Proper derivation of subspace mapping from whole space mapping in boson expansion theory

TL;DR

The paper addresses how to derive subspace mapping from whole-space mapping in boson expansion theory by employing the norm operator method, which renormalizes contributions from phonons not adopted as boson excitations. It formalizes a mapping framework with $U_\xi = \\hat{Z}^{\xi-\frac12} \\widetilde{U}$ and projection operators that connect WFSM and FSSM, showing that non-adopted phonon effects can be consistently incorporated and that FSSM can be obtained from WFSM under appropriate renormalization, contrary to conventional claims. It demonstrates that the Park operator is effective for both WFSM and FSSM when NAMD is satisfied and clarifies the relationship between small-parameter expansions and NAMD, revealing when expansions are infinite versus finite. Overall, the norm-operator approach resolves long-standing confusion in BET, generalizes skeleton boson realizations beyond the Dyson framework, and provides a practical route to derive subspace mappings with controlled Pauli effects.

Abstract

The norm operator method, which was recently proposed as a new formulation of the boson expansion theory (BET), is used to show that the subspace mapping is properly derived from the whole space mapping. This derivation requires the appropriate renormalization of the contribution of phonons that are not adopted as boson excitations in the subspace mapping. This was impossible with conventional BETs (which ignore these contributions), and is only made possible for the first time by the norm operator method, which treats these contributions appropriately. We also correct the confusion in the claims of conventional BETs. Namely, contrary to conventional claims, we show that when the phonon excitations not adopted as boson excitations make no contribution at all, the subspace mapping is obtained simply by discarding those excitations. Furthermore, we demonstrate that the Park operator, which had been considered effective only in the whole space mapping, is also effective in the subspace mapping.