v-Representability on a one-dimensional torus at elevated temperatures

We extend a previous result [Sutter et al., J. Phys. A: Math. Theor. 57, 475202 (2024)] to give an explicit form of the set of $v$-representable densities on the one-dimensional torus with any fixed number of particles in contact with a heat bath at finite temperature. The particle interaction has to satisfy some mild assumptions but is kept entirely general otherwise. For densities, we consider the Sobolev space $H^1$ and exploit the convexity of the functionals. This leads to a broader set of potentials than the usual $L^p$ spaces and encompasses distributions. By including temperature and thus considering all excited states in the Gibbs ensemble, Gâteaux differentiability of the thermal universal functional is guaranteed. This yields $v$-representability and it is demonstrated that the given set of $v$-representable densities is even maximal.

💡 Research Summary

The paper addresses the long‑standing v‑representability problem in density‑functional theory (DFT) for a system of N spin‑½ fermions confined to a one‑dimensional torus, extending previous zero‑temperature results to finite (thermal) temperatures. The authors consider a very general class of two‑body interactions W that satisfy the Kato–Lions–Miyazawa–Nelson (KLMN) condition, ensuring that the full Hamiltonian H_v = –½Δ + W + V (with V the external one‑body potential) is self‑adjoint on the same quadratic form domain as the kinetic energy.

The central innovation is the choice of functional spaces. Densities are taken from the Sobolev space H¹(𝕋) and restricted to the strictly positive subset

X_{>0} = { ρ ∈ L²(𝕋) | ∇ρ ∈ L²(𝕋), ∫_𝕋 ρ = N, ρ(x) > 0 ∀x }.

Because H¹ embeds continuously into all L^p (p ≥ 2) in one dimension, X_{>0} is an open set, which resolves the “openness” problem that plagues the usual L³ setting. External potentials are represented by the dual space X* = {