An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability

Abstract

Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v -representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N -representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

Keywords

• one-particle reduced density matrix
• functional theory
• N-representability
• bosons
• Hohenberg–Kohn theorem
• convex analysis