Strictly correlated electrons in a quantum ring: from Kohn-Sham to Kantorovich potentials

TL;DR

This paper characterizes interactions in quantum rings affecting optimal transport plans and derives asymptotics of the adiabatic connection potential for strongly interacting electrons.

Contribution

It extends the class of interactions for which optimal transport structures are understood and rigorously derives the asymptotics of the adiabatic connection potential in quantum systems.

Findings

Characterized pairwise interactions for symmetric multimarginal optimal transport.

Proved convergence of the Lieb density functional to the optimal transport functional.

Extended results to periodic systems in arbitrary dimensions.

Abstract

Our goal in this paper is twofold. First, we characterize the class of pairwise interactions for which the Seidl conjecture on the structure of optimal plans for the symmetric multimarginal optimal transport problem with one-dimensional marginal holds. This extends previous results by Colombo, De Pascale, and Di Marino [Can. Jou. Math., 67 (2015), https://doi.org/10.4153/CJM-2014-011-x], which treated the case of translation-invariant, convex and decreasing interactions. In particular, our results apply to physically relevant interactions for electrons living on a quantum ring. The second main goal of the paper is to rigorously derive the leading order asymptotics of the adiabatic connection potential for strongly interacting systems. More precisely, we show that for electrons in a quantum ring (or one-dimensional interval), not only the Lieb density functional converges to the optimal transport (or strictly correlated) functional in the semiclassical limit, but also the representing potential converges to a regular Kantorovich potential. As an intermediate step, we also extend previous results on the strongly interacting limit of the Lieb functional to periodic systems in arbitrary dimensions.