openfermion.hamiltonians.mean_field_dwave

Return symbolic representation of a BCS mean-field d-wave Hamiltonian.

openfermion.hamiltonians.mean_field_dwave(
    x_dimension: int,
    y_dimension: int,
    tunneling: float,
    sc_gap: float,
    chemical_potential: Optional[float] = 0.0,
    periodic: bool = True
) -> openfermion.ops.FermionOperator

Used in the notebooks

Used in the tutorials

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. Each site on the grid can have an "up" fermion and a "down" fermion. Therefore, there are a total of 2N spin-orbitals, where N = x_dimension * y_dimension is the number of sites.

The Hamiltonian for this 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})

    - \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma}
    \\
    &- \sum_{\langle i,j \rangle} \Delta_{ij}
      (a^\dagger_{i, \uparrow} a^\dagger_{j, \downarrow} -
       a^\dagger_{i, \downarrow} a^\dagger_{j, \uparrow} +
       a_{j, \downarrow} a_{i, \uparrow} -
       a_{j, \uparrow} a_{i, \downarrow})
\end{align}

$$

where

- 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
- \\(\Delta_{ij}\\) is equal to \\(+\Delta/2\\) for
  horizontal edges and \\(-\Delta/2\\) for vertical edges,
  where \\(\Delta\\) is the superconducting gap.
- \\(\mu\\) is the chemical potential

x_dimension int

The width of the grid.

y_dimension int

The height of the grid.

tunneling float

The tunneling amplitude \(t\).

sc_gap float

The superconducting gap \(\Delta\)

chemical_potential float, optional

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

periodic bool, optional

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

mean_field_dwave_model An instance of the FermionOperator class.