Directional Kagome Network

• Directional Kagome network is an interconnected lattice system combining kagome geometry with symmetry breaking, topological, and chiral features.
• It enables robust, unidirectional transport and edge modes across diverse platforms such as quantum magnets, photonic metasurfaces, and moiré-engineered van der Waals systems.
• Applications include realizing exotic quantum phases, designing valleytronic circuits, and engineering metamaterials with flat bands, Dirac points, and non-reciprocal behavior.

A directional Kagome network refers to an interconnected structure in which the underlying Kagome lattice geometry combines with symmetry-breaking, topological, and/or chiral features that enforce a preferred directionality in physical phenomena such as electronic, photonic, or magnonic transport. The term has been formalized in several complementary contexts, most notably in quantum spin liquids with Dzyaloshinskii–Moriya interactions, topological photonic metasurfaces, quantum-graph models with chiral vertex coupling, and twistronic van der Waals systems featuring topologically protected 1D channel networks. These systems realize macroscopic consequences of the Kagome geometry—enhanced degeneracies, flat bands, and strong response to discrete symmetry breaking—while supporting directional channels, chiral edge modes, or networks of domain-wall states that exhibit non-reciprocal or valley-polarized transport.

1. Topological and Chiral Kagome Networks in Quantum Magnets

The directional Kagome network in magnetism emerges from the unification of the Heisenberg antiferromagnet (HAF), the XXZ anisotropic model, and the Dzyaloshinskii–Moriya (DM) ferromagnet on the Kagome lattice. The key Hamiltonians are:

• Heisenberg antiferromagnet:

$$H_{\rm HAF} = J\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j, \qquad J>0$$

• XXZ model:

$$H_{\rm XXZ} = J\sum_{\langle ij\rangle} (S_i^xS_j^x + S_i^yS_j^y + \Delta S_i^zS_j^z) , \quad J>0,\;\Delta\in\mathbb R$$

• DM ferromagnet:

$$H_{\rm FDM} = J_F\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j + \sum_{\langle ij\rangle}D \hat z\cdot (\mathbf S_i\times \mathbf S_j)$$

where the DM vectors impose the same chirality on each triangle.

A three-fold mapping—site-dependent rotation by $\pm 2\pi/3$ about $\hat z$ for each sublattice—connects these models and reveals that their spectra and extensive ground-state degeneracies are preserved. The phase diagram in the $(\Delta, D/J)$ plane encompasses continuous branches of spin liquids and chiral phases. The network topology arises because each of the three Hamiltonian lines (one XXZ, two chiral analogues $\rm XXZ^\pm$) is related by these site rotations, and the end points connect to chiral ferromagnetic states with finite scalar chirality $$\chi_{ijk} = \mathbf S_i\cdot (\mathbf S_j\times\mathbf S_k)\neq 0$$. The Ising limit $\Delta\to\infty$ supports quantum disorder, forming the central node in the network. This structure provides routes for realizing chiral or vector-chiral Kagome magnets and for searching for novel quantum spin liquids through parameter tuning or optical lattice emulation (Essafi et al., 2015).

2. Directional Vertex Coupling in Kagome Quantum Graphs

Quantum graphs with Kagome connectivity support directionality by imposing a “preferred orientation” at each vertex through non-time-reversal-invariant coupling. The general vertex boundary condition is: $(U - I)\,\psi(v) + i\ell (U + I)\,\psi'(v) = 0$ where $U$ is taken as the cyclic shift matrix (not symmetric), $\ell > 0$ sets a length scale, and $\psi(v), \psi'(v)$ are the wavefunction and derivative vectors on $N$ incident edges.

This coupling enforces circulation around the node, breaking time-reversal symmetry and producing a directional (chiral) network. The band structure exhibits infinite families of flat bands and continuous Bloch bands, with universal properties—specifically, the probability $P_\sigma$ that a random positive energy is in the spectrum is $P_\sigma \approx 0.639$, independent of commensurability in the generic case. The directionality materially affects high-energy transport, generating semi-transparent, but non-trivial, S-matrix behavior at large $k$. Flat bands and gap closures (“Dirac-like” points) occur at specific quasimomenta, mirroring features known from tight-binding Kagome models but with altered spectral embedding due to chiral matching (Baradaran et al., 2021).

3. Moiré-Engineered Kagome Networks of Chiral Channels in Graphene/hBN

In doubly aligned graphene/hexagonal boron nitride (hBN) heterostructures, a superlattice of 1D topological chiral channels forms a real-space Kagome network. The underlying mechanism is local inversion symmetry breaking by parallel-polarity hBN encapsulation, yielding a sublattice gap $\Delta_{uv}(\tau)$ modulated in the plane by the offset vector $\tau$ between the two hBN layers. At domain boundaries where $\Delta_{vv'}(\tau(r))=0$—i.e., the minigap between adjacent moiré minibands closes—the Chern number $Q$ jumps by $\pm 1$. The resulting networks of domain walls support Jackiw–Rebbi-type chiral modes with valley-contrasting propagation. The network’s geometry is set by the condition $\Delta_{vv'}(\tau(r))=0$, resulting in straight lines intersecting at $120^\circ$ and forming the skeleton of a Kagome lattice.

Key properties include:

• Percolating, topologically protected, unidirectional 1D modes supporting metallic conductivity in an otherwise gapped system.
• Aharonov–Bohm oscillations in network hexagons, with period set by the flux quantum.
• Strong anisotropy and nonlocality in valley-resolved transport, with suppressed backscattering.
• Direct observability via local density-of-states mapping at network domain walls using scanning probes.

This realizes a macroscopic, electrically addressable Kagome network of directional (valley-polarized) channels in a solid-state platform (Moulsdale et al., 2022).

4. Directionality and Chirality in 3D Topological Kagome Networks

Structural chirality in a Kagome network can be realized by extending the 2D Kagome lattice into three dimensions with a screw operation—a translation of sublattices by $c/3$ per layer, combined with $C_3$ rotation. The resulting 3D chiral Kagome lattice possesses:

• Preservation of Kagome flat bands, van Hove singularities, and Dirac–Weyl points in $k_z = 0$ and $k_z = \pi$ planes.
• Quantized Chern numbers at multi-fold Weyl points; the sign of chirality (right- or left-handed lattice) sets the Chern number’s sign.
• One-dimensional Fermi-arc surface states that are dispersive in one in-plane direction and flat in the other, enforcing unidirectional, non-local surface transport.
• At domain walls between regions of opposite handedness, electronic transport is forced along 1D interface channels, with directionality controlled by domain chirality. Conductance between remote leads can be enhanced compared to direct connections due to enforced surface channeling.

Several real materials (e.g., CePO$_4$) display these features in both electronic and phononic band structures, providing platforms for studying chirality-driven topological transport and correlated electron instabilities within a Kagome network framework (You et al., 2023).

5. Directional Photonic and Spin Topological Kagome Metasurfaces

Electromagnetic realizations of directional Kagome networks exploit the lattice’s Dirac cones at $K/K'$, symmetry-breaking perturbations, and resulting valley Hall or photonic spin Hall effects. In all-dielectric photonic crystals, breaking inversion symmetry by radial displacements in the Kagome motif opens a band gap and induces nontrivial valley Chern numbers. At domain boundaries between regions of opposite sign perturbations:

• Topologically protected, unidirectional edge states appear.
• Edge waveguides are immune to backscattering, even through sharp $60^\circ$ or $120^\circ$ bends.
• Realistic 3D metasurface platforms (e.g., InGaAsP slabs with air holes) realize low-loss, broadband, telecommunication-band waveguides compatible with integrated photonics.

In spin-photonic Kagome metasurfaces, a complementary-pattern bi-anisotropy breaks duality symmetry, enabling robust spin-polarized edge modes. Armchair interfaces support leaky-wave antenna action with strong directional (forward/backward) beam control; scan ranges up to $50^\circ$ over X-band frequencies are demonstrated. Parametric tuning of lattice constant and border width (aperture) allows for precise engineering of both mode spectrum and edge channel directionality (Wong et al., 2019, Abtahi et al., 5 Feb 2025).

6. Unified View: Features of Directional Kagome Networks

The directional Kagome network concept is unified by several core features appearing across physical realizations:

Feature	Physical Realization	Topological/Chiral Mechanism
Flat and dispersive bands	Quantum magnetism, photonic crystals, 3D electronics	Kagome symmetry; network geometry
Chiral/valley/domain wall modes	Moiré graphene/hBN, photonic edges, spin liquids	Inversion or time-reversal breaking
Directional S-matrix/coupling	Quantum-graph models, metasurfaces	Preferred orientation, bi-anisotropy
Percolating or non-local channels	Moiré and 3D chiral systems, edge states in PTIs	Topological invariants (Chern)
Robustness to disorder/bending	Photonic/metasurface networks, magnonic bands	Protection by chirality/valley

This synthetic framework supports exploring quantum disorder, chiral phases, and advanced device functionality (e.g., leaky-wave antennas, robust photonic routing, valleytronic circuits) using the geometric, topological, and symmetry features unique to the directional Kagome network paradigm. Each physical context exploits the interplay between lattice-induced degeneracies (flat/Dirac bands) and chiral or directional mechanisms (e.g., DM interaction, inversion-breaking potentials, preferred vertex matching) to achieve highly nontrivial transport and correlation effects (Essafi et al., 2015, Baradaran et al., 2021, Moulsdale et al., 2022, You et al., 2023, Wong et al., 2019, Abtahi et al., 5 Feb 2025).