Real-space topological characterization of quasiperiodic quantum walks: Boundary-dependent phases and the Schur index

Abstract

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 $\pi$ 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 $\pi$-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.