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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
12/2021

An effective solution to convex 1-body N-representability.

Federico Castillo, Jean-Philippe Labbe, Julia Liebert, Arnau Padrol, Eva Philippe and Christian Schilling

Abstract

From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of 1-fermion and N-fermion density matrices, therefore it coincides with the convex hull of the pure N-representable 1-fermion density matrices. Consequently, the description of ground state physics through 1-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the 1-body N-representability problem to ensemble states with fixed spectrum w, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the 1-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of w. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes.

Received:
May 15, 2021
Published:
May 15, 2021
MSC Codes:
81, 52, 05
Keywords:
N-representability, Symmetric polytopes, Quantum marginal problem

Related publications

inJournal
2023 Repository Open Access
Federico Castillo, Jean-Philippe Labbé, Julia Liebert, Arnau Padrol, Eva Philippe and Christian Schilling

An effective solution to convex \(1\)-body \(N\)-representability

In: Annales Henri Poincaré, (2023)