2021/05/13 by Federico Castillo, Jean‐Philippe Labbé, Castillo, Federico +9 · 2 citations
Physics and Astronomy · #47L07 #52A27 #81C99 #81R05 #81V73 #81V74 #90C25 #Advanced Chemical Physics Studies #Atomic and Molecular Physics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nuclear physics research studies #Primary 52B12 #Quantum Physics (quant-ph) #Secondary 81V45
paper · pdf · doi:10.48550/arxiv.2105.06459
openalex publication_date 2021/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.