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

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.

Foundation of one-particle reduced density matrix functional theory for excited states

“…) Thus, we obtain in total five (non-trivial) constraints on the natural occupation number vector λ ↓ , in agreement with Table I. Hence (see also [67]), for arbitrary fermion number N and arbitrary basis set size d, a 1RDM γ is relaxed w-ensemble N -representable for r = 4, γ ∈ E 1 N (w), if and only if its decreasingly ordered natural occupation numbers fulfill the following constraints…”

“…A more comprehensive mathematical analysis presented in our forthcoming work [67] reveals an even stronger connection between the spectral polytopes Σ(w) for different weight vectors w. Comparing their hyperplane representations for different values r reveals a hierarchical generalization of Pauli's exclusion principle. For r = 1, Σ(w) is described by the Pauli exclusion principle constraint λ ↓ 1 ≤ 1.…”

“…Most importantly, in analogy to Pauli's exclusion principle and in striking contrast to the generalized Pauli constraints [54,55,64], these conditions are effectively independent of the particle number N and the dimension d of the underlying one-particle Hilbert space. To be more specific (as it will be explained and proven in a more mathematical context in [67]), calculating these conditions for some specific (smaller) setting (N, d) would immediately imply the corresponding conditions for arbitrary N, d, including the important complete basis set limit, i.e., d → ∞.…”

“…This is a pleasant feature of the relaxed w-ensemble N -representability constraints compared to the generalized Pauli constraints [54,55] whose complexity depends strongly on the setting (N, d). Hence, the hierarchy of generalized exclusion principle constraints is independent of the values of N and d. Actually, not only the number of inequalities is independent of N and d but even their form (see also the examples in the next section and [67]). We complete the present section by presenting the number of generating vectors v (i) and inequalities for r up to twelve in Table I…”

“…We present the mathematical formalism for this in Ref. [67] and provide in the following only the inequalities. For the setting (N, d) = (3,6), the minimal hyperplane representation consists of all facet-defining inequalities for r = 3, complemented by the following two new constraints…”