  The exact study of small systems can guide us toward measures for extracting
information about many-body physics as we move to more complex systems capable
of quantum information processing or quantum analog simulation. We use exact
diagonalization to study many electrons in short 1-D atom chains represented by
long-range extended Hubbard-like models. We introduce a novel measure, the
Single-Particle Excitation Content (SPEC) of an eigenstate and show that the
dependence of SPEC on state number reveals the nature of the ground state, and
the onset and saturation of correlation between the electrons as Coulomb
interaction strength increases. We use this SPEC behavior to identify five
regimes as interaction is increased: a non-interacting single-particle regime,
a regime of perturbative Coulomb interaction in which the SPEC is a nearly
universal function of state number, the onset and saturation of correlation, a
regime of fully correlated states in which hopping is a perturbation and SPEC
is a different universal function of state number, and the regime of no
hopping. In particular, the behavior of the SPEC shows that when
electron-electron correlation plays a minor role, all of the lowest energy
states are made up primarily of single-particle excitations of the ground
state, and as the Coulomb interaction increases, the lowest energy states
increasingly contain many-particle excitations. In addition, the SPEC
highlights a fundamental, distinct difference between a non-interacting system
and one with minute, very weak interactions. While SPEC is a quantity that can
be calculated for small exactly diagonalizable systems, it guides our intuition
for larger systems, suggesting the nature of excitations and their distribution
in the spectrum. Thus, this function, like correlation functions or order
parameters, provides us with a window of intuition about the behavior of a
physical system.
