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

\[ \begin{aligned} H = &- t \sum_{\langle i,j \rangle} \sum_\sigma (a^\dagger_{i, \sigma} a_{j, \sigma} + a^\dagger_{j, \sigma} a_{i, \sigma}) - \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma} \\ &- \sum_{\langle i,j \rangle} \Delta_{ij} (a^\dagger_{i, \uparrow} a^\dagger_{j, \downarrow} - a^\dagger_{i, \downarrow} a^\dagger_{j, \uparrow} + a_{j, \downarrow} a_{i, \uparrow} - a_{j, \uparrow} a_{i, \downarrow}) \end{aligned} \]

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.