Quasi-1D Planar Magnetic Topological Heterostructure

Abstract

We theoretically introduce a quasi-1D magnetic heterostructure of alternating 2D topological and normal insulator strips. Its low-energy physics is governed by a hybrid Hamiltonian intertwining the Su-Schrieffer-Heeger and Shockley models, with spin-momentum locking and local Zeeman splitting. Symmetry analysis places it in class AIII, characterized by chiral symmetry and a $\mathbb{Z}$ topological invariant. Computing the winding number from the block-off-diagonal structure of the Hamiltonian reveals topological phases characterized by invariants $ν= 0$, $1$, and $2$. Furthermore, a single magnetic defect acts as a sensitive local probe, whose in-gap spectrum provides a spectroscopic fingerprint to distinguish topological phases. Extending the platform to a multilayer geometry uncovers a nonsymmorphic projective symmetry that gives rise to Möbius band topology, with the Brillouin zone compactifying into a Klein bottle. Our work establishes a platform for higher-order topology via heterostructure design and magnetic patterning.

Figures (7)

• Figure 1: Schematic illustration of the proposed model. The device is a quasi-1D ribbon comprising an alternating sequence of 2D topological insulator (TI) and normal insulator (NI) segments. Magnetic impurities (green) are assumed to be present at each TI-NI interface. The effective coupling between the counter-propagating chiral edge modes is described by the tight-binding Hamiltonian in Eq. \ref{['Hamiltonian']}, with the hopping parameters defined as follows: $\Delta_S$ (tunneling across a TI segment), $\Delta_D$ (tunneling across a NI segment), and $\Delta_F$ (direct on-edge spin-flip tunneling).
• Figure 2: Electronic spectrum \ref{['eq:spectrum']} calculated for various Hamiltonian parameters. Note that the magnetic terms $\Delta_F$ and $\Delta_Z$ produce equivalent band splittings.
• Figure 3: Topological phase diagram of the hybrid SSH--Shockley model at $k_y=0$. The dimensionless parameters are $x=|\Delta_S|/\Delta_D$ and $y=R/\Delta_D$ with $R=\sqrt{\Delta_F^2+\Delta_z^2}$. Phases are characterized by the winding number $\nu$. The dashed line indicates a cut at fixed $x=0.5$, along which increasing the magnetic parameters ($y$) drives the transitions $\nu=2\to1\to0$. The dot marks the triple point $(x,y)=(1,0)$.
• Figure 4: (Color online) Energy spectra of in-gap bound states induced by a single magnetic defect, calculated at $k_y = 0$ as a function of the Zeeman splitting strength $\Delta_Z$ and intra-edge intermodes tunneling magnitude $\Delta_F$. (a) Topological phase ($\Delta_S < \Delta_D$). The introduction of the defect gives rise to four in-gap states. The initially degenerate levels split, and all states exhibit dispersion. Notably, the lower pair of states cross at a critical value of $\Delta_Z$. (b) Trivial phase ($\Delta_S > \Delta_F$). Only two in-gap states emerge. While the degeneracy is also lifted, one branch is dispersionless and is pinned precisely at the edge of the bulk band gap. In contrast to the topological phase, the bound states in the trivial phase exhibit no crossing. The spectra are obtained from the analytical expression in Eq. (\ref{['ingap states']}).
• Figure 5: (Color online) A comparison of two distinct global topologies arising from closing a quasi-1D ribbon. (a) Möbius strip geometry, formed by connecting the ends of the ribbon with a half-twist. (b) Cylindrical geometry, formed by direct connection of the ends. The fundamental difference manifests in the boundary conditions for the chiral edge modes at the junction (highlighted in red). (c) In the Möbius strip, the connection forces edge modes of the same chirality to meet at the seam, leading to their hybridization and intersection in real space. (d) In the cylinder, the connection brings together edge modes of opposite chirality, as is the case at all other interfaces in the array. This results in two spatially separated edge modes, localized on opposite physical boundaries and propagating without intersection.