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Return symbolic representation of a Bose-Hubbard Hamiltonian.
openfermion.hamiltonians.bose_hubbard(
x_dimension,
y_dimension,
tunneling,
interaction,
chemical_potential=0.0,
dipole=0.0,
periodic=True
)
Used in the notebooks
| Used in the tutorials |
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In this model, bosons move around on a lattice, and the energy of the model depends on where the bosons are.
The lattice is described by a 2D grid, with dimensions
x_dimension x y_dimension. It is also possible to specify
if the grid has periodic boundary conditions or not.
The Hamiltonian for the Bose-Hubbard model has the form
$$ H = - t \sum_{\langle i, j \rangle} (b_i^\dagger b_j + b_j^\dagger b_i)
+ V \sum_{\langle i, j \rangle} b_i^\dagger b_i b_j^\dagger b_j
+ \frac{U}{2} \sum_i b_i^\dagger b_i (b_i^\dagger b_i - 1)
- \mu \sum_i b_i^\dagger b_i.
$$
where
- The indices \\(\langle i, j \rangle\\) run over pairs
\\(i\\) and \\(j\\) of nodes that are connected to each other
in the grid
- \\(t\\) is the tunneling amplitude
- \\(U\\) is the on-site interaction potential
- \\(\mu\\) is the chemical potential
- \\(V\\) is the dipole or nearest-neighbour interaction potential
Args | |
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| x_dimension (int): The width of the grid. y_dimension (int): The height of the grid. tunneling (float): The tunneling amplitude \(t\). interaction (float): The attractive local interaction strength \(U\). 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. dipole (float): The attractive dipole interaction strength \(V\). |
Returns | |
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bose_hubbard_model
|
An instance of the BosonOperator class. |