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.

Deriving density-matrix functionals for excited states

“…3 and 4 we introduce in this section the required foundational concepts of w -ensemble RDMFT which has recently been proposed by us for bosons in Ref. [10] and for fermions in Ref. [8,9].…”

“…One of the main achievements of Refs. [8][9][10]18] was to overcome the too intricate wensemble N -representability constraints that define the domain E 1 N (w ) of the universal functional F w . In analogy to Valone's ground state RDMFT [21] this was achieved by performing an exact convex relaxation.…”

“…It was exactly this convex relaxation which allowed us in Refs. [8][9][10] to obtain a feasible functional theory thanks to a comprehensive characterization of the set E 1 N (w ) for bosons and fermions. To be more specific, we derived a compact description of the corresponding spectral set Σ(w…”

“…This and other recent progress in the field of ground state RDMFT for bosons [3][4][5][6][7] and w -ensemble RDMFT for excited states [8][9][10] urges us to systematically derive in this paper the first functionals for w -ensemble RDMFT. To initiate the development of w -ensemble functionals in different fields of physics we derive analytically the universal functional for both the symmetric Hubbard dimer with on-site interaction and the homogeneous Bose gas in the Bogoliubov regime.…”

Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

“…In recent times, EDFT has undergone significant developments that are crucial to its advancement. − However, it suffers from the disadvantages that, to treat a high-lying excited state, it is usually required to include all lower-lying states in the ensemble and that the weight dependence of the exchange-correlation functional is hard to model. Another ensemble theory that has been receiving increasing attention and shares some of the problems of EDFT is w -ensemble one-body reduced density matrix functional theory. − …”