qit.hamiltonian.bose_hubbard¶
- qit.hamiltonian.bose_hubbard(C, U=1, mu=0, m=10)¶
Bose-Hubbard model, bosons 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)
m (int) – boson Hilbert space truncation dimension
- Returns
Hamiltonian, dimension vector
- Return type
tuple
The Bose-Hubbard model consists of spinless bosons confined in a graph defined by the symmetric connection matrix C (only upper triangle is used). The bosons interact with other bosons at the same site with interaction strength U, as well as with an external chemical potential mu. The Hamiltonian has been normalized by the boson hopping constant t.
\[H = -\sum_{\langle i,j \rangle} b^\dagger_i b_{j} +\frac{U}{2t} \sum_i n_i (n_i-1) -\frac{\mu}{t} \sum_i n_i\]The dimensions of the boson Hilbert spaces (infinite in principle) are truncated to m.