Non-Hermitian quantum many-body systems are a fascinating subject to be explored. Using the generalized density matrix renormalisation group method and complementary exact diagonalization, we elucidate the many-body ground states and dynamics of a 1D interacting non-Hermitian Aubry-Andre-Harper model for bosons. We find stable ground states in the superfluid and Mott insulating regimes under wide range of conditions in this model. We reveal a skin superfluid state induced by the non-Hermiticity from the nonreciprocal hopping. We investigate the topology of the Mott insulating phase and find its independence of the non-Hermiticity. The topological Mott insulators in this non-Hermitian system are characterized by four equal Chern numbers and a quantized shift of biorthogonal many-body polarizations. Furthermore, we show generic asymmetric expansion and correlation dynamics in the system.
By introducing loss to one sublattice of a dimer chain described by the extended Aubry-André or Harper (AAH) model, we study the topological features including the edge states, spectrum and winding number of the chain. We find that the parameter region for the system to have real band-gap-closing is increased due to the loss, and the average displacement of the single excitation can still witness the topological features of the chain in the presence of loss. The robustness of the zero energy eigenstate against four kinds of disorders is also examined. A feasible experiment setup based on coupl…
Aubry-Andre Harper (AAH) lattice models, characterized by reflection-asymmetric, sinusoidally varying nearest-neighbor tunneling profile, are well-known for their topological properties. We consider the fate of such models in the presence of balanced gain and loss potentials $\pm iγ$ located at reflection-symmetric sites. We predict that these models have a finite $\mathcal{PT}$ breaking threshold only for {\it specific locations} of the gain-loss potential, and uncover a hidden symmetry that is instrumental to the finite threshold strength. We also show that the topological edge-states remain…
We study the damping dynamics of the single-particle correlation for an open system under periodic and aperiodic order, which is dominated by the Lindblad master equation. In the absence of the aperiodic order, the Liouvillian superoperator exhibits the non-Hermitian skin effect, which leads to unidirectional damping dynamics, dubbed as "chiral damping". Due to the non-Hermitian skin effect, the damping dynamics is boundary sensitive: The long-time damping of such open systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. We reveal the phase tr…
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…
Due to the topological nature of Aubry-André-Harper (AAH) model, exotic edge states have been found existing in one-dimensional periodic and quasiperiodic lattices. In this article, we investigate continuous-time quantum walks of identical particles initially located on either edge of commensurate AAH lattices in detail. It is shown that the quantum walker is delocalized among the whole lattice until the strength of periodic modulation is strong enough. The inverse participation ratios (IPRs) for all of the eigenstates are calculated. It is found that the localization properties of the quantum…
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 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 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-André-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 ar…
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…
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.
We study one-dimensional optical lattices described by generalized Aubry-André models that include both commensurate and incommensurate modulations of the hopping amplitude. This brings together two interesting features of this class of systems: Anderson localization and the existence of topological edge states. We follow changes of the single-particle energy spectrum induced by variations of the system parameters, with focus on the survival of topological states in the localized regime.
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 …
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 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…
Statistical delocalization in the Anyonic Aubry-André model
Donostia International Physics Center (DIPC)（多诺斯蒂亚国际物理中心）；University of the Basque Country (UPV/EHU)（巴斯克大学）；Technische Universität Berlin（柏林工业大学）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）
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…
In this work, we investigate the Anderson localization problems of the generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasi-periodic potential where the parameter $|α|\geq1$. The Lyapunov exponent $γ(E)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasi-periodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $λ-E$ plane. The critical region consists of critical states. In comparison with localized and extended states, the fluctuation of spatial extensions of the critical states is …
We study quench dynamics in an interacting spin chain with a quasi-periodic on-site field, known as the interacting Aubry-André model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to $L = 50$. We find that the choice of periodicity $Φ$ of the quasi-periodic field influences the dynamics. For $Φ= (\sqrt{5}-1)/2$ (the inverse golden ratio) and interaction $Δ= 1$, the model most frequently considered in the literature, we obtain the critical disorder $W_c = 4.8 \pm 0.5$ in units where the non-interacting transition is at …
We introduce and explore a family of self-dual models of single-particle motion in quasiperiodic potentials, with hopping amplitudes that fall off as a power law with exponent $p$. These models are generalizations of the familiar Aubry-Andre model. For large enough $p$, their static properties are similar to those of the Aubry-Andre model, although the low-frequency conductivity in the localized phase is sensitive to $p$. For $p \leq 2.1$ the Aubry-Andre localization transition splits into three transitions; two distinct intermediate regimes with both localized and delocalized states appear ne…
Whether disordered and quasiperiodic many-body quantum systems host a long-lived localized phase in the thermodynamic limit has been the subject of intense recent debate. While in one dimension substantial evidence for the existence of such a many-body localized (MBL) phase exists, the behavior in higher dimensions remains an open puzzle. In two-dimensional disordered systems, for instance, it has been argued that rare regions may lead to thermalization of the whole system through a mechanism dubbed the avalanche instability. In quasiperiodic systems, rare regions are altogether absent and the…
We study the effects of disorder on the topological Chern insulating phase in the Harper--Hofstadter--Hatsugai (HHH) model. The model with half flux has a bulk band gap and thus exhibits a nontrivial topological phase. We consider two typical types of disorder: on-site random disorder and the Aubry--Andre type quasi-periodic potential. Using the coupling matrix method, we clarify the global topological phase diagram in terms of the Chern number. The disorder modifies the gap closing behavior of the system. This modification induces the Chern insulating phase even in the trivial phase parameter…
We investigate a system of fermions trapped in a quasi-periodic potential from an open quantum system theory perspective, designing a protocol in which an impurity atom (a two level system) is coupled to a trapped fermionic cloud described by the non-interacting Aubry-André model. The Fermi system is prepared in a charge density wave state before it starts its relaxation. In this work we focus our attention on the time evolution of the impurity in such an out of equilibrium environment, and study whether the induced dynamics can be classified as Markovian or non-Markovian. We find how the loca…
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…
We theoretically study a one-dimensional (1D) mutually incommensurate bichromatic lattice system which has been implemented in ultracold atoms to study quantum localization. It has been universally believed that the tight-binding version of this bichromatic incommensurate system is represented by the well-known Aubry-Andre model. Here we establish that this belief is incorrect and that the Aubry-Andre model description, which applies only in the extreme tight-binding limit of very deep primary lattice potential, generically breaks down near the localization transition due to the unavoidable ap…
We study the localization properties of weakly interacting Bose gas in a quasiperiodic potential commonly known as Aubry-André model. Effect of interaction on localization is investigated by computing the `superfluid fraction' and `inverse participation ratio'. For interacting Bosons the inverse participation ratio increases very slowly after the localization transition due to `multisite localization' of the wave function. We also study the localization in Aubry-André model using an alternative approach of classical dynamical map, where the localization is manifested by chaotic classical dynam…
We investigate the nature of the superfluid-insulator quantum phase transition driven by disorder for non-interacting ultracold atoms on one-dimensional lattices. We consider two different cases: Anderson-type disorder, with local energies randomly distributed, and pseudo-disorder due to a potential incommensurate with the lattice, which is usually called the Aubry-André model. A scaling analysis of numerical data for the superfluid fraction for different lattice sizes allows us to determine quantum critical exponents characterizing the disorder-driven superfluid-insulator transition. We also …
We discuss the systematic engineering of quasicrystals in open quantum systems where quasiperiodicity is introduced through purely dissipative processes. While the resulting short-time dynamics is governed by non-Hermitian variants of the Aubry-Andre-Harper model, we demonstrate how phases and phase transitions pertaining to the non-Hermitian quasicrystals fundamentally change the long-time, steady-state-approaching dynamics under the Lindblad master equation. Our schemes are based on an exact mapping between the eigenspectrum of the Liouvillian superoperator with that of the non-Hermitian Ham…
