Tight-Binding Models

Overview

The nearest-neighbor tight-binding (TB) Hamiltonian on a periodic lattice is one of the simplest quantum models, yet it already captures essential physics: Dirac cones (graphene), flat bands (kagome, Lieb), frustration-induced band asymmetry (triangular), and Van Hove singularities.

\[H = -t \sum_{\langle i,j \rangle} \bigl(c^\dagger_i c_j + c^\dagger_j c_i\bigr)\]

where $t > 0$ is the nearest-neighbor hopping amplitude and the sum runs over all nearest-neighbor pairs $\langle i,j \rangle$ determined by the lattice topology.

Parameters: Hopping amplitude $t$ (default 1.0).

Boundary conditions: All TB spectra in QAtlas use periodic boundary conditions (PBC) in both directions on an $L_x \times L_y$ supercell of unit cells.

Two Computation Paths

QAtlas implements two independent routes to the TB spectrum, enabling cross-validation:

1. Hardcoded Bloch formulas (src/models/quantum/tightbinding/): For each lattice with a dedicated source file, the closed-form dispersion relation is evaluated analytically at each allowed $\mathbf{k}$-point. This is maximally efficient and transparent.

2. Generic Bloch builder (test/util/bloch.jl): Given any Lattice2D topology, the function bloch_tb_spectrum reads the unit cell definition (sublattice positions and connections) via get_unit_cell, constructs the $n_{\text{sub}} \times n_{\text{sub}}$ Bloch Hamiltonian

   \[H_{\alpha\beta}(\mathbf{k}) = -t \sum_{\text{connections}\; \beta \to \alpha} e^{i(d_x \theta_1 + d_y \theta_2)} + \text{h.c.}\]

   at each of the $L_x \cdot L_y$ allowed momenta, and diagonalizes it numerically. This requires no model-specific formulas and serves as a fully generic cross-check.

When both paths agree (verified in test_bloch_generic.jl), the hardcoded formulas and the generic builder are mutually validated.

For lattices without dedicated source files (Dice, Shastry–Sutherland, Union Jack, Square), only the generic builder and real-space ED are used; both agree in every tested case.

Lattice Catalogue

Bipartite: The lattice graph admits a two-colouring (A/B sublattice decomposition). Bipartite lattices have chiral (sublattice) symmetry, so the spectrum is symmetric about $E = 0$.

Flat band: A completely dispersionless band (zero bandwidth). Flat bands arise from destructive interference in the hopping network, typically associated with a sublattice imbalance in bipartite lattices (Lieb, Dice) or geometric frustration (Kagome).

Generic Bloch Builder

The generic builder in test/util/bloch.jl works for any 2D periodic topology that Lattice2D supports:

include("test/util/bloch.jl")

# Spectrum of the 3×3 honeycomb TB model
λ = bloch_tb_spectrum(Honeycomb, 3, 3, 1.0)

# Works identically for any topology
λ_dice = bloch_tb_spectrum(Dice, 4, 4, 1.0)
λ_ss   = bloch_tb_spectrum(ShastrySutherland, 3, 3, 1.0)

The builder reads Lattice2D.get_unit_cell(Topology), which returns the sublattice positions and a list of Connection objects $(s_{\text{src}}, s_{\text{dst}}, d_x, d_y)$. No hand-derived formulas are needed.

Verification