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The exact study of small systems can guide us toward measures for extractinginformation about many-body physics as we move to more complex systems capableof quantum information processing or quantum analog simulation. We use exactdiagonalization to study many electrons in short 1-D atom chains represented bylong-range extended Hubbard-like models. We introduce a novel measure, theSingle-Particle Excitation Content (SPEC) of an eigenstate and show that thedependence of SPEC on state number reveals the nature of the ground state, andthe onset and saturation of correlation between the electrons as Coulombinteraction strength increases. We use this SPEC behavior to identify fiveregimes as interaction is increased: a non-interacting single-particle regime,a regime of perturbative Coulomb interaction in which the SPEC is a nearlyuniversal function of state number, the onset and saturation of correlation, aregime of fully correlated states in which hopping is a perturbation and SPECis a different universal function of state number, and the regime of nohopping. In particular, the behavior of the SPEC shows that whenelectron-electron correlation plays a minor role, all of the lowest energystates are made up primarily of single-particle excitations of the groundstate, and as the Coulomb interaction increases, the lowest energy statesincreasingly contain many-particle excitations. In addition, the SPEChighlights a fundamental, distinct difference between a non-interacting systemand one with minute, very weak interactions. While SPEC is a quantity that canbe calculated for small exactly diagonalizable systems, it guides our intuitionfor larger systems, suggesting the nature of excitations and their distributionin the spectrum. Thus, this function, like correlation functions or orderparameters, provides us with a window of intuition about the behavior of aphysical system.

Many-Body Physics in Small Systems: Observing the Onset and Saturation of Correlation in Linear Atomic Chains