This is the simplest single-impurity single-orbital Anderson model with conserved total charge (Q) and total spin (S) quantum numbers.


\[H_\mathrm{imp} = \epsilon_1 n + U_1 n_\uparrow n_\downarrow\]

where \(n_\sigma = d^\dagger_{\sigma} d_{\sigma}\) and \(n=\sum_\sigma n_\sigma\) with \(\sigma=\uparrow,\downarrow\).


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

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

The number 1 is an orbital index. The choice of including an index (which is not strictly needed here) is to ease writing generic codes that support an arbitrary orbital degeneracy.

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, \(f_0\) (i.e., the local orbital of

the bath at the position of the impurity)

Structure of Green’s functions

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

Dynamic susceptibilities

Dynamic spin and charge susceptibilities are calculated.