Point-Group Symmetry and the Impurity Problem

It is shown that in a pure crystal the density of states of a given point-group symmetry (that is, belonging to a given row of a given irreducible representation of the point group) is simply (d_Γp)g⁰(E), where g⁰(E) is the familiar over-all density of states, d_Γ is the dimensionality of the representation, and p is the order of the group. Point-group symmetry is incorporated into Green's-function theory to define Green's functions which propagate excitations of particular symmetries from shell to shell (rather than from site to site). Both simple crystal Green's functions and two-time thermodynamic Green's functions are considered, and the ferromagnetic-impurity problem is formulated in terms of such Green's functions (at $T=0$ and at arbitrary temperatures, respectively). The analysis is illustrated explicitly for a fcc ferromagnet with first- and second-nearest-neighbor exchange.