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

TL;DR

This paper develops an exact one-particle functional theory for bosonic excitations, extending the Hohenberg-Kohn theorem and solving the N-representability problem with a convex relaxation, revealing a hierarchy of bosonic exclusion principles.

Contribution

It introduces a generalized Hohenberg-Kohn theorem for bosonic systems and provides a convexified solution to the N-representability problem, including a hierarchy of bosonic exclusion constraints.

Findings

Proves a one-to-one correspondence between the one-particle reduced density matrix and system properties.

Develops a convex relaxation approach to the bosonic N-representability problem.

Discovers a hierarchy of bosonic exclusion principles analogous to fermionic Pauli constraints.

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 $\boldsymbol{w}$ 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 $\boldsymbol{w}$-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 $\boldsymbol{w}$-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.