# openfermion.hamiltonians.FermiHubbardModel

A general, parameterized Fermi-Hubbard model.

The general (AKA 'multiband') Fermi-Hubbard model has k degrees of freedom per site in a lattice. For a lattice with n sites, there are N = k * n spatial orbitals. Additionally, in what we call the "spinful" model each spatial orbital is associated with "up" and "down" spin orbitals, for a total of 2N spin orbitals; in the spinless model, there is only one spin-orbital per site for a total of N.

For a lattice with only one type of site and edges from each site only to itself and its neighbors, the Hamiltonian for the spinful model has the form

.. math::

\begin{align}
H = &- \sum_{a < b} t_{a, b}^{(\mathrm{onsite})}
\sum_{i} \sum_{\sigma}
(a^\dagger_{i, a, \sigma} a_{i, b, \sigma} +
a^\dagger_{i, b, \sigma} a_{i, a, \sigma})
\\
&- \sum_{a} t_{a, a}^{(\mathrm{nghbr})}
\sum_{\{i, j\} } \sum_{\sigma}
(a^\dagger_{i, a, \sigma} a_{j, a, \sigma} +
a^\dagger_{j, a, \sigma} a_{i, a, \sigma})

- \sum_{a < b} t_{a, b}^{(\mathrm{nghbr})}
\sum_{(i, j)} \sum_{\sigma}
(a^\dagger_{i, a, \sigma} a_{j, b, \sigma} +
a^\dagger_{j, b, \sigma} a_{i, a, \sigma})
\\
&+ \sum_{a < b} U_{a, b}^{(\mathrm{onsite}, +)}
\sum_{i} \sum_{\sigma}
n_{i, a, \sigma} n_{i, b, \sigma}
\\
&+ \sum_{a} U_{a, a}^{(\mathrm{nghbr}, +)}
\sum_{\{i, j\} } \sum_{\sigma}
n_{i, a, \sigma} n_{j, a, \sigma}
+ \sum_{a < b} U_{a, b}^{(\mathrm{nghbr}, +)}
\sum_{(i, j)} \sum_{\sigma}
n_{i, a, \sigma} n_{j, b, \sigma}
\\
&+ \sum_{a \leq b} U_{a, b}^{(\mathrm{onsite}, -)}
\sum_{i} \sum_{\sigma}
n_{i, a, \sigma} n_{i, b, -\sigma}
\\
&+ \sum_{a} U_{a, a}^{(\mathrm{nghbr}, -)}
\sum_{\{ i, j \} } \sum_{\sigma}
n_{i, a, \sigma} n_{j, a, -\sigma}
+ \sum_{a < b} U_{a, b}^{(\mathrm{nghbr}, -)}
\sum_{( i, j )} \sum_{\sigma}
n_{i, a, \sigma} n_{j, b, -\sigma}
\\
&- \sum_{a} \mu_a
\sum_i \sum_{\sigma} n_{i, a, \sigma}
\\
&- h \sum_{i} \sum_{a}
\left(n_{i, a, \uparrow} - n_{i, a, \downarrow}\right)
\end{align}


where

- The indices :math:(i, j) and :math:\{i, j\} run over ordered and
unordered pairs, respectively of sites :math:i and :math:j of
neighboring sites in the lattice,
- :math:a and :math:b index degrees of freedom on each site,
- :math:\sigma \in \{\uparrow, \downarrow\} is the spin,
- :math:t_{a, b}^{(\mathrm{onsite})} is the tunneling amplitude
between spin orbitals on the same site,
- :math:t_{a, b}^{(\mathrm{nghbr})} is the tunneling amplitude
between spin orbitals on neighboring sites,
- :math:U_{a, b}^{(\mathrm{onsite, \pm})} is the Coulomb potential
between spin orbitals on the same site with the same (+) or different
(-) spins,
- :math:U_{a, b}^{(\mathrm{nghbr, \pm})} is the Coulomb potential
betwen spin orbitals on neighborings sites with the same (+) or
different (-) spins,
- :math:\mu_{a} is the chemical potential, and
- :math:h is the magnetic field.


One can also construct the Hamiltonian for the spinless model, which has the form

.. math::

\begin{align}
H = &- \sum_{a < b} t_{a, b}^{(\mathrm{onsite})}
\sum_{i}
(a^\dagger_{i, a} a_{i, b} +
a^\dagger_{i, b} a_{i, a})
\\
&- \sum_{a} t_{a, a}^{(\mathrm{nghbr})}
\sum_{\{i, j\} }
(a^\dagger_{i, a} a_{j, a} +
a^\dagger_{j, a} a_{i, a})

- \sum_{a < b} t_{a, b}^{(\mathrm{nghbr})}
\sum_{(i, j)}
(a^\dagger_{i, a} a_{j, b} +
a^\dagger_{j, b} a_{i, a})
\\
&+ \sum_{a < b} U_{a, b}^{(\mathrm{onsite})}
\sum_{i}
n_{i, a} n_{i, b}
\\
&+ \sum_{a} U_{a, a}^{(\mathrm{nghbr})}
\sum_{\{i, j\} }
n_{i, a} n_{j, a}
+ \sum_{a < b} U_{a, b}^{(\mathrm{nghbr})}
\sum_{(i, j)}
n_{i, a} n_{j, b}
\\
&- \sum_{a} \mu_a
\sum_i n_{i, a}
\end{align}


lattice (HubbardLattice): The lattice on which the model is defined. tunneling_parameters (Iterable[Tuple[Hashable, Tuple[int, int], float]], optional): The tunneling parameters. interaction_parameters (Iterable[Tuple[Hashable, Tuple[int, int], float, int?]], optional): The interaction parameters. potential_parameters (Iterable[Tuple[int, float]], optional): The potential parameters. magnetic_field (float, optional): The magnetic field. Default is 0.
particle_hole_symmetry If true, each number operator :math:n is replaced with :math:n - 1/2.

## Methods

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### tunneling_terms

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