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 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 study the effects of electron-electron interactions on the charge excitation spectrum of the spinful Su-Schrieffer-Heeger (SSH) model, a prototype of a one-dimensional bulk obstructed topological insulator. In view of recent progress in the fabrication of dopant-based quantum simulators we focus on experimentally detectable signatures of interacting topology in finite lattices. To this end we use Lanczos-based exact diagonalization to calculate the single-particle spectral function in real space which generalizes the local density of states to interacting systems. Its spatial and spectral resolution allows for the direct investigation and identification of edge states. By studying the noninteracting limit, we demonstrate that the topological in-gap states on the boundary are robust against both finite-size effects as well as random bond and onsite disorder which suggests the feasibility of simulating the SSH model in engineered dopant arrays in silicon. While edge excitations become zero-energy spinlike for any finite interaction strength, our analysis of the spectral function shows that the single-particle charge excitations are gapped out on the boundary. Despite the loss of topological protection, we find that these edge excitations are quasiparticlelike as long as they remain within the bulk gap. Above a critical interaction strength of ${U}_{c}\ensuremath{\approx}5t$ these quasiparticles on the boundary lose their coherence which is explained by the merging of edge and bulk states. This is in contrast to the many-body edge excitations which survive the limit of strong coupling, as established in the literature. Our findings show that for moderate repulsive interactions the nontrivial phase of the interacting SSH model can be detected through remnant signatures of topological single-particle states using single-particle local measurement techniques such as scanning tunneling spectroscopy.
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.
Model lattices such as the kagome and Lieb lattices have been widely investigated to elucidate the properties of interacting flat-band systems. While a quasicrystal does not have proper bands, the non-interacting density of states of several of them displays the typical signature of a flat band pinned at the Fermi level: a delta-function zero-energy peak. Here, we employ quantum Monte Carlo simulations to determine the effect of onsite repulsion on these quasicrystals. While global properties such as the antiferromagnetic structure factor and the specific heat behave similarly as in the case of periodic lattices undergoing a Mott transition, the behavior of the local density of states depends on the coordination number of the site. In particular, sites with the smallest coordination number, which give the dominant spectral-weight contribution to the zero-energy peak, are the ones most strongly impacted by the interaction. Besides establishing site-selective correlations in quasicrystals, our work also points to the importance of the real-space structure of flat bands in interacting systems.
Low-energy properties of half-filled Fibonacci Hubbard models are studied by weak-coupling renormalization group and density matrix renormalization group methods. In the case of diagonal modulation, weak Coulomb repulsion is irrelevant and the system behaves as a free Fibonacci chain, while for strong Coulomb repulsion the charge sector becomes a Mott insulator and the spin sector behaves as a uniform Heisenberg antiferromagnetic chain. The off-diagonal modulation always drives the charge sector to a Mott insulator and the spin sector to a Fibonacci antiferromagnetic Heisenberg chain.
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.