qit.hamiltonian.holstein¶
- qit.hamiltonian.holstein(C, omega=1, g=1, m=10)¶
Holstein model, electrons on a lattice coupled to phonons.
- Parameters
C (array) – Connection matrix of the interaction graph
omega (float) – phonon frequency (normalized)
g (float) – electron-phonon coupling constant (normalized)
m (int) – phonon Hilbert space truncation dimension
- Returns
Hamiltonian, dimension vector
- Return type
tuple
The Holstein model consists of spinless electrons confined in a graph defined by the symmetric connection matrix C (only upper triangle is used), coupled to phonon modes represented by a harmonic oscillator at each site. The dimensions of phonon Hilbert spaces (infinite in principle) are truncated to m.
The order of the subsystems is [e1, …, en, p1, …, pn]. The Hamiltonian has been normalized by the electron hopping constant t.
\[H = -\sum_{\langle i,j \rangle} c_i^\dagger c_j +\frac{\omega}{t} \sum_i b_i^\dagger b_i -\frac{g \omega}{t} \sum_i (b_i + b_i^\dagger) c_i^\dagger c_i\]