Fermi-Hubbard Chain

Purpose and Hamiltonian

The spinful Fermi-Hubbard chain is

$$ H=-t\sum_{\langle i,j\rangle,\sigma} (c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.}) +U\sum_i n_{i\uparrow}n_{i\downarrow} -\mu\sum_{i,\sigma}n_{i\sigma}. $$

The orbital order is $(0\uparrow,0\downarrow,1\uparrow,1\downarrow,\ldots)$.

Fermionic hopping sign

Basis and scaling

There are $2N$ binary orbitals and dimension $2^{2N}$. Dense and CSR builders are available. Fixed spin-resolved particle numbers reduce the dimension to $\binom{N}{N_\uparrow}\binom{N}{N_\downarrow}$ while preserving explicit full-basis occupation-state mappings.

Package use

from quantum_lattice_models import fermi_hubbard_chain_sparse

H = fermi_hubbard_chain_sparse(n_sites=4, hopping=1.0, interaction=4.0)

from quantum_lattice_models import fermi_hubbard_chain_sector

sector = fermi_hubbard_chain_sector(
    n_sites=8, n_up=4, n_down=4, interaction=4.0
)

Parameters

Builder	Parameter	Type	Default	Constraint
`fermi_hubbard_chain`	`n_sites`	`int`	`3`	>= 1
`fermi_hubbard_chain`	`hopping`	`float`	`0.5`
`fermi_hubbard_chain`	`interaction`	`float`	`3.0`
`fermi_hubbard_chain`	`chemical_potential`	`float`	`0.0`
`fermi_hubbard_chain`	`periodic`	`bool`	`False`
`fermi_hubbard_chain_sparse`	`n_sites`	`int`	`3`	>= 1
`fermi_hubbard_chain_sparse`	`hopping`	`float`	`0.5`
`fermi_hubbard_chain_sparse`	`interaction`	`float`	`3.0`
`fermi_hubbard_chain_sparse`	`chemical_potential`	`float`	`0.0`
`fermi_hubbard_chain_sparse`	`periodic`	`bool`	`False`

Validation and cautions

Single-site energies and explicit fermionic parity-sign cases are tested. Sector matrices are validated against full-space blocks for open and periodic chains. The full occupation basis grows faster than the spin-chain basis.