Additional Benchmark Lattices

Graphene

graphene_lattice is the nearest-neighbor limit of the finite honeycomb construction. With zero onsite terms it is bipartite and its finite spectrum is symmetric around zero.

Two-dimensional Anderson model

anderson_square_lattice adds reproducible uniform onsite disorder in $[-W/2,W/2]$ to a square-lattice hopping Hamiltonian. The random seed and boundary conditions are explicit parameters.

Checkerboard Chern insulator

checkerboard_chern_insulator uses a two-orbital Qi-Wu-Zhang-type real-space representation,

$$ H(\mathbf k)=t\sin k_x,\sigma_x+t\sin k_y,\sigma_y+ (m+t\cos k_x+t\cos k_y)\sigma_z. $$

The orbital ordering is row-major unit cells with two orbitals per cell.

Dice or $T_3$ lattice

dice_lattice has one hub and two rim sites per unit cell. Its bipartite imbalance produces a zero-energy flat-band subspace in finite open systems.

Package use

from quantum_lattice_models import (
    anderson_square_lattice_sparse,
    checkerboard_chern_insulator,
    dice_lattice,
    graphene_lattice,
)

graphene = graphene_lattice(3, 4)
disordered = anderson_square_lattice_sparse(12, 12, disorder=4.0, seed=7)
chern = checkerboard_chern_insulator(4, 4, mass=1.0)
dice = dice_lattice(3, 4)

Parameters

Builder Parameter Type Default Constraint
graphene_lattice n_rows int 3 >= 1
graphene_lattice n_cols int 3 >= 1
graphene_lattice hopping float 1.0
graphene_lattice periodic_x bool False
graphene_lattice periodic_y bool False
graphene_lattice_sparse n_rows int 3 >= 1
graphene_lattice_sparse n_cols int 3 >= 1
graphene_lattice_sparse hopping float 1.0
graphene_lattice_sparse periodic_x bool False
graphene_lattice_sparse periodic_y bool False
anderson_square_lattice n_rows int 4 >= 1
anderson_square_lattice n_cols int 4 >= 1
anderson_square_lattice hopping float 1.0
anderson_square_lattice disorder float 1.0
anderson_square_lattice seed int 0
anderson_square_lattice periodic_x bool False
anderson_square_lattice periodic_y bool False
anderson_square_lattice_sparse n_rows int 4 >= 1
anderson_square_lattice_sparse n_cols int 4 >= 1
anderson_square_lattice_sparse hopping float 1.0
anderson_square_lattice_sparse disorder float 1.0
anderson_square_lattice_sparse seed int 0
anderson_square_lattice_sparse periodic_x bool False
anderson_square_lattice_sparse periodic_y bool False
checkerboard_chern_insulator n_rows int 3 >= 1
checkerboard_chern_insulator n_cols int 3 >= 1
checkerboard_chern_insulator hopping float 1.0
checkerboard_chern_insulator mass float 1.0
checkerboard_chern_insulator periodic_x bool False
checkerboard_chern_insulator periodic_y bool False
checkerboard_chern_insulator_sparse n_rows int 3 >= 1
checkerboard_chern_insulator_sparse n_cols int 3 >= 1
checkerboard_chern_insulator_sparse hopping float 1.0
checkerboard_chern_insulator_sparse mass float 1.0
checkerboard_chern_insulator_sparse periodic_x bool False
checkerboard_chern_insulator_sparse periodic_y bool False
dice_lattice n_rows int 3 >= 1
dice_lattice n_cols int 3 >= 1
dice_lattice hopping float 1.0
dice_lattice periodic_x bool False
dice_lattice periodic_y bool False
dice_lattice_sparse n_rows int 3 >= 1
dice_lattice_sparse n_cols int 3 >= 1
dice_lattice_sparse hopping float 1.0
dice_lattice_sparse periodic_x bool False
dice_lattice_sparse periodic_y bool False

Validation and cautions

Dense and sparse builders are cross-checked. The package validates graphene spectral symmetry, Anderson reproducibility, checkerboard Hermiticity, and the dice zero-energy flat-band subspace. Finite models do not by themselves imply thermodynamic or bulk topological conclusions.