qit.hamiltonian.hubbard¶
- qit.hamiltonian.hubbard(C, U=1, mu=0)¶
Hubbard model, fermions on a lattice.
- Parameters
C (array[bool]) – Connection matrix of the interaction graph
U (float) – Fermion-fermion interaction strength (normalized)
mu (float) – External chemical potential (normalized)
- Returns
Hamiltonian, dimension vector
- Return type
tuple
The Hubbard model consists of spin-1/2 fermions confined in a graph defined by the symmetric connection matrix C (only upper triangle is used). The fermions interact with other fermions at the same site with interaction strength U, as well as with an external chemical potential mu. The Hamiltonian has been normalized by the fermion hopping constant t.
\[H = -\sum_{\langle i,j \rangle, \sigma} c^\dagger_{i,\sigma} c_{j,\sigma} +\frac{U}{t} \sum_i n_{i,up} n_{i,\downarrow} -\frac{\mu}{t} \sum_i (n_{i,up}+n_{i,\downarrow})\]