qat.fermion.hamiltonians.make_anderson_model

qat.fermion.hamiltonians.make_anderson_model(u: float, mu: float, v: ndarray, epsilon: ndarray) → ElectronicStructureHamiltonian

Returns the canonical second quantized form

\[H_{\mathrm{CSQ}} = \sum_{p,q} h_{pq} f_p^\dagger f_q + \frac{1}{2}\sum_{p,q,r,s} h_{pqrs} f_p^\dagger f_q^\dagger f_r f_s\]

of a single impurity coupled with \(n_b\) bath modes Anderson model Hamiltonian

\[\begin{split}H_{\mathrm{SIAM}} = U c_{\uparrow}^\dagger c_{\uparrow} c_{\downarrow}^\dagger c_{\downarrow} - \mu(c_{\uparrow}^\dagger c_{\uparrow}+c_{\downarrow}^\dagger c_{\downarrow}) + \sum_{i=1..n_b} \sum_{\sigma=\uparrow,\downarrow} V_i (c_{\sigma}^\dagger a_{i,\sigma} + \mathrm{h.c.}) \\ + \sum_{i=1..n_b} \sum_{\sigma=\uparrow,\downarrow} \epsilon_i a_{i,\sigma}^\dagger a_{i,\sigma}.\end{split}\]

Parameters

• U (float) – Coulomb repulsion intensity.

• mu (float) – Chemical potential.

• V (np.ndarray) – Tunneling energies. This vector has the same size as the number of bath mode.

• epsilon (np.ndarray) – Bath modes energies. This vector has the same size as the number of bath mode.

Returns

ElectronicStructureHamiltonian object constructed from \(h_{pq}\) (matrix of size \((2n_b+2) \times (2n_b+2)\)) and \(h_{pqrs}\) (4D tensor with size \(2n_b+2\) in each dimension)

Note

Convention: \(f_0\) corresponds to \(c_{\uparrow}\) (annihilation in the ‘up’ mode of the impurity), \(f_1\) corresponds to \(c_{\downarrow}\) (annihilation in the ‘down’ mode of the impurity), \(f_2\) corresponds to \(a_{1,\uparrow}\) (annihilation in the ‘up’ mode of the 1st bath mode), \(f_3\) corresponds to \(a_{1,\downarrow}\) (annihilation in the ‘down’ mode of the 1st bath mode), and so on.