e-Harper Model Research Articles

Multiple localization transitions and novel quantum phases induced by a staggered on-site potential

We propose a one-dimensional generalized Aubry-Andr\'e-Harper (AAH) model with off-diagonal hopping and staggered on-site potential. We find that the localization transitions could be multiple reentrant with the increasing of staggered on-site potential. The multiple localization transitions are verified by the quantum static and dynamic measurements such as the inverse or normalized participation ratios, fractal dimension, and survival probability. Based on the finite-size scaling analysis, we also obtain an interesting intermediate phase where the extended, localized, and critical states are coexistent in certain regimes of model parameters. These results are quite different from those in the generalized AAH model with off-diagonal hopping, and can help us to find novel quantum phases and new localization phenomena in the disordered systems.

Power law hopping of single particles in one-dimensional non-Hermitian quasicrystals

In this paper, a non-Hermitian Aubry-Andr\'e-Harper model with power law hoppings ($1/{s}^{a}$) and quasiperiodic parameter $\ensuremath{\beta}$ is studied, where $a$ is the power law index, $s$ is the hopping distance, and $\ensuremath{\beta}$ is a member of the metallic mean family. We find that the number of the quasiperiodic parameter $\ensuremath{\beta}$-dependent regimes depends on the strength of the non-Hermiticity. Under a particularly weak non-Hermitian effect, there preserves ${P}_{\ensuremath{\ell}=1,2,3,4}$ regimes where the fraction of ergodic eigenstates is $\ensuremath{\beta}$ dependent as ${\ensuremath{\beta}}^{\ensuremath{\ell}}L$ ($L$ is the system size), similar to those in the Hermitian case. However, ${P}_{\ensuremath{\ell}}$ regimes are ruined by the strong non-Hermitian effect. Moreover, by analyzing the fractal dimension, we find that there are two types of edges aroused by the power law index $a$ in the single-particle spectrum, i.e., an ergodic-to-multifractal edge for the long-range hopping case ($a<1$), and an ergodic-to-localized edge for the short-range hopping case ($a>1$). Meanwhile, the existence of these two types of edges is found to be robust against the non-Hermitian effect. By employing the Simon-Spence theory, we analyzed the absence of the localized states for $a<1$. For the short-range hopping case, with the Avila's global theory and the Sarnak method, we consider a specific example with $a=2$ to reveal the presence of the intermediate phase and to analytically locate the intermediate regime and the ergodic-to-localized edge, which are self-consistent with the numerically results.

Topological invariants for interacting systems: From twisted boundary conditions to center-of-mass momentum

Beyond the well-known topological band theory for single-particle systems, it is a great challenge to characterize the topological nature of interacting multiparticle quantum systems. Here, we uncover the relation between topological invariants defined through the twisted boundary condition (TBC) and the center-of-mass (c.m.) momentum state in multiparticle systems. We find that the Berry phase defined through the TBC can be equivalently obtained from the multiparticle Wilson loop formulated by c.m. momentum states. As the Chern number can be written as the winding of the Berry phase, we consequently prove the equivalence of Chern numbers obtained via TBC and c.m. momentum state approaches. As a proof-of-principle example, we study topological properties of the Aubry-Andr\'e-Harper model. Our numerical results show that the TBC approach and c.m. approach are well consistent with each other for both the many-body case and the few-body case. Our work lays a concrete foundation and provides insights for exploring multiparticle topological states.

Diagnostics of nonergodic extended states and many body localization proximity effect through real-space and Fock-space excitations

We provide real-space and Fock-space (FS) characterizations of ergodic, nonergodic extended (NEE) and many-body localized (MBL) phases in an interacting quasiperiodic system, namely, the generalized Aubry-Andr\'e-Harper model, which possesses a mobility edge in the noninteracting limit. We show that a mobility edge in the single-particle (SP) excitations survives even in the presence of interaction in the NEE phase. In contrast, all single-particle excitations get localized in the MBL phase due to the MBL proximity effect. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the nonlocal FS propagators. Based on a finite-size scaling analysis of the typical local self-energy across the NEE to ergodic transition, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the distribution of local self-energy and the decay of the nonlocal propagator in the FS, whereas the typical local FS self-energy cannot tell them apart.

Topological phonon polariton enhanced radiative heat transfer in bichromatic nanoparticle arrays mimicking Aubry-André-Harper model

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, the NP chain can be regarded as an extension of the off-diagonal Aubry-Andr\'e-Harper (AAH) model. By calculating the eigenstate spectra with respect to the modulation phase that creates a synthetic dimension, we demonstrate that under this type of modulation the chain supports nontrivial topological modes localized over the boundaries, since the present system inherits the topological property of two-dimensional integer quantum Hall systems. In this circumstance the gap-labeling theorem and corresponding Chern number can be used to characterize the features of band gaps and topological edge modes. Based on many-body radiative heat transfer theory for a set of dipoles, we theoretically show the presence of topological gaps and midgap TPhPs can substantially enhance radiative heat transfer for an array much longer than the wavelength of radiation. We show how the modulation phase that acts as the synthetic dimension can tailor the radiative heat transfer rate by inducing or annihilating topological modes. We also discuss the role of dissipation in the enhancement of radiative heat transfer. These findings therefore provide a fascinating route for tailoring near-field radiative heat transfer based on the concept of topological physics.

Localization transitions and winding numbers for non-Hermitian Aubry-André-Harper models with off-diagonal modulations

We study localization and topological properties, as well as the fate of the critical phase, of non-Hermitian generalizations of the Aubry-Andr\'e-Harper model with both on-site and off-diagonal incommensurate modulations. Non-Hermiticity arises from nonreciprocal hopping and a complex phase in the potential. In the absence of nonreciprocal hopping, we compute analytically the localization length of single-particle states by applying Avila's global theory. The system has the same phase diagram as Hermitian cases, except that the complex phase renormalizes the strength of the potential. In the presence of nonreciprocal hopping, the phase diagram is analytically determined by a similarity transformation. Due to the presence of the skin effect, induced by nonreciprocity, the skin phase turns into extended and critical phases when the boundary condition changes from open to periodic, while states are localized asymmetrically in the boundary-independent localized phase. The nonreciprocal hopping is in favor of the critical phase under a periodic boundary condition. The spectra are complex, and loops always exist. A winding number is not a proper indicator of the presence of loops, but the topological transition is in agreement with the localization transition.

Floquet band engineering with Bloch oscillations

This work provides a convenient and powerful means towards the engineering of Floquet bands via Bloch oscillations, by adding a tilted linear potential to periodically driven lattice systems. The added linear field not only restricts the spreading of a time-evolving wavepacket but also, depending on the ratio between the Bloch oscillation frequency and the modulation frequency of the periodic driving, dramatically modifies the band profile and topology. Specifically, we consider a driven Aubry-Andr\'e-Harper model as a working example, in the presence of a linear field. Almost flat Floquet bands or Floquet bands with large Chern numbers due to the interplay between the periodic driving and Bloch oscillations can be obtained, with the band structure and topology extensively tunable by adjusting the ratio of two competing frequencies. To confirm our finding, we further execute the Thouless pumping of one and two interacting bosons in such a lattice system and establish its connection with the topological properties of single- and two-particle Floquet bands.

Controllable topological edge mode in an optically excited exciton-polariton lattice

We propose an all-optical scheme of topological lasing and switching based on the Aubry-Andr\'e-Harper (AAH) model of an exciton-polariton chain. We theoretically show that the phase parameter of the optical potential, with a tunable effective quasimomentum, allows the system to exhibit nontrivial topological properties which are attributed to higher dimensions. The topological modes emerging within the bulk band gaps are spatially localized at the edges of the polariton lattice, and their topological properties are characterized by the nonzero Chern numbers of the bulk bands. Polariton lasing in topological edge modes exhibits a higher efficiency and better robustness than in bulk modes, and can be switched between two opposite edges of the lattice by nonresonant excitation, which paves a way for topologically protected optical circuits.

Breakdown of the correspondence between the real-complex and delocalization-localization transitions in non-Hermitian quasicrystals

The correspondence between the real-complex transition in energy and delocalization-localization transition is well-established in a class of Aubry-Andr'e-Harper model with exponential non-Hermitian on-site potentials. In this paper, we study a generalized Aubry-Andr'e model with off-diagonal modulation and non-Hermitian on-site potential. We find that, when there exists an incommensurate off-diagonal modulation, the correspondence breaks down, although the extended phase is maintained in a wide parameter range of the strengths of the on-site potential and the off-diagonal hoppings. An additional intermediate phase with a non-Hermitian mobility edge emerges when the off-diagonal hoppings become commensurate. This phase is characterized by the real and complex sections of the energy spectrum corresponding to the extended and localized states. In this case, the aforementioned correspondence reappears due to the recovery of the PT-symmetry.

Quantum phase transition between hyperuniform density distributions

We study an electron distribution under a quasiperiodic potential in light of hyperuniformity, aiming to establish a classification and analysis method for aperiodic but orderly density distributions realized in, e.g., quasicrystals. Using the Aubry-Andre-Harper model, we first reveal that the electron-charge distribution changes its character as the increased quasiperiodic potential alters the eigenstates from extended to localized ones. While these changes of the charge distribution are characterized by neither multifractality nor translational-symmetry breaking, they are characterized by hyperuniformity class and its order metric. We find a nontrivial relationship between the density of states at the Fermi level, a charge-distribution histogram, and the hyperuniformity class. The change to a different hyperuniformity class occurs as a first-order phase transition except for an electron-hole symmetric point, where the transition is of the third order. Moreover, we generalize the hyperuniformity order metric to a function, to capture more detailed features of the density distribution, in some analogy with a generalization of the fractal dimension to a multifractal one.

Persistent homology analysis of a generalized Aubry-André-Harper model

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-Andr\'e-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 critical phases. However, we find that the persistent entropy also clearly distinguishes ordered from disordered regimes of the model. The persistent homology approach can be applied to both the energy eigenstates and the wave packet propagation dynamics.

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.

Localization and multifractal properties of the long-range Kitaev chain in the presence of an Aubry-André-Harper modulation

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-Andr\'e-Harper model in the presence of long-range hopping and superconducting pairing amplitudes. These amplitudes decay with power-law exponents $\xi$ and $\alpha$ 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 $\alpha<1$ and energy-dependent transitions from ergodic to localized states with an intermediate multifractal region for $\alpha>1$. The size of the intermediate multifractal region depends not only on the value of the superconducting pairing term $\Delta$, 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.

Dimerization-induced mobility edges and multiple reentrant localization transitions in non-Hermitian quasicrystals

Non-Hermitian effects could create rich dynamical and topological phase structures. In this work, we show that the collaboration between lattice dimerization and non-Hermiticity could generally bring about mobility edges and multiple localization transitions in one-dimensional quasicrystals. Non-Hermitian extensions of the Aubry-Andr\'e-Harper model with staggered on-site potential and dimerized hopping amplitudes are introduced to demonstrate our results. Reentrant localization transitions due to the interplay between quasiperiodic gain/loss and lattice dimerization are found. Quantized winding numbers are further adopted as topological invariants to characterize transitions among phases with distinct spectrum and transport nature. Our study thus enriches the family of non-Hermitian quasicrystals by incorporating effects of lattice dimerization, and offering a convenient way to modulate localization transitions and mobility edges in non-Hermitian systems.

Dynamics of spectral correlations in the entanglement Hamiltonian of the Aubry-André-Harper model

We numerically study the evolution of spectral correlations in the entanglement Hamiltonian (EH) of non-interacting fermions in the Aubry-Andr\'e-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 make a systematic study of the emergence of these three timescales for various initial states and find that the number of time intervals remains three unless the Hamiltonian is tuned in the localized phase or when the initial state is maximally entangled, then there is a featureless time evolution. We find a broad direct correlation between the entanglement entropy and the length of the ramp of the SFF. We also find that in the delocalized phase the spectral correlations are stronger in the center of the spectrum and grow progressively weaker as more and more of the spectrum is considered.

Dynamical transitions in aperiodically kicked tight-binding models

If a localized quantum state in a tight-binding model with structural aperiodicity is subject to noisy evolution, then it is generally expected to result in diffusion and delocalization. In this work, it is shown that the localized phase of the kicked Aubry-Andr\'e-Harper (AAH) model is robust to the effects of noisy evolution, for long times, provided that some kick is delivered once every time period. However, if strong noisy perturbations are applied by randomly missing kicks, a sharp dynamical transition from a ballistic growth phase at initial times to a diffusive growth phase for longer times is observed. Such sharp transitions are seen even in translationally invariant models. These transitions are related to the existence of flat bands, and using a 2-band model we obtain analytical support for these observations. The diffusive evolution at long times has a mechanism similar to that of a random walk. The time scale at which the sharp transition takes place is related to the characteristics of noise. Remarkably, the wavepacket evolution scales with the noise parameters. Further, using kick sequence modulated by a 'coin toss', it is argued that the correlations in the noise are crucial to the observed sharp transitions.

Dynamics of a quantum phase transition in the Aubry-André-Harper model with p -wave superconductivity

We investigate the nonequilibrium dynamics of the one-dimension Aubry-Andr\'e-Harper model with $p$-wave superconductivity by changing the potential strength with slow and sudden quench. Firstly, we study the slow quench dynamics from the localized phase to the 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 $z\ensuremath{\nu}$, giving the correlation length $\ensuremath{\nu}=1$ and dynamical exponent $z=1.373\ifmmode\pm\else\textpm\fi{}0.023$, which are different from the Aubry-Andr\'e model. Secondly, we also study the sudden quench dynamics between three different phases: localized phase, critical phase, and extended phase. In the limit of $V=0$ and $V=\ensuremath{\infty}$, we analytically study the sudden quench dynamics via the Loschmidt echo. The results suggest that, if the initial state and the post-quench Hamiltonian are in different phases, the Loschmidt echo vanishes at some time intervals. Furthermore, we found that, if the initial value is in the critical phase, the direction of quenching is the same as one of the two limits mentioned before, and similar behaviors will occur.

Phase transitions in a non-Hermitian Aubry-André-Harper model

The Aubry-Andr\'e-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 the dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent $\ensuremath{\delta}$ of wave-packet spreading, with $\ensuremath{\delta}=1$ in the delocalized phase $V&lt;{V}_{c}$ (ballistic transport), $\ensuremath{\delta}\ensuremath{\simeq}1/2$ at the critical point $V={V}_{c}$ (diffusive transport), and $\ensuremath{\delta}=0$ in the localized phase $V&gt;{V}_{c}$ (dynamical localization). However, the phase transition turns out to be smooth when one measures, as a dynamical variable, the speed $v(V)$ of excitation transport in the lattice, which is a continuous function of potential amplitude $V$ and vanishes as the localized phase is approached. Here we consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, in which hopping along the lattice is asymmetric, and show that the dynamical localization-delocalization transition is discontinuous, not only in the diffusion exponent $\ensuremath{\delta}$, but also in the speed $v$ of ballistic transport. This means that even very close to the spectral phase transition point, rather counterintuitively, ballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballistic velocity can increase as $V$ is increased above zero, i.e., surprisingly, disorder in the lattice can result in an enhancement of transport.

Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model

We uncover a systematic structure in the single particle phase-diagram of the quasiperiodic Aubry-Andr\'e-Harper(AAH) model with power-law hoppings ($\sim \frac{1}{r^\sigma}$) 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 $\sigma<1$ ($\sigma>1$). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical property of the `metallic mean family'. The entanglement entropy of the noninteracting many-body ground states respects the area-law if the Fermi level belongs in the localized regime while logarithmically violating it if the Fermi-level belongs in the delocalized or multifractal regimes. The prefactor of logarithmically violating term shows interesting behavior in different phases. Entanglement entropy shows the area-law even in the delocalized regime for special filling fractions, which are related to the metallic means.

Self-consistent theory of mobility edges in quasiperiodic chains

We introduce a self-consistent theory of mobility edges in nearest-neighbour tight-binding chains with quasiperiodic potentials. Demarcating boundaries between localised and extended states in the space of system parameters and energy, mobility edges are generic in quasiperiodic systems which lack the energy-independent self-duality of the commonly studied Aubry-Andr\'e-Harper model. The potentials in such systems are strongly and infinite-range correlated, reflecting their deterministic nature and rendering the problem distinct from that of disordered systems. Importantly, the underlying theoretical framework introduced is model-independent, thus allowing analytical extraction of mobility edge trajectories for arbitrary quasiperiodic systems. We exemplify the theory using two families of models, and show the results to be in very good agreement with the exactly known mobility edges as well numerical results obtained from exact diagonalisation.