system size and disorder strength, where all eigenvalues are real despite the Hamiltonian being non-hermitian. This robust $\mathcal{PT}$-symmetric phase can support an Anderson localization transition, giving a rich phase diagram as a result of the interplay between disorder and $\mathcal{PT}$-symmetry. Our model provides a perfect platform to study disorder-driven localization phenomena in a $\mathcal{PT}$-symmetric system.
In this paper, we investigate the driven dynamics of the localization transition in the non-Hermitian Aubry-André model with the periodic boundary condition. Depending on the strength of the quasi-periodic potential $λ$, this model undergoes a localization-delocalization phase transition. We find that the localization length $ξ$ satisfies $ξ\sim \varepsilon^{-ν}$ with $\varepsilon$ being the distance from the critical point and $ν=1$ being a universal critical exponent independent of the non-Hermitian parameter. In addition, from the finite-size scaling of the energy gap between the ground sta…
Off-diagonal Aubry-André (AA) model has recently attracted a great deal of attention as they provide condensed matter realization of topological phases. We numerically study a generalized off-diagonal AA model with p-wave superfluid pairing in the presence of both commensurate and incommensurate hopping modulations. The phase diagram as functions of the modulation strength of incommensurate hopping and the strength of the p-wave pairing is obtained by using the multifractal analysis. We show that with the appearance of the p-wave pairing, the system exhibits mobility-edge phases and critical p…
The analysis of level statistics provides a primary method to detect signatures of chaos in the quantum domain. However, for experiments with ion traps and cold atoms, the energy levels are not as easily accessible as the dynamics. In this work, we discuss how properties of the spectrum that are usually associated with chaos can be directly detected from the evolution of the number operator in the one-dimensional, noninteracting Aubry-André model. Both the quantity and the model are studied in experiments with cold atoms. We consider a single-particle and system sizes experimentally reachable.…
We study a one-dimensional quasiperiodic system described by the off-diagonal Aubry-André model and investigate its phase diagram by using the symmetry and the multifractal analysis. It was shown in a recent work ({\it Phys. Rev. B} {\bf 93}, 205441 (2016)) that its phase diagram was divided into three regions, dubbed the extended, the topologically-nontrivial localized and the topologically-trivial localized phases, respectively. Out of our expectation, we find an additional region of the extended phase which can be mapped into the original one by a symmetry transformation. More unexpectedly,…
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…
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 …
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 study the quantum localization phenomena of noninteracting particles in one-dimensional lattices based on tight-binding models with various forms of hopping terms beyond the nearest neighbor, which are generalizations of the famous Aubry-André and noninteracting Anderson model. For the case with deterministic disordered potential induced by a secondary incommensurate lattice (i.e. the Aubry-André model), we identify a class of self dual models, for which the boundary between localized and extended eigenstates are determined analytically by employing a generalized Aubry-André transformation.…
