Lieb Lattice
Overview
The nearest-neighbor tight-binding model on the Lieb lattice (line-centred square lattice) is the canonical example of a flat band arising from bipartite sublattice imbalance. The Lieb lattice has three sublattices per unit cell: A (corner site) and B, C (edge-centre sites). Only A–B and A–C bonds exist; there is no direct B–C bond. The lattice is therefore bipartite between $\{A\}$ and $\{B, C\}$.
\[H = -t \sum_{\langle i,j \rangle} \bigl(c^\dagger_i c_j + c^\dagger_j c_i\bigr)\]
Lattice properties: 3 sublattices per unit cell (A corner, B right edge, C top edge), bipartite, coordination number 4 (A) / 2 (B, C), flat band at $E = 0$.
Key physics: Because the B and C sublattices have no mutual hopping, a localized state on B and C sites can be constructed at every momentum with zero energy, giving a completely flat band at $E = 0$. This is a consequence of Lieb's theorem on bipartite Hubbard models.
Bloch Spectrum
Statement
The $3 \times 3$ Bloch Hamiltonian in the sublattice basis is
\[H(\mathbf{k}) = -2t \begin{pmatrix} 0 & \cos(\theta_1/2) & \cos(\theta_2/2) \\ \cos(\theta_1/2) & 0 & 0 \\ \cos(\theta_2/2) & 0 & 0 \end{pmatrix}\]
where $\theta_1 = 2\pi m/L_x$ and $\theta_2 = 2\pi n/L_y$. The zero B–C block yields an exact closed-form spectrum:
\[E_{mn} \in \left\{-E(\mathbf{k}),\; 0,\; +E(\mathbf{k})\right\}\]
\[E(\mathbf{k}) = 2t\,\sqrt{\cos^2\!\left(\frac{\pi m}{L_x}\right) + \cos^2\!\left(\frac{\pi n}{L_y}\right)}\]
The $E = 0$ eigenvalue appears at every momentum, forming the dispersionless flat band. The spectrum is symmetric about zero (bipartite chiral symmetry).
Flat band and M-point band touching
For a generic $(L_x, L_y)$, the flat band contributes exactly $L_x L_y$ zero eigenvalues. However, when both $L_x$ and $L_y$ are even, the M-point $(\theta_1, \theta_2) = (\pi, \pi)$ is an allowed momentum, and there $E(\mathbf{k}) = 2t\sqrt{\cos^2(\pi/2) + \cos^2(\pi/2)} = 0$. Both dispersive bands collapse to zero at this point, contributing two extra zero modes. The total count of $E = 0$ eigenvalues is:
\[N_{E=0} = L_x L_y + \begin{cases} 2 & \text{if both } L_x, L_y \text{ even} \\ 0 & \text{otherwise} \end{cases}\]
Band structure summary
| Band | Energy range | Notes |
|---|---|---|
| Lower dispersive | $[-2\sqrt{2}\,t,\; 0)$ | Minimum at $\Gamma$: $E = -2t\sqrt{2}$ |
| Flat band | $0$ | $L_x L_y$ states |
| Upper dispersive | $(0,\; +2\sqrt{2}\,t]$ | Maximum at $\Gamma$: $E = +2t\sqrt{2}$ |
Derivation
See Bloch Lieb Flat Band for the derivation of $H(\mathbf{k})$, the closed-form eigenvalues, and the proof that $E = 0$ is an exact eigenvalue at every momentum.
References
- E. H. Lieb, "Two Theorems on the Hubbard Model", Phys. Rev. Lett. 62, 1201 (1989) – Lieb's theorem on flat bands in bipartite lattices.
- H. Tasaki, "From Nagaoka's Ferromagnetism to Flat-Band Ferromagnetism and Beyond", Prog. Theor. Phys. 99, 489 (1998) – review of flat-band physics.
QAtlas API
# Sorted single-particle spectrum, 3×3 Lieb PBC
λ = QAtlas.fetch(QAtlas.Lieb(), TightBindingSpectrum(); Lx=3, Ly=3, t=1.0)
# → 27 eigenvalues; exactly 9 zero modes (flat band, no M-point bonus)Verification
| Test file | Method | What is checked |
|---|---|---|
test_lieb_tight_binding.jl | Real-space ED via Lattice2D | $\lambda_{\text{real}} = \lambda_{\text{Bloch}}$ for $2 \times 2$ through $4 \times 4$ |
test_lieb_tight_binding.jl | Bipartite symmetry | $\sum \lambda_i = 0$ and $\{\lambda\} = \{-\lambda\}$ |
test_lieb_tight_binding.jl | Flat band count | $N_{E=0} = L_x L_y$ (odd sizes) or $L_x L_y + 2$ (both even) |
test_lieb_tight_binding.jl | Ground state | $E_{\min} = -2\sqrt{2}\,t$ (at $\Gamma$) |
test_lieb_tight_binding.jl | $t$ scaling | $\lambda(t) = t \cdot \lambda(1)$ |
test_lieb_tight_binding.jl | $3 \times 3$ M-point | No extra zero modes (both dimensions odd) |
test_bloch_generic.jl | Generic Bloch builder | Closed form $=$ bloch_tb_spectrum |
Connections
- Lattice family: Part of the tight-binding model catalogue. The Lieb lattice is bipartite like Honeycomb but has a flat band due to its sublattice imbalance, unlike the equal-sublattice honeycomb.
- Flat-band contrast: Flat band at $E = 0$ (bipartite origin), compared to $E = +2t$ for the Kagome lattice (frustration origin).
- Dice lattice: The Dice ($T_3$) lattice shares the bipartite flat-band mechanism with the Lieb lattice (flat band at $E = 0$, hub vs rim sublattice imbalance).
- Methods: Computed via the Bloch Hamiltonian method.