Due to the topological nature of the Aubry-Andr\'e-Harper (AAH) model, interesting 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 walker is mainly determined by the IPRs of the topologically protected edge states. More interestingly, the edge states are shown to have an evident ``repulsion'' effect on quantum walkers initiated from the lattice sites inside the bulk. Furthermore, we examine the role of nearest-neighbor interaction on the quantum walks of two identical fermions. Clear enhancement of the repulsion effect by strong interaction has been shown.
Due to the topological nature of the Aubry-Andr'e-Harper (AAH) model, interesting 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 walker is mainly determined by the IPRs of the topologically protected edge states. More interestingly, the edge states are shown to have an evident ``repulsion'' effect on quantum walkers initiated from the lattice sites inside the bulk. Furthermore, we examine the role of nearest-neighbor interaction on the quantum walks of two identical fermions. Clear enhancement of the repulsion effect by strong interaction has been shown.
Abstract We study the quantum walk on the off-diagonal Aubry-André-Harper (AAH) lattice with periodic modulation using a digital quantum computer. We investigate various initial states at the single-particle level, considering different hopping modulation strengths and phase factors. Initiating the quantum walk with a particle at the lattice edge reveals the robustness of the edge state, attributed to the topological nature of the AAH model, and displays the influence of the phase factor on this edge state. On the other hand, when the quantum walk begins with a particle in the lattice bulk, we observe a repulsion of the bulk walker from the edge, especially under strong hopping modulation. Furthermore, we extend our investigation to the quantum walk of two particles with nearest-neighbour (NN) interaction. We show the repulsion effect in the quantum walk when two walkers originate from the edge and bulk of the lattice due to the interaction. Additionally, when two particles are positioned at nearest-neighbor sites with strong hopping modulation, they unexpectedly form a local bound state at very small interaction strength, highlighting the unique interplay between hopping modulation and interaction in our quantum walk setup. We analyze these phenomena by examining physical quantities such as density evolution, two-particle correlation, and participation entropy, and discuss their potential applications in quantum technologies.
Non-Hermitian extensions of the Aubry-André-Harper (AAH) model reveal a rich variety of phase transitions arising from the interplay of quasiperiodicity and nonhermiticity. Despite their theoretical significance, experimental explorations remain challenging due to intricacies in realizing controlled nonhermiticity. Here, we present the first experimental realization of the unitary almost-Mathieu operator that simulates the AAH model by employing single-photon quantum walks. We systematically observe a phase transition between localized and delocalized regimes in the Hermitian limit. Subsequently, by introducing a nonreciprocal hopping, we experimentally probe the parity-time symmetry-breaking transition that is characterized by the emergence of complex quasienergies. Moreover, we identify a novel spectral transition exclusive to discrete-time settings, beyond which the real parts of all quasienergies vanish. These results clarify the interplay between localization, symmetry breaking, and topology in non-Hermitian quasicrystals, thereby paving the way for future exploration of synthetic quantum matter.
Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.
We investigate the one-dimensional longer-range Kitaev chain with an on-site complex Aubry-Andr\'e-Harper (AAH) potential. For the incommensurate (commensurate) case, by calculating the real-space (Bloch) winding number and energy spectrum we discover the AAH potential-induced phases and present the related phase diagrams. We also find that the imaginary phase shift in AAH potentials can suppress these topological phases for both cases. For the incommensurate case, the inverse participate ratios of the eigenstates with the smallest ∣E∣s indicate the existence of the incommensurate AAH potential-induced phases, and the result of the mean inverse participate ratio shows that these incommensurate AAH potential-induced phases are not always strongly localized.
We investigate the dynamics of two identical spinless fermions on a one-dimensional lattice with open boundary conditions (OBC), subject to quasiperiodic long-range interactions. Using numerical exact diagonalization (ED), we study this non-integrable system as a continuous-time quantum walk and uncover a robust correlated dynamical regime. This regime, characterized by an approximately constant inter-particle distance, emerges under sufficiently strong quasiperiodic modulation of the long-range interactions. Further, the study shows that the behavior is determined by the nature of the interaction and the choice of boundary condition. Notably, by tuning the phase of the quasiperiodic modulation, we observe three distinct manifestations of this phenomenon: localization, nearest-neighbor separation oscillations, and next-nearest-neighbor separation transitions -- each arising for specific initial separations. Furthermore, we identify the suppression of entanglement entropy in the system, including instances of oscillatory behavior. Our results highlight how quasiperiodic long-range interactions shape few-body quantum dynamics.
We investigate the quantum transport of delocalized states in continuous-time quantum walks (CTQWs) on a one-dimensional lattice containing a single defect. The defect is modeled by assigning complex-valued hopping amplitudes to the edges that connect the site corresponding to the mean position of the delocalized initial state to its nearest neighbors. We find that this single defective site is sufficient to enhance the ballistic spreading of an initially Gaussian wave packet. Extending these results, we implement a time-dependent alternation protocol between two distinct defect configurations, each individually yielding poor propagation of the state. The combination of these two unfavorable configurations improves the transport efficiency of the quantum walker, revealing a manifestation of Parrondo's paradox in CTQWs with delocalized initial states. This study provides insights into the role of complex-phase defects and time-dependent protocols in CTQWs, demonstrating that the interplay between quantum interference and graph engineering can effectively enhance quantum transport in discrete lattices.