Magnetic-Field Tunable M\"{o}bius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals

We propose that three-dimensional layered octagonal quasicrystals can host magnetic-field-tunable M\"{o}bius insulators and various higher-order topological insulators (HOTIs), enabled by the interplay of quasicrystalline symmetry and magnetic order. By constructing a minimal model based on stacked Ammann-Beenker tilings with magnetic exchange coupling and octagonal warping, we demonstrate that an A-type antiferromagnetic (AFM) configuration yields a topological phase protected by an effective time-reversal symmetry $\mathcal{S}=\mathcal{T}\tau_{1/2}$. Breaking $\mathcal{S}$ via an in-plane magnetic field induced canting of the AFM order while preserving a nonsymmorphic glide symmetry $\mathcal{G}_n=\tau_{1/2}\mathcal{M}_n$ leads to M\"{o}bius-twisted surface states, realizing a M\"{o}bius insulator in an aperiodic 3D system. Furthermore, we show that the quasicrystal with a general magnetic configuration supports multiple HOTI phases characterized by distinct hinge mode configurations that can be switched by rotating the magnetic field. A low-energy effective theory reveals that these transitions are driven by mass kinks between adjacent surfaces. Our work establishes a platform for realizing symmetry-protected topological phases unique to quasicrystals and highlights the tunability of hinge and surface states via magnetic control.