# 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

\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.

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