where c i † † and c i are the creation and annihilation operators, respectively, of a fermion on the impurity. The index i collects the spin index and possibly other quantum numbers such as orbital (in the case of a multi-orbital impurity) and cluster site (in the case of multi-site impurity). f p † † and f p are the corresponding fermion operators on the non-interacting bath, where the bath quantum number p will typically be continuous.