Antihelical Edge States in 2D Topological Materials

• Antihelical edge states are topological boundary modes exhibiting reversed spin–momentum locking, where spin-polarized channels propagate in the same direction along parallel boundaries.
• They emerge in 2D quantum materials like graphene heterostructures, photonic crystals, and circuit lattices through staggered spin–orbit coupling and modified Kane–Mele/Haldane models.
• Their unique behavior challenges conventional bulk-boundary correspondence and offers tunable topological transport opportunities, with implications for spintronics and higher-order topological phases.

Antihelical edge states are a fundamentally distinct class of topological boundary modes observed in a range of two-dimensional (2D) quantum materials and metamaterials. Unlike conventional helical states—where counterpropagating edge channels at a given boundary are associated with opposite spin or pseudospin projections—antihelical edge states are characterized by an inversion or modification of the spin–momentum locking and propagation direction. Recent theoretical and experimental developments have established antihelical edge states as robust excitations in a variety of platforms, with physical realizations ranging from photonic crystals and circuit lattices to graphene heterostructures and higher-order topological metals.

1. Definition and Phenomenology

Antihelical edge states are edge-localized electronic or photonic states whose propagation direction and internal pseudospin or spin degree of freedom manifest an atypical correlation compared to helical or chiral edge modes. In several prominent realizations—such as the modified Kane–Mele model and topological photonic metals—antihelical states can be summarized by the following traits:

• Edge states with a particular spin (or pseudospin) all propagate in the same direction along parallel boundaries, in contrast to the counterpropagating pairs of standard helical states (Dai et al., 8 Aug 2024, Liu et al., 4 Sep 2025).
• In several symmetry-protected settings, these propagating modes co-exist with gapless bulk excitations, so the total topological invariant is nontrivial but the bulk remains metallic (e.g., in 2D Weyl nodal-line semimetals).
• The antihelical edge states may be interpreted as a superposition of two antichiral edge channels related by time-reversal, where spin or polarization is locked to a single propagation direction along both edges (Dai et al., 8 Aug 2024).
• Unlike the edge conductance quantization observed for helical quantum spin Hall states, antihelical edge states typically do not yield quantized spin Hall transport due to their strong hybridization with gapless bulk states (Dai et al., 8 Aug 2024, Liu et al., 4 Sep 2025).

2. Formation Mechanisms and Topological Origin

The emergence of antihelical edge states depends sensitively on both the underlying topology of the bulk bands and the symmetry structure of the lattice Hamiltonians:

• Staggered Spin-Orbit Coupling (SOC) in Graphene-like Lattices: In systems with intrinsic spin–orbit coupling that differs in sign between two sublattices (such as $t_I^A = -t_I^B$), the resulting band structure produces displaced Weyl cones and a topologically nontrivial metallic phase. The edge states in zigzag nanoribbons show antihelical behavior, distinguished by co-directional spin-polarized currents on opposite edges (Dai et al., 8 Aug 2024, Liu et al., 4 Sep 2025).
• Modified Haldane and Kane–Mele Models: In both the modified Haldane model (with symmetrized next-nearest neighbor hopping) and the modified Kane–Mele model (with staggered intrinsic SOC), the spin or pseudospin polarization associated with the edge modes aligns in the same direction along both edges, giving rise to antihelical or antichiral edge states, respectively (Colomés et al., 2017, Dai et al., 8 Aug 2024). The antihelical structure is robust as long as the underlying topological invariant remains nontrivial.
• Topological Metals and Chern Number: The topological protection of antihelical edge states is typically governed by a nonzero $Z_2$ invariant or spin-Chern number, even in the absence of a full bandgap. In photonic and circuit lattice experiments, this protection manifests in the resilience of the edge transport to moderate disorder, though the coexistence with bulk metallic states leads to greater fragility than in conventional gapped topological insulators (Xie et al., 2023, Dai et al., 8 Aug 2024).

3. Mathematical Description and Distinguishing Signatures

The bulk-edge correspondence for antihelical edge states fundamentally differs from that of helical or chiral modes. Key analytical and numerical signatures include:

• Hamiltonians and Dispersion: For the case of staggered intrinsic SOC in the absence of Rashba coupling, the effective edge state energies on zigzag boundaries are given by

$$E_s^{(l)}(k) = 6s\, \lambda_{I}^B \sin(\sqrt{3}ka), \quad E_s^{(u)}(k) = -6s\, \lambda_{I}^A \sin(\sqrt{3}ka)$$

where $s$ is the spin index and $a$ is the lattice constant (Dai et al., 8 Aug 2024). The reversed sign relation between lower and upper boundary dispersions is a haLLMark of antihelical edge states.

• Protection and Fragility: In the presence of moderate on-site nonmagnetic disorder, transmission through antihelical edge channels decays exponentially due to their strong mixing with gapless bulk states, in contrast to the more robust quantized transport of helical and pseudohelical edge states (Dai et al., 8 Aug 2024, Liu et al., 4 Sep 2025). This outcome is a direct consequence of the metallic bulk acting as a compensating channel for backscattering.
• Quantized Transport Coefficients: Near zero energy, transmission in the antihelical regime is predicted to be $T_{\alpha} = 2 T_{\uparrow} = 2 T_{\downarrow} = 4$, supported by numerical calculations within the Landauer–Büttiker formalism, provided that dissipation into the bulk remains negligible (Liu et al., 4 Sep 2025).

4. Interplay with Higher-Order Topology and Bulk Phase Transitions

Application of additional symmetry-breaking terms, notably an in-plane Zeeman field, fundamentally alters the nature of antihelical edge states:

• Gap Opening and Corner States: An in-plane Zeeman field introduces a mass term to the edge Hamiltonian, opening a gap in what were formerly antihelical edge dispersions while leaving the bulk bands primarily metallic (gapless at the $K$ and $K'$ valleys). At the interfaces between edges with different effective mass signs (i.e., at the sample corners), localized zero-dimensional states—corner states—emerge due to the Jackiw–Rebbi mechanism (Liu et al., 4 Sep 2025).
• Higher-Order Topological Metal: This transition converts the first-order (edge-localized) antihelical topological metal into a higher-order topological metal (e.g., second order for 2D systems), where the robust boundary states are now confined to the corners. Quantized edge conductance is suppressed, but midgap states at the sample corners persist, which can be confirmed by numerical diagonalization and analytic continuum modeling (Liu et al., 4 Sep 2025).
• Theoretical Formulation: The low-energy continuum model near the $K$ valley captures this evolution:

$$H_\mathrm{eff} = \frac{3t}{2}(\sigma_x k_x + \sigma_y k_y)s_0 + 3\sqrt{3} t_I \sigma_0 s_z + \lambda \sigma_0 s_x$$

with corresponding eigenvalues exhibiting both bulk band crossings and Zeeman-induced edge (and corner) gaps (Liu et al., 4 Sep 2025).

5. Experimental Realizations and Material Platforms

Antihelical edge states have been theoretically predicted and, in some platforms, experimentally realized in a range of systems:

Platform Type	Lattice Model	Key Ingredients	Reference(s)
Graphene nanoribbons	Modified Kane–Mele	Staggered intrinsic SOC	(Dai et al., 8 Aug 2024, Liu et al., 4 Sep 2025)
Transition-metal dichalcogenide monolayers	Modified Haldane	Strong spin–orbit; sublattice asymmetry	(Colomés et al., 2017)
Photonic crystals	Coupled ring resonators	Anisotropic couplings, pseudospins	(Xie et al., 2023)
Circuit lattices	Braided LC networks	NN and NNN “braided” couplings	(Yang et al., 2020)

Significant points include:

• In conventional graphene-on-TMDC stacks, antihelical states are typically not realized directly, as Rashba SOC generated by substrate symmetry breaking favors pseudohelical edge states instead (Dai et al., 8 Aug 2024).
• Photonic and circuit platforms offer flexible means to tune couplings and symmetry, enabling controlled realization and manipulation of antihelical edge transport (Colomés et al., 2017, Yang et al., 2020, Xie et al., 2023).
• Experimental verification is based on real-space mapping (such as local photonic field distribution or voltage profiles), observation of co-directional edge modes on parallel boundaries, and robust propagation around corners or Möbius geometries.

6. Theoretical and Practical Implications

Antihelical edge states reveal several important aspects germane to the theory and application of topological systems:

• Breakdown of Edge–Bulk Decoupling: The coexistence of antihelical edge states with a gapless metallic bulk disrupts the conventional bulk-boundary correspondence, placing strict limits on expected conductance quantization and disorder resilience.
• Spin-Polarized Signals and Topological Purification: The unidirectional, spin-polarized transport of antihelical edge states along parallel boundaries, with the possibility of manipulating spin currents independently of charge, opens perspectives for topological spintronics and robust optical routing (Xie et al., 2023, Bao et al., 2022).
• Higher-Order Topological Phases: By gapping out the edge modes via Zeeman fields, higher-order topological phases emerge, with zero-dimensional boundary-localized corner states that persist even in a metallic bulk (Liu et al., 4 Sep 2025).
• Tunability and External Control: The antihelical–to–helical transition can be externally controlled through gate-controlled SOC engineering, Zeeman-like perturbations, or structural domain engineering, providing a tunable platform for realizing distinct topological regimes.

7. Outlook, Limitations, and Open Problems

While antihelical edge states enrich the taxonomy of topological phases, several points warrant attention for future inquiry:

• Disorder Fragility: Antihelical edge modes—due to their hybridization with bulk metallic states—are particularly susceptible to nonmagnetic disorder, unlike conventional (fully gapped) helical states (Dai et al., 8 Aug 2024).
• Topological Protection Mechanisms: The precise classification and protection of antihelical states, especially in the absence of a bulk bandgap, require a refined approach that goes beyond the framework of integer or $\mathbb{Z}_2$ invariants applicable to insulators.
• Experimental Identification: Detection distinctions between pseudohelical, antihelical, and antichiral edge states remain a subject of ongoing research, especially in platforms where Rashba SOC is present or can be controlled.
• Generalization to Higher Dimensions: The construction of higher-order topological metals with antihelical boundary or hinge states provides a roadmap for observing robust low-dimensional excitations embedded within metallic host materials (Liu et al., 4 Sep 2025, Cheng et al., 2021).

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In conclusion, antihelical edge states represent a novel frontier in topological matter, characterized by the co-directional propagation of spin-polarized edge channels along both boundaries, intricate interplay with bulk metallicity, and susceptibility to symmetry-breaking perturbations. Their realization expands the possible functionalities of topologically protected transport, with implications for spintronics, photonics, and higher-order topological phases.