On the extensivity of the roots of effective Hamiltonians in many-body formalisms employing incomplete model spaces

We analyze the current controversies regarding the extensivity of energies computed from an effective Hamiltonian defined over an incomplete model space (IMS). We show that the recently developed formalism in Fock space, using a size-extensive normalization for a valence universal operator Ω, generates both a connected Heff and size-extensive energies. In contrast, the corresponding Hilbert space formalisms, with intermediate normalization for Ω, produce size-inextensive energies. It is emphasized that the extensivity of energies for the Fock space theory stems not just from the connectivity of Heff but also due to the existence of certain special null matrix-elements in the matrix of Heff demanded by the decoupling conditions defined in Fock space