Kagome Lattice
Overview
The nearest-neighbor tight-binding model on the kagome lattice is the archetypal example of a flat band arising from geometric frustration. The kagome lattice has three sublattices (A, B, C) forming corner-sharing triangles. It is not bipartite.
\[H = -t \sum_{\langle i,j \rangle} \bigl(c^\dagger_i c_j + c^\dagger_j c_i\bigr)\]
Lattice properties: 3 sublattices per unit cell, non-bipartite (frustrated), coordination number 4, flat band at $E = +2t$.
Key physics: One of the three bands is completely dispersionless at $E = +2t$. The flat band touches the upper dispersive band at the $\Gamma$-point ($\mathbf{k} = 0$), where the spectrum degenerates to $\{-4t, +2t, +2t\}$. The flat band arises from destructive interference of hopping amplitudes around the corner-sharing triangles.
The dispatch tag Kagome is not exported from QAtlas to avoid a name clash with the Lattice2D.Kagome topology type. Use the fully qualified form QAtlas.Kagome() in user code.
Bloch Hamiltonian
Statement
In the three-sublattice basis, the Bloch Hamiltonian is the $3 \times 3$ real symmetric matrix
\[H(\mathbf{k}) = -2t \begin{pmatrix} 0 & \cos(\theta_1/2) & \cos(\theta_2/2) \\ \cos(\theta_1/2) & 0 & \cos\bigl((\theta_2 - \theta_1)/2\bigr) \\ \cos(\theta_2/2) & \cos\bigl((\theta_2 - \theta_1)/2\bigr) & 0 \end{pmatrix}\]
where $\theta_1 = \mathbf{k} \cdot \mathbf{a}_1 = 2\pi m/L_x$ and $\theta_2 = \mathbf{k} \cdot \mathbf{a}_2 = 2\pi n/L_y$, with $\mathbf{a}_1, \mathbf{a}_2$ the primitive vectors of Lattice2D's kagome topology.
Each off-diagonal element collects both the intra-cell and inter-cell hoppings of a sublattice pair (A–B, A–C, or B–C), yielding the factor of 2 and the half-angle arguments.
Flat band
Regardless of $\mathbf{k}$, one eigenvalue of $H(\mathbf{k})$ is always $+2t$. This can be verified directly: the vector $(1, -e^{i\theta_1/2}\cos(\theta_2/2)/\cos(\theta_1/2), \ldots)$ lies in the kernel of $H - 2tI$ at every momentum (appropriately regularized at singular points).
For an $L_x \times L_y$ lattice, the flat band contributes $L_x L_y$ eigenvalues at $+2t$. The upper dispersive band also reaches $+2t$ at the $\Gamma$-point, contributing one additional eigenvalue there, for a total degeneracy of $L_x L_y + 1$.
Band structure summary
| Band | Energy range | Degeneracy at $\Gamma$ |
|---|---|---|
| Lower dispersive | $[-4t, +2t)$ | 1 (at $-4t$) |
| Upper dispersive | $(-4t, +2t]$ | 1 (at $+2t$, touching flat band) |
| Flat band | $+2t$ | $L_x L_y$ (all $\mathbf{k}$) |
Derivation
See Bloch Kagome Flat Band for the derivation of $H(\mathbf{k})$ and the proof that one eigenvalue is identically $+2t$.
References
- I. Syozi, "Statistics of Kagome Lattice", Prog. Theor. Phys. 6, 306 (1951) – original lattice definition.
- D. L. Bergman, C. Wu, L. Balents, "Band touching from real-space topology in frustrated hopping models", Phys. Rev. B 78, 125104 (2008) – flat band analysis.
QAtlas API
# Sorted single-particle spectrum, 3×3 kagome PBC
λ = QAtlas.fetch(QAtlas.Kagome(), TightBindingSpectrum(); Lx=3, Ly=3, t=1.0)
# → 27 eigenvalues; 10 of them equal to +2.0 (9 flat band + 1 Γ-touch)Verification
| Test file | Method | What is checked |
|---|---|---|
test_kagome_tight_binding.jl | Real-space ED via Lattice2D | $\lambda_{\text{real}} = \lambda_{\text{Bloch}}$ for $2 \times 2$ through $4 \times 4$ |
test_kagome_tight_binding.jl | Flat band count | Exactly $L_x L_y + 1$ eigenvalues at $+2t$ |
test_kagome_tight_binding.jl | Ground state | $E_{\min} = -4t$ (unique, at $\Gamma$) |
test_kagome_tight_binding.jl | Structural | $\text{tr}\,H = 0$ |
test_kagome_tight_binding.jl | $t$ scaling | $\lambda(t) = t \cdot \lambda(1)$ |
test_bloch_generic.jl | Generic Bloch builder | Hardcoded $3 \times 3$ diagonalization $=$ bloch_tb_spectrum |
Connections
- Lattice family: Part of the tight-binding model catalogue. The kagome lattice is frustrated (non-bipartite) like Triangular, but features a flat band that the triangular lattice lacks.
- Flat-band contrast: Flat band at $E = +2t$, compared to $E = 0$ for the Lieb lattice. The kagome flat band arises from frustration, while the Lieb flat band arises from bipartite sublattice imbalance.
- Methods: Computed via the Bloch Hamiltonian method.