openfermion.hamiltonians.mean_field_dwave

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

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

.. math::

\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 :math:`\langle i, j \rangle` run over pairs
  :math:`i` and :math:`j` of sites that are connected to each other
  in the grid
- :math:`\sigma \in \{\uparrow, \downarrow\}` is the spin
- :math:`t` is the tunneling amplitude
- :math:`\Delta_{ij}` is equal to :math:`+\Delta/2` for
  horizontal edges and :math:`-\Delta/2` for vertical edges,
  where :math:`\Delta` is the superconducting gap.
- :math:`\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 :math:t. sc_gap (float): The superconducting gap :math:\Delta chemical_potential (float, optional): The chemical potential :math:\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.