Many-body physics in small systems: Observing the onset and saturation of correlation in linear atomic chains

The exact study of small systems can guide us toward relevant measures for extracting information about many-body physics as we move to larger and more complex systems capable of quantum information processing or quantum analog simulation. We use exact diagonalization to study many electrons in short one-dimensional atom chains represented by long-range extended Hubbard-like models. We introduce a measure, the single-particle excitation content (SPEC) of an eigenstate and show that the functional dependence of the SPEC on the eigenstate number reveals the nature of the ground state (ordered phases), 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 noninteracting single-particle regime, a regime of perturbative Coulomb interaction in which the SPEC is a nearly universal function of eigenstate number, the onset and saturation of correlation, a regime of fully correlated states in which hopping is a perturbation, and the 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 eigenstates are made up primarily of single-particle excitations of the ground state, and as the Coulomb interaction increases, the lowest-energy eigenstates increasingly contain many-particle excitations. In addition, the SPEC highlights a fundamental distinct difference between a noninteracting system and one with minute very weak interactions. Although the 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, such as correlation functions or order parameters, provides us with a window of intuition about the behavior of a physical system.