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

$$ \begin{align} H = &- t \sum_{\langle i,j \rangle} \sum_{\sigma} (a^\dagger_{i, \sigma} a_{j, \sigma} + a^\dagger_{j, \sigma} a_{i, \sigma}) + U \sum_{i} a^\dagger_{i, \uparrow} a_{i, \uparrow} a^\dagger_{i, \downarrow} a_{i, \downarrow} \\ &- \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma} - h \sum_i (a^\dagger_{i, \uparrow} a_{i, \uparrow} - a^\dagger_{i, \downarrow} a_{i, \downarrow}) \end{align} $$


- 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.