ffsim.fermi_hubbard_1d¶
- ffsim.fermi_hubbard_1d(norb, tunneling, interaction, *, chemical_potential=0, nearest_neighbor_interaction=0, periodic=False)[source]¶
One-dimensional Fermi-Hubbard model Hamiltonian.
The Hamiltonian for the one-dimensional Fermi-Hubbard model with \(N\) spatial orbitals is given by
\[\begin{split}H = -t \sum_{\substack{\braket{pq} \\ \sigma}} (a^\dagger_{p\sigma} a_{q\sigma} + a^\dagger_{q\sigma} a_{p\sigma}) + U \sum_p n_{p\alpha} n_{p\beta} - \mu \sum_p (n_{p\alpha} + n_{p\beta}) + V \sum_{\substack{\braket{pq} \\ \sigma \tau}} n_{p\sigma} n_{q\tau}\end{split}\]where \(n_{p\sigma} = a_{p\sigma}^\dagger a_{p\sigma}\) is the number operator on orbital \(p\) with spin \(\sigma\) and the index \(\braket{pq}\) runs over pairs of orbitals \(p\) and \(q\) that are connected on the line. If periodic boundary conditions are chosen, then the first and last orbitals are connected. More explicitly:
For open boundary conditions, \(\braket{pq}\) runs over pairs \((p, p + 1)\) for \(p = 1, \ldots, N - 1\).
For periodic boundary conditions, \(\braket{pq}\) runs over pairs \((p, p + 1 \bmod N)\) for \(p = 1, \ldots, N\).
In the case that \(N = 2\), using periodic boundary conditions will cause the connection between the two vertices to be counted twice, forming a “ring with two edges”.
References
- Parameters:
norb (
int) – The number of spatial orbitals \(N\).tunneling (
float) – The tunneling amplitude \(t\).interaction (
float) – The onsite interaction strength \(U\).chemical_potential (
float) – The chemical potential \(\mu\).nearest_neighbor_interaction (
float) – The nearest-neighbor interaction strength \(V\).periodic (
bool) – Whether to use periodic boundary conditions.
- Return type:
- Returns:
The one-dimensional Fermi-Hubbard model Hamiltonian.