# SIAM (QS)

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

## Hamiltonian

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

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

## Parameters

• $$\epsilon_1$$, 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.