ffsim.fermi_hubbard_2d

ffsim.fermi_hubbard_2d(norb_x, norb_y, tunneling, interaction, *, chemical_potential=0, nearest_neighbor_interaction=0, periodic=False, periodic_x=False, periodic_y=False)[source]

Two-dimensional Fermi-Hubbard model Hamiltonian on a square lattice.

The Hamiltonian for the two-dimensional Fermi-Hubbard model on a square lattice with \(N_x\) columns and \(N_y\) rows 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 lattice. If periodic boundary conditions are chosen, then orbitals in the first and last columns are connected, as are orbitals in the first and last rows, so that the square lattice forms a torus. In the case that one of the dimensions has size 2, using periodic boundary conditions will cause the connection along that dimension to be counted twice.

References

Parameters:

  • norb_x (int) – The number of spatial orbitals in the x-direction \(N_x\).

  • norb_y (int) – The number of spatial orbitals in the y-direction \(N_y\).

  • 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 in all dimensions. This argument overrides the periodic_x and periodic_y arguments.

  • periodic_x (bool) – Whether to use periodic boundary conditions in the X dimension.

  • periodic_y (bool) – Whether to use periodic boundary conditions in the Y dimension.

Return type:

FermionOperator

Returns:

The two-dimensional Fermi-Hubbard model Hamiltonian.