Thermodynamic limit of the Pieces' Model V2

TL;DR

This paper analyzes the ground states of the Pieces' Model V2 in the thermodynamic limit, revealing a near-factorization of configurations and improving the energy expansion accuracy for interacting fermions in a large interval.

Contribution

It introduces a new approach to approximate ground states and refines the thermodynamic limit expansion for the Pieces' Model V2 with finite-range interactions.

Findings

Ground state configurations nearly factorize into groups of one or two particles.

Improved the energy expansion of the ground state per particle to an error of O(ρ^{2−δ}).

Provided an approximate ground state for the Pieces' Model V2.

Abstract

We study the ground states of the pieces' model in the Fermi-Dirac statistics in the thermodynamic limit. In other words, we consider the minimizing configurations of $ n $ interacting fermions in an interval $ \Lambda $ divided into pieces by a Poisson point process, when $ \frac{n}{\vert \Lambda\vert}\to \rho>0 $ as $ \vert \Lambda \vert \to \infty $. We notice that a decomposition into groups of pieces arises from the hypothesis of finite-range pairwise interaction. Under assumptions of convexity and non-degeneracy of the subsystems, we get an almost complete factorization of any ground state. This method applies at least for groups comprising one or two particles. It improves the expansion of the thermodynamic limit of the ground state energy per particle up to the error $ O(\rho^{2-\delta}) $, with $ 0<\delta<1 $. It also provides an approximate ground state for the pieces' model.