Tight-Binding Chain

Purpose and Hamiltonian

The generic one-dimensional single-particle model is

$$ H=-t\sum_{\langle i,j\rangle}(|i\rangle\langle j|+\mathrm{h.c.}) +\sum_i\epsilon_i|i\rangle\langle i|. $$

onsite may be a scalar or one value per site. Dense and CSR builders are available.

Package use

from quantum_lattice_models import tight_binding_chain_sparse

H = tight_binding_chain_sparse(
    n_sites=128, hopping=1.0, onsite=0.2, periodic=False
)

Parameters

Builder	Parameter	Type	Default	Constraint
`tight_binding_chain`	`n_sites`	`int`	`8`	>= 1
`tight_binding_chain`	`hopping`	`float`	`1.0`
`tight_binding_chain`	`onsite`	`float`	`0.0`
`tight_binding_chain`	`periodic`	`bool`	`False`
`tight_binding_chain_sparse`	`n_sites`	`int`	`32`	>= 1
`tight_binding_chain_sparse`	`hopping`	`float`	`1.0`
`tight_binding_chain_sparse`	`onsite`	`float`	`0.0`
`tight_binding_chain_sparse`	`periodic`	`bool`	`False`

Validation and cautions

The open uniform-chain spectrum is checked against $E_m=-2t\cos[m\pi/(N+1)]$. For n_sites=2, periodic construction includes both directed wrap contributions, consistent with the package bond convention.