Return symbolic representation of a Fermi-Hubbard Hamiltonian.

Used in the notebooks

Used in the tutorials

The idea of this model is that some fermions move around on a grid and the energy of the model depends on where the fermions are. The Hamiltonians of this model live on a grid of dimensions x_dimension x y_dimension. The grid can have periodic boundary conditions or not. In the standard Fermi-Hubbard model (which we call the "spinful" model), there is room for an "up" fermion and a "down" fermion at each site on the grid. In this model, there are a total of 2N spin-orbitals, where N = x_dimension * y_dimension is the number of sites. In the spinless model, there is only one spin-orbital per site for a total of N.

The Hamiltonian for the spinful model has the form

$$ \begin{align} H = &- t \sum{\langle i,j \rangle} \sum{\sigma} (a^\dagger{i, \sigma} a{j, \sigma} + a^\dagger{j, \sigma} a{i, \sigma})

     + U \sum_{i} a^\dagger_{i, \uparrow} a_{i, \uparrow}
                 a^\dagger_{i, \downarrow} a_{i, \downarrow}
    &- \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma}
     - h \sum_i (a^\dagger_{i, \uparrow} a_{i, \uparrow} -
               a^\dagger_{i, \downarrow} a_{i, \downarrow})



- The indices \\(\langle i, j \rangle\\) run over pairs
  \\(i\\) and \\(j\\) of sites that are connected to each other
  in the grid
- \\(\sigma \in \{\uparrow, \downarrow\}\\) is the spin
- \\(t\\) is the tunneling amplitude
- \\(U\\) is the Coulomb potential
- \\(\mu\\) is the chemical potential
- \\(h\\) is the magnetic field

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

$$ H = - t \sum_{\langle i, j \rangle} (a^\dagger_i a_j + a^\dagger_j a_i)

    + U \sum_{\langle i, j \rangle} a^\dagger_i a_i a^\dagger_j a_j
    - \mu \sum_i a_i^\dagger a_i.


x_dimension int

The width of the grid.

y_dimension int

The height of the grid.

tunneling float

The tunneling amplitude \(t\).

coulomb float

The attractive local interaction strength \(U\).

chemical_potential float, optional

The chemical potential \(\mu\) at each site. Default value is 0.

magnetic_field float, optional

The magnetic field \(h\) at each site. Default value is 0. Ignored for the spinless case.

periodic bool, optional

If True, add periodic boundary conditions. Default is True.

spinless bool, optional

If True, return a spinless Fermi-Hubbard model. Default is False.

particle_hole_symmetry bool, optional

If False, the repulsion term corresponds to:

    U \sum_{k=1}^{N-1} a_k^\dagger a_k a_{k+1}^\dagger a_{k+1}

If True, the repulsion term is replaced by:

    U \sum_{k=1}^{N-1} (a_k^\dagger a_k - \frac12)
                       (a_{k+1}^\dagger a_{k+1} - \frac12)

which is unchanged under a particle-hole transformation.
Default is False

hubbard_model An instance of the FermionOperator class.