Chiral Topological Edge Modes

• Chiral topological edge modes are spatially localized, unidirectional excitations emerging from nontrivial band topology and characterized by nonzero Chern numbers.
• These modes are realized in diverse systems such as magnonic, photonic, and electronic platforms, facilitating fault-tolerant transport and potential quantum computing applications.
• Experimental and theoretical frameworks validate the role of symmetry protection and bulk-edge correspondence in maintaining mode robustness despite interactions and non-Hermitian effects.

Chiral topological edge modes are spatially localized, unidirectional excitations at the boundaries or interfaces of topological matter, characterized by robust propagation that is protected against elastic backscattering by bulk topological invariants such as Chern numbers. Their physical realization spans a wide variety of quantum materials and engineered metamaterials, including electronic, photonic, magnonic, spin, and anyonic systems. The following presents a comprehensive account of their theoretical foundation, classification principles, experimental context, and mathematical characterization.

1. Topological Band Structure and Edge Mode Emergence

Chiral topological edge modes are fundamentally a consequence of nontrivial band topology in crystalline or engineered periodic systems. When the bulk bands of a material acquire nonzero Chern numbers or related topological invariants, the bulk–edge correspondence principle dictates that robust, gapless modes must appear at boundaries or interfaces between regions of distinct topology. For instance, in magnonic crystals, a periodic array of magnetic materials, the inclusion of both exchange and dipolar interactions in the linearized Landau–Lifshitz equation yields bosonic Bogoliubov–de Gennes (BdG) bands with nonzero Chern numbers, leading to chiral spin-wave edge modes that are immune to backscattering (Shindou et al., 2012).

The general mathematical setting involves a bulk Bloch Hamiltonian $H(\mathbf{k})$ whose eigenstates, when projected into topologically nontrivial bands, acquire a Berry curvature $\mathcal{B}_j(\mathbf{k})$. The integral over the Brillouin zone,

$$C_j = \frac{1}{2\pi}\int_{\mathrm{BZ}} d^2k \: \mathcal{B}_j(\mathbf{k}),$$

gives the Chern number $C_j$—a central integer-valued topological invariant. A nonzero Chern number enforces the appearance of $|C_j|$ chiral edge channels traversing the gap between adjacent bands.

2. Classification, Protection, and Bulk-Edge Correspondence

Chiral edge modes are classified by both the topology of the bulk and the symmetries present. In electronic and magnonic Chern insulators, edge modes propagate unidirectionally and are protected as long as the bulk gap and topological order persist. In spin models, time-reversal symmetry breaking—in p+ip or d+id wave states—produces protected chiral states at the edge, with their number and directionality set by net Chern number or $\mathbb{Z}_2$ indices (Li et al., 2012, Zeng et al., 2023).

Nontrivial extensions arise in non-Hermitian systems, where chiral topological edge modes can be classified according to two winding numbers: one analog of the Berry-phase (directional) winding, and a uniquely non-Hermitian phase winding, capturing fractional “exceptional point” (EP) charges. Interfaces between media differing in mass or non-Hermitian “charge” ($s$) display three edge mode families: Hermitian-like, genuinely non-Hermitian, and mixed, with the difference in winding numbers $\Delta w_1$ and $\Delta w_2$ determining the number and type of chiral edge states (Leykam et al., 2016). The edge modes in such systems remain protected so long as the underlying topological indices are conserved.

3. Explicit Models and Mathematical Frameworks

Magnonic Crystals

In magnonic crystals, the linearized dynamics is governed by

$$\frac{1}{|\gamma|\mu_0} \frac{d}{dt} m_+ = \pm 2i M_s (\nabla \cdot Q \nabla) m_+ \mp 2i m_+ (\nabla \cdot Q \nabla) M_s \mp i H_0 m_+ \pm i h_+ M_s$$

with $Q$ arising from exchange, and $h_+$ from dipolar interactions. After Holstein–Primakoff transformation, a bosonic BdG eigenproblem is formulated:

$$i \frac{d}{dt} \begin{pmatrix} \beta_{\mathbf{k}} \ \beta_{-\mathbf{k}}^\dagger \end{pmatrix} = \sigma_3 H_{\mathbf{k}} \begin{pmatrix} \beta_{\mathbf{k}} \ \beta_{-\mathbf{k}}^\dagger \end{pmatrix}$$

Eigenstates allow explicit computation of Berry connection and curvature, leading to a rigorous evaluation of Chern numbers (Shindou et al., 2012).

Spin Chains and Lattice Models

In trimer lattices and other 1D models, chiral edge states may appear at a single edge in the inversion-symmetry broken phase—contrary to the SSH model where edge states always form left/right pairs. Their persistence even under strong disorder, and the connection of their appearance to a Zak phase quantized at $\pi$, show that their origin is deeply rooted in a (hidden) 2D topological invariant obtainable by mapping to e.g. the Aubry–André–Harper model (Alvarez et al., 2018).

Chiral edge states in nonlinear 1D topological lattices require a generalized chiral symmetry: the nonlinear Hamiltonian $H_\psi$ must act off-diagonally with respect to a sublattice decomposition (e.g., $|\psi\rangle_B = 0 \implies (H_\psi \psi)_A = 0$), ensuring the edge modes stay at zero energy and remain localized as long as the effective, amplitude-dependent topological indices remain nontrivial (Jezequel et al., 2021).

Anyon Chains and Modular Tensor Categories

Lattice models built with modular tensor category (MTC) symmetry at the microscopic level—even when not exactly soluble—can support chiral edge modes corresponding to non-Abelian anyons (e.g., Ising or Fibonacci). Simulations using matrix product states and exact MTC symmetry (guaranteed at the lattice scale) find entanglement spectra and central charges matching those of chiral CFTs, confirming the realization of chiral edge theories on the boundary of bulk non-Abelian topological states (Ueda et al., 5 Aug 2024).

4. Experimental Platforms and Probing Techniques

A range of platforms allow for the creation and interrogation of chiral topological edge modes:

• Optical Lattices: In cold atom setups, the Haldane model can be engineered by controlling sublattice offset $\Delta_{AB}(x,y)$, with spatially varying potentials generating internal “topological interfaces” with sharply localized and tunable chiral edge modes. Differential and disorder-enhanced measurements, along with wavepacket tracking, can detect topological transport with high fidelity (Goldman et al., 2016).
• Photonic and Magnonic Systems: Photonic Chern insulators support chiral edge photons that can be hybridized with flat-band resonances (edge-attached resonators), giving broadband, ultra-slow light regimes with the group velocity $v_g$ remaining nearly constant and small over a wide frequency window, all while maintaining topological protection. This requires only edge engineering rather than bulk modification (Yu et al., 2020). In magnonics, materials such as yttrium iron garnet (YIG) provide long coherence lengths for direct edge mode observation via Brillouin scattering, Kerr microscopy, and infrared thermography (Shindou et al., 2012).
• Transport and Resistance Distributions: In QAH–SC junctions, the presence and type of superconducting chiral edge modes (including Majorana states) can be detected through full probability distributions of the charge transmission fraction $T = T_{ee} - T_{eh}$ and resistance. For example, in the $N=2$ phase, $T$ becomes uniformly distributed for long junctions, while for a chiral Majorana ($N=1$), the arcsine law arises. These signatures remain robust against disorder and decoherence (Huang et al., 14 Mar 2025).
• Optical Conductivity: The number of in-gap resonance peaks in the optical absorption of a chiral TSC equals the half-integer part of the BdG Chern number, as direct consequences of vertical, momentum-conserving transitions among edge-banded Bogoliubov excitations. This property differentiates topological superconductors from trivial or collective-mode-dominated responses (Huang et al., 17 Jul 2024).

5. Deviations, Non-Hermitian Effects, and Limitations of Topological Protection

While the notion of topological protection underpins the robustness of chiral edge modes, a series of works uncover subtle breakdown mechanisms:

• Magnonic Edge Mode Fragility: Beyond linear spin-wave theory, magnon-magnon interactions (especially cubic terms breaking $U(1)$ conservation) induce finite edge mode lifetimes, hybridization with bulk magnons, and even edge-to-edge coupling. These processes undermine localization and quantization and may completely suppress the chiral resonance unless large magnetic fields are applied to energetically separate the edge modes from the two-magnon continuum (Habel et al., 2023).
• Non-Hermitian Skin Effect: Dissipation, modeled as an imaginary Fermi velocity, induces the so-called non-Hermitian skin effect, with chiral edge states localizing at spatially distinct boundaries for positive/negative mode energies. Only the kinetic (Fermi velocity) NH term, not on-site loss/gain, results in such spatial separation. The energy spectra become straight lines in the complex plane, and the spatial localization is fundamentally tied to chirality (Yang et al., 2023).
• Chiral Edge Magnon Interactions: In strongly localized edge regimes, magnon-magnon scattering deviates from bare XXZ chain predictions due to resonant coupling to bulk two-magnon bound states—a “Feshbach resonance” mechanism. These features can be directly probed using protocols based on spin-polarized scanning tunneling spectroscopy, and the effective edge Hamiltonian is renormalized by bulk interactions, resulting in momentum-dependent scattering phases (Birnkammer et al., 7 Oct 2024).

6. Field Theoretical Perspective and Symmetry Protection

Chiral edge mode theories can be constructed within the framework of chiral conformal field theory and modular tensor categories. In 2+1D topological phases, Abelian edges correspond to CFTs with integer central charge, while non-Abelian anyonic systems have fractional central charge and more exotic statistics. Constrained fermions, obtained via the quantum wires approach, realize the coset structure of edge CFTs and allow for fractionalized edge excitations. Chiral anomalies and gauge anomalies are handled by mapping ill-defined chiral determinants to Wess–Zumino–Witten actions through regularization schemes that preserve left-right decoupling (Hernaski et al., 2017).

In topological field theory (e.g., abelian BF theory on a curved 3D manifold), chiral edge modes are described as decoupled Luttinger liquids, with local, position-dependent velocities encoding metric information from the bulk. However, positive-definite Hamiltonian constraints preclude physical realization of parallel (“same-direction”) edge mode configurations, restricting the BF framework (and thus its potential as a model of parallel Hall response) to systems with single or helical (opposite-velocity) edge modes—mirroring the physics of topological insulators and the quantum anomalous Hall phase (Bertolini et al., 2022).

7. Interfaces, Disorder, and Anomalous Chiral States

Chiral edge modes can exist not only at edges but also at engineered topological interfaces within the bulk or at boundaries between distinct topological regions:

• Engineered Interfaces: Spatially varying parameters such as $\Delta_{AB}(x,y)$ in cold atom optical lattices can create internal boundaries supporting highly localized and tunable chiral edge channels (Goldman et al., 2016).
• Random Networks and Floquet Systems: In periodically driven or discrete-time scattering networks—represented as Eulerian graphs—anomalous chiral edge modes exist even when bulk Chern numbers vanish, enforced by phase rotation symmetry and topological winding (circulation) of network loops. The existence and quantization of net directional flow—encapsulated by a winding number—remain well-defined even in randomly connected, non–periodically driven networks (Delplace, 2019).
• Momentum-Space Arc States: On the 2D surface of a 3D TI in contact with a magnetic insulator, a chiral edge mode emerges along the interface whenever the magnetization is tilted or the chemical potential is mismatched. This edge state manifests as a momentum-space arc connecting the Dirac point to the magnetically gapped band edge, with propagation velocity $v_\text{edge} = v \cos(\theta-\phi)$ tunable by the magnetization and electrostatic potential (Beenakker, 19 Aug 2024).

8. Applications and Prospects

Chiral topological edge modes exemplify the interface between fundamental quantum topology and practical device functionality. Their immunity to backscattering makes them ideal for fault-tolerant interconnects, robust slow-light photonic waveguides (Yu et al., 2020), quantum Hall interferometry, spintronics applications (including current splitters and logic gates) (Shindou et al., 2012), and the detection and manipulation of Majorana quasiparticles and non-Abelian anyons for topological quantum computation (Ueda et al., 5 Aug 2024, Zeng et al., 2023). Distinct signatures in transport measurements (e.g., arcsine and uniform resistance distributions), optical absorption (number and position of in-gap resonance peaks), and entanglement spectra (central charge and level spacing ratios) serve as operational fingerprints for distinguishing topological phases.

Nevertheless, bosonic topological systems—especially those lacking number conservation—are generically more fragile, with interactions and decoherence providing channels for breakdown of edge protection. Strategies such as energy separation (via applied fields), symmetry engineering (chiral nonlinearities), and real-space edge engineering continue to be refined to harness the unique transport and coherence properties of chiral topological edge modes across platforms.