Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals

TL;DR

This work demonstrates that a 3D layered octagonal quasicrystal built from stacked Ammann-Beenker tilings can host symmetry-protected topological phases that are tunable by magnetic fields. An AFM TI phase is protected by an effective time-reversal symmetry $\mathcal{S}=\mathcal{T}\tau_{1/2}$, and an in-plane field can induce a glide-symmetry-protected Möbius insulator, with Möbius surface states and hinge modes forming a higher-order phase. Introducing octagonal warping couples with magnetic order to realize multiple HOTIs, $\alpha$, $\beta$, and $\gamma$, whose hinge configurations can be switched by the field orientation; an effective $k\cdot p$ theory shows mass terms on surfaces generate domain walls that host hinge modes. The results establish quasicrystals as a platform for unconventional symmetry-protected topological phases and reveal magnetic-field-tunable control over surface and hinge states with potential applications in topological electronics.

Abstract

We propose that three-dimensional layered octagonal quasicrystals can host magnetic-field-tunable Möbius 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 $\mathcal{S}=\mathcal{T}τ_{1/2}$. Breaking $\mathcal{S}$ via an in-plane magnetic field induced canting of the AFM order while preserving a nonsymmorphic glide symmetry $\mathcal{G}_n=τ_{1/2}\mathcal{M}_n$ leads to Möbius-twisted surface states, realizing a Möbius 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.

Figures (6)

• Figure 1: (a) Schematic illustration of a 3D AFM quasicrystal lattice. Each layer in the $x$–$y$ plane forms an Ammann-Beenker tiling (ABT) quasicrystal, exhibiting eightfold rotational symmetry. The ABT quasicrystal layers are stacked along the $z$-axis in an AA stacking configuration. Light blue and gray indicate opposite spin orientations in adjacent layers, representing A-type AFM ordering. (b) When an in-plane magnetic field is applied, the AFM spins cant toward the field direction, breaking certain crystalline symmetries and enabling the realization of Möbius insulator or various higher-order topological insulators.
• Figure 2: Energy spectrum and spatial distribution of the surface state for the AFM TI phase. (a) Band structure along the $k_z$ direction for the AFM TI in an octagonal prime geometry. The red lines represent surface states. (b) (Top view along the $z$ direction) Wavefunction amplitude distribution of the surface state corresponding to $k_z=0$. The calculation is performed using Eq. \ref{['Ham']} and \ref{['Ham_comp']} with the parameters $C_1=0.5,\ M_1=0.5,\ v=0.5,\ M_0 = -2$, $g = 0$, $m = 0.4$, $\theta = 0$, and $\phi = 0$.
• Figure 3: Energy spectrum and spatial distribution of the surface state for the Möbius insulator phase. (a) Band structure along $k_z$ for an octagonal prism geometry with periodic boundary conditions along the $z$ direction. The red lines represent gapless surface states. (b) (Top view along the $z$ direction) Wavefunction amplitude distribution of the surface states corresponding to the Dirac cone at $k_z=0$, which exist only on the front and rear side facets. Since $\mathbf{B}=B\hat{\mathbf{x}}$, the AFM cants along the field direction, which is modeled by Eq. \ref{['m_l']} with parameters $\theta = 0.3\pi$ and $\phi = 0$. Other parameters are the same as Fig. \ref{['fig:AFM-TI']}.
• Figure 4: Higher-order Möbius insulator in a finite octagonal prism geometry with open boundary conditions along the stacking direction. The system is composed of 5 layers of quasicrystal lattices, each comprising 577 sites. (a) (Top view along the $z$ direction) Wavefunction amplitude distribution of the hinge states. (b) Schematic illustration of the higher-order Möbius insulator. The red lines with arrows represent hinge modes connecting Möbius surface states on the front and rear side surfaces.
• Figure 5: Magnetic-field tunable HOTI phases. (a),(c),(e) Wavefunction amplitude distribution of HOTI $\alpha$, $\beta$, and $\gamma$ phase, respectively. (b),(d),(f) Schematic illustration of the chiral current in three-dimensional space for the hinge modes shown in the left figure. The arrow in the lower-left corner indicates the direction of the magnetization $\mathbf{m}$. The parameters used for (a) $g = 0.25$, $\theta = 0$, and $\phi = 0$; (c) $g = 0$, $\theta = 0.3\pi$, and $\phi = \pi/8$; and (e) $g = 0.25$, $\theta = 0.3\pi$, and $\phi = 0$. Other parameters are the same for the three cases, which are $C_1=0.5,\ M_1=0.5,\ v=0.5,\ M_0 = -2$, and $m = 0.4$.