Higher-Order Topological Metals

• Higher-Order Topological Metals are 2D systems featuring metallic bulk bands coexisting with topologically protected corner states due to staggered intrinsic SOC and Zeeman field effects.
• They exhibit unique antihelical edge states with spin-split Weyl cones, leading to quantized conductance and robust transport signatures in a modified graphene lattice.
• The model is validated through continuum Dirac theory and transport simulations, indicating promising experimental platforms such as engineered graphene and cold-atom systems.

Two-dimensional higher-order topological metals are characterized by the coexistence of metallic bulk bands (crossed conduction and valence bands at Fermi energy) and topologically protected zero-dimensional boundary states (corner modes), realized here through the interplay of staggered intrinsic spin–orbit coupling (SOC) and an in-plane Zeeman field in modified graphene. This model provides an explicit platform where higher-order topology and metallic band structure are present simultaneously, producing quantized transport signatures, gapped edge spectra with corner-localized modes, and persistent topological metal behavior.

1. Model Construction and Bulk Band Topology

The starting point is a graphene-based lattice described by a modified Kane–Mele model, but with a crucial departure from the canonical case: the intrinsic SOC, $t_I$, is staggered between the two sublattices ($t_I^A = -t_I^B$). The full Hamiltonian in momentum space is

$$H(\mathbf{k}) = [f_x(\mathbf{k})\,\sigma_x + f_y(\mathbf{k})\,\sigma_y]\,s_0 + f_I(\mathbf{k})\,\sigma_0\,s_z + \lambda\,\sigma_0\,s_x,$$

where $\sigma_{x,y,z}$ and $s_{x,y,z}$ are Pauli matrices in sublattice and spin space, respectively, $f_I(\mathbf{k})$ encodes the staggered intrinsic SOC, and $\lambda$ is the in-plane Zeeman field strength. The energy spectrum in the absence of the Zeeman field ($\lambda = 0$) consists of two pairs of bands, with each spin branch experiencing a distinct energy shift at the $K$ and $K'$ valleys; this results in the spin-up and spin-down Weyl cones being displaced in energy, leading to band crossing at the Fermi level and realizing a topological metallic state.

The addition of the Zeeman field ($\lambda \neq 0$) does not gap out the bulk crossings at $K$, $K'$; the system remains metallic. However, this Zeeman term modifies the boundary (edge) states decisively.

2. Antihelical Edge State Structure

For zigzag ribbon geometries without a Zeeman field, the system supports boundary-localized antihelical edge states. These states connect the $K$ and $K'$ valleys and are characterized as follows:

• For a given zigzag edge, both spin-up and spin-down edge states propagate in the same direction (unlike conventional helical edge states of a quantum spin Hall insulator, where opposite spins propagate in opposite directions).
• On opposite zigzag edges, the propagation direction of the same-spin edge mode is reversed—hence the term “antihelical”.

The presence of staggered SOC is essential in achieving this configuration, as it produces spin splitting and opposite band shifts for the two sublattices, while preserving metallicity in the bulk. The antihelical edge states are robust and yield quantized conductance signatures.

3. Quantized Transport and Landauer–Büttiker Analysis

The presence of robust conducting channels is confirmed via transport calculations using the Landauer–Büttiker formula. The authors compute the spin-resolved conductance

$G_\sigma(E) = (e^2/h) T_\sigma(E),$

where $T_\sigma(E)$ is the transmission coefficient, explicitly given by the trace over the product of the left/right lead coupling matrices and the retarded Green’s function. When the bulk conduction is suppressed (by tuning the coupling parameter $\eta_2$ between sample and leads), the system exhibits quantized plateaus in the transmission: $T_\alpha = 2 T_\uparrow = 2 T_\downarrow = 4$, corresponding to two parallel edge channels per spin, a direct fingerprint of antihelical edge transport in the metallic phase.

4. Emergence and Characterization of Higher-Order Topology

The introduction of an in-plane Zeeman field, irrespective of its orientation, gaps out the antihelical edge spectrum. The bulk Dirac crossings at the $K$, $K'$ valleys persist, but the one-dimensional edge channels are destroyed. This scenario establishes the precondition for higher-order topology: at the intersection of two edges (the system’s corners), the effective Dirac mass term changes sign, creating Jackiw–Rebbi domain walls that host localized zero-energy corner states.

For a finite-size nanoflake (diamond geometry with zigzag edges), the Zeeman field projects onto the edge subspace as a mass term whose sign changes from edge to edge: $$m_\text{I} = m_\text{II} = -m_\text{III} = -m_\text{IV}$$. The crossing of mass sign at corners thus hosts four robust, zero-dimensional corner states, directly confirmed by the calculation of the energy spectrum, spatially resolved wave functions (corner-localized density distributions), and by the persistent gapless crossing in the bulk bands.

5. Continuum Theory and Analytic Confirmation

The low-energy properties are analytically confirmed by expanding around the $K$ valley and deriving an effective Dirac Hamiltonian: $$H_\text{eff} = \frac{3t}{2}(\sigma_x k_x + \sigma_y k_y)s_0 + 3\sqrt{3} t_I \sigma_0 s_z + \lambda \sigma_0 s_x.$$ Solving for the band crossings ($\epsilon = 0$), the resulting condition,

$$k_x^2 + k_y^2 = \frac{4\lambda^2}{9t^2} + \frac{12 t_I^2}{t^2},$$

demonstrates that the Dirac cones are shifted in energy by both SOC and Zeeman field, and the bulk remains metallic (the energy vanishes on rings in the Brillouin zone), guaranteeing coexistence of higher-order topological boundary states and metallic bulk.

6. Distinctions from Conventional Topological Metals

In contrast to conventional two-dimensional metals or first-order topological insulators/metals:

• The metallic phase here is driven explicitly by a bulk band crossover induced by staggered SOC, with the edge states protected by a valley-dependent topology.
• The higher-order boundary states (corner-localized modes) arise even though the bulk remains metallic: an essential requirement is the gapping of edge dispersions by the Zeeman field, which leaves only domain-wall protected corner zero modes.
• Quantized transport in the absence of backscattering is realized via antihelical edge channels before the Zeeman field is applied; after the Zeeman field, the higher-order topology is evidenced by the discrete zero-dimensional corner states.

7. Experimental Implications and Signatures

The transport quantization, robust presence of corner states (localized in spatial density), and analytic confirmation via the continuum Dirac theory together establish the observability of this higher-order topological metallic phase. This suggests that material realizations featuring sublattice-staggered intrinsic SOC and tunable in-plane Zeeman fields (such as engineered graphene or cold-atom emulators) are promising platforms for realizing and probing 2D higher-order topological metals. A plausible implication is that future experiments in patterned graphene nanostructures could directly observe quantized edge conductance and corner-state spectroscopy in the metallic regime (Liu et al., 4 Sep 2025).