Triangular Lattice
Overview
The nearest-neighbor tight-binding model on the triangular lattice is the simplest frustrated lattice hopping problem. Each site has 6 nearest neighbours, the lattice has a single sublattice per unit cell, and the resulting single band exhibits a characteristic asymmetric dispersion that is the hallmark of geometric frustration.
\[H = -t \sum_{\langle i,j \rangle} \bigl(c^\dagger_i c_j + c^\dagger_j c_i\bigr)\]
Lattice properties: 1 sublattice per unit cell, 6 nearest neighbours, not bipartite (frustrated), no flat band.
Key physics: Because the triangular lattice is non-bipartite, there is no chiral (particle-hole) symmetry, and the spectrum is not symmetric about $E = 0$. The band ranges from $-6t$ (at $\Gamma$) to $+3t$ (at the $K$-points), an asymmetric window that is a direct consequence of frustration. The density of states exhibits a Van Hove singularity within the band.
Bloch Spectrum
Statement
Since the triangular lattice has one sublattice, the Bloch "Hamiltonian" is a scalar at each $\mathbf{k}$-point:
\[E_{mn} = -2t\left[\cos\theta_1 + \cos\theta_2 + \cos(\theta_2 - \theta_1)\right]\]
where $\theta_1 = 2\pi m/L_x$ and $\theta_2 = 2\pi n/L_y$, with $\mathbf{a}_1 = (1, 0)$, $\mathbf{a}_2 = (1/2, \sqrt{3}/2)$ the primitive vectors of Lattice2D's triangular topology.
The full spectrum has $L_x L_y$ eigenvalues, one per allowed momentum.
Band edges
| Point | $(m/L_x, n/L_y)$ | Energy |
|---|---|---|
| $\Gamma$ | $(0, 0)$ | $-6t$ (global minimum, unique) |
| $K$ | $(1/3, 2/3)$ | $+3t$ (global maximum) |
| $K'$ | $(2/3, 1/3)$ | $+3t$ (global maximum) |
The band range $[-6t, +3t]$ has total width $9t$, compared to $[-4t, +4t]$ (width $8t$) for the square lattice. The asymmetry $|E_{\min}| = 6t \neq |E_{\max}| = 3t$ is a direct manifestation of the absence of bipartite symmetry.
Van Hove singularity
At certain energies within the band, the density of states $g(E) = \sum_{\mathbf{k}} \delta(E - E(\mathbf{k}))$ diverges logarithmically due to saddle points in the dispersion $E(\mathbf{k})$. These Van Hove singularities are responsible for electronic instabilities (magnetism, superconductivity) in triangular-lattice materials.
K-point commensurability
The $K$-point eigenvalue $E = +3t$ appears in the finite-size spectrum only when both $L_x$ and $L_y$ are divisible by 3, since the $K$-point momenta $(1/3, 2/3)$ and $(2/3, 1/3)$ must be commensurate with the discrete Brillouin zone grid. When this condition is met, exactly 2 eigenvalues sit at $+3t$.
References
- G. H. Wannier, "Antiferromagnetism. The Triangular Ising Net", Phys. Rev. 79, 357 (1950) – triangular lattice frustration.
- T. Koretsune, M. Ogata, "Electronic structures of triangular lattice models", J. Phys. Soc. Jpn. 76, 074706 (2007) – NN tight-binding spectrum and Van Hove singularity.
QAtlas API
# Sorted single-particle spectrum, 6×6 triangular PBC
λ = QAtlas.fetch(QAtlas.Triangular(), TightBindingSpectrum(); Lx=6, Ly=6, t=1.0)
# → 36 eigenvalues, ranging from -6.0 to +3.0Verification
| Test file | Method | What is checked |
|---|---|---|
test_triangular_tight_binding.jl | Real-space ED via Lattice2D | $\lambda_{\text{real}} = \lambda_{\text{Bloch}}$ for $3 \times 3$ through $6 \times 6$ |
test_triangular_tight_binding.jl | Band edges | $E_{\min} = -6t$ (unique), $E_{\max} = +3t$ (when 3 |
test_triangular_tight_binding.jl | Frustration | Spectrum is NOT symmetric about zero |
test_triangular_tight_binding.jl | $K$-point degeneracy | $\geq 2$ eigenvalues at $+3t$ when both $L_x, L_y \equiv 0 \pmod{3}$ |
test_triangular_tight_binding.jl | Structural | $\text{tr}\,H = 0$ |
test_triangular_tight_binding.jl | $t$ scaling | $\lambda(t) = t \cdot \lambda(1)$ |
test_bloch_generic.jl | Generic Bloch builder | Scalar formula $=$ bloch_tb_spectrum |
Connections
- Lattice family: Part of the tight-binding model catalogue. The triangular lattice is frustrated (non-bipartite) like Kagome, but has only one sublattice and no flat band.
- Bipartite contrast: The square lattice (also 1 sublattice) has a symmetric band $[-4t, +4t]$ because it is bipartite. The triangular band $[-6t, +3t]$ breaks this symmetry.
- Magnetic frustration: The non-bipartite structure underlies the classical Ising antiferromagnet frustration problem on the triangular lattice (Wannier 1950).
- Methods: Computed via the Bloch Hamiltonian method.