Holstein (QS)

This is the single-orbital Anderson-Holstein model with a single phonon mode. The phonon cutoff is part of the Hamiltonian specification and it is hard-coded to 10; the model name is then e.g. Holstein/Nph=10. Conserved total charge (Q) and total spin (S) quantum numbers.


\[H_\mathrm{imp} = \epsilon_1 n + U_1 n_\uparrow n_\downarrow + g_1 (a+a^\dagger) (n - n_1) + \omega a^\dagger a\]


  • \(\epsilon_1\), eps1, energy level

  • \(U_1\), U1, electron-electron interaction

  • \(g_1\), g1, electron-phonon coupling

  • \(n_1\), n1, offset (charge reference point for e-ph coupling)

  • \(\omega\), omega, phonon frequency

Expectation values

  • \(\langle n \rangle\), n_d, impurity occupancy

  • \(\langle n^2 \rangle\), n_d^2, impurity occupancy squared

  • \(\langle \sum_\sigma d^\dagger_\sigma f_{0\sigma} + \text{h.c.} \rangle\), hop0, hopping between the impurity and the zero-th site of the Wilson chain

  • \(\langle a^\dagger a \rangle\), nph, phonon number expectation value

  • \(\langle a+a^\dagger \rangle\), displ, phonon displacement

  • \(\langle (a+a^\dagger)^2 \rangle\), displ^2, phonon displacement squared

Structure of Green’s functions

Single block imp, scalar-valued (1x1 matrix)

Dynamic susceptibilities

Dynamic spin and charge susceptibilities are calculated.

Additional quantities

Phonon spectral function.