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

Abstract

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.