We investigate how a perpendicular space analysis of quasiperiodic systems may be used to gain insight into their topological properties. In particular, for the Fibonacci chain, we show that the number of times quantities such as the charge density and the entanglement entropy oscillate in perpendicular space is directly related to the integer topological labels of its spectral gaps. We show that these oscillations have two distinct interpretations, obtained by extrapolating results from the two extreme limits of the Fibonacci chain -- the valence bond picture in the strong modulation limit, and perturbation around the periodic chain in the weak modulation limit. This effect is found to remain robust at moderate interactions, as well as in the presence of disorder. It thus suggests a straightforward way to experimentally measure topological invariants of such systems without invoking phason flips.
We show how measuring real space properties such as the charge density in a quasiperiodic system can be used to gain insight into their topological properties. In particular, for the Fibonacci chain, we show that the total on-site charge oscillates when plotted in the appropriate coordinates, and the number of oscillations is given by the topological label of the gap in which the Fermi level lies. We show that these oscillations have two distinct interpretations, obtained by extrapolating results from the two extreme limits of the Fibonacci chain---the valence bond picture in the strong modulation limit, and perturbation around the periodic chain in the weak modulation limit. This effect is found to remain robust at moderate interactions, as well as in the presence of disorder. We conclude that experimental measurement of the real space charge distribution can yield information on topological properties in a straightforward way.
We study the topological properties of one-dimensional discrete-time quantum walks with Fibonacci quasiperiodic modulation. Spectral analysis under open boundary conditions reveals isolated edge modes that coexist at both zero and
π energies within bulk gaps. Using the mean chiral displacement (MCD) as a dynamical bulk probe, we obtain a fractal, butterfly-like phase diagram, indicating a nontrivial bulk topology. However, since the MCD involves wave-packet averaging, its direct correspondence with boundary-localized states remains ambiguous. To resolve this issue, we determine the integer topological invariant by employing the Schur function formalism, a scattering-based approach defined on the unit disk. This analysis identifies a topological phase with winding number
W=2 inside the ``wings'' of the butterfly diagram, demonstrating that the coexistence of zero- and
π-energy edge modes is an intrinsic bulk property. Moreover, we find that the topological index crucially depends on the phason degree of freedom: the winding number can take values of
W=0,
2, and even
4 depending on the local surface termination. This behavior originates from the surface impedance, which yields an effective winding number ranges from complete masking by reflective boundaries (
W=0) to enhancement through surface resonances (
W=4). Finally, we show that although individual surface terminations lead to anisotropic phase diagrams, their ensemble average restores the overall geometric structure observed in the MCD phase diagram. Our results establish a complete bulk-edge correspondence in quasiperiodic quantum walks and provide a guiding principle for surface topological engineering, where edge transport can be controlled purely through boundary manipulation.
Recently, the relation between one-dimensional (1D) quasicrystals and 2D topological insulators has been demonstrated theoretically and also experimentally by using optical waveguides.It has been shown that 1D quasicrystals can be classified in terms of topology for 2D integer quantum Hall systems.Such quasicrystals can be also realized in ultracold atoms loaded in optical superlattices.In this study, we consider a 1D Bose-Hubbard model in a quasiperiodic superlattice.It is known that a gap in the excitation spectrum is induced by the interaction, and the resulting Mott insulating phase is characterized by a nonzero Chern number.The system is called a topological Mott insulator.In the non-interacting case, topological equivalence between the Fibonacci quasicrystal and the Harper model is already known.However, it has not been clarified whether such topological equivalence exists in topological Mott insulators.We show numerically that the topological equivalence exists in a wide range of quasiperiodic Bose-Hubbard model.
We introduce a characterization of topological order based on bulk oscillations of the entanglement entropy and the definition of an ``entanglement gap,'' showing that it is generally applicable to pure and disordered quantum systems. Using exact diagonalization and the strong disorder renormalization group method, we demonstrate that this approach gives results in agreement with the use of traditional topological invariants, especially in cases where topological order is known to persist in the presence of off-diagonal bond disorder. The entanglement gap is then used to analyze classes of quantum systems with alternating bond types, allowing us to construct their topological phase diagrams. The validity of these phase diagrams is verified in limiting cases of dominant bond types, where the solution is known.
The non-Hermitian skin effect (NHSE), which drives bulk states toward system boundaries, modifies bulk-boundary correspondence and complicates the identification of topological edge modes. Although breaking translational symmetry is known to influence this behavior, a systematic comparison of different structural classes remains limited. Here we investigate periodic, random, and quasiperiodic (Fibonacci) systems using a one-dimensional non-Hermitian quantum walk model. By matching the local scattering parameters in a topologically nontrivial regime, we isolate the role of spatial structure in the presence of the NHSE. We find that periodic systems exhibit pronounced boundary accumulation of bulk states. Random systems suppress this accumulation through Anderson localization, but the topological gap becomes partially filled with localized in-gap states. In contrast, the Fibonacci quasiperiodic system suppresses large-scale boundary accumulation while maintaining a well-defined topological gap. Analysis of the wave functions suggests that the hierarchical quasiperiodic structure fragments bulk states across multiple length scales, thereby mitigating the NHSE. These results identify deterministic quasiperiodicity as a distinct structural regime for controlling non-Hermitian skin dynamics and isolating topological boundary modes.
Abstract We study non-local measures of spectral correlations and their utility in characterizing and distinguishing between the distinct eigenstate phases of quantum chaotic and many-body localized systems. We focus on two related quantities, the spectral form factor and the density of all spectral gaps, and show that they furnish unique signatures that can be used to sharply identify the two phases. We demonstrate this by numerically studying three one-dimensional quantum spin chain models with (i) quenched disorder, (ii) periodic drive (Floquet), and (iii) quasiperiodic detuning. We also clarify in what ways the signatures are universal and in what ways they are not. More generally, this thorough analysis is expected to play a useful role in classifying phases of disorder systems.