Strong local bosonic fluctuations: The key to understanding strongly correlated metals

In this paper, we present a theoretical framework for understanding the extremely correlated Fermi liquid (ECFL) phenomenon within the $U=\ensuremath{\infty}$ Hubbard model. Our approach involves deriving equations of motion for the single-particle Green's function $G$ and its associated self-energy $\mathrm{\ensuremath{\Sigma}}$, which involves the product of the bosonic correlation function comprising both density $({D}_{N})$ and spin $({D}_{S})$ correlations with $G$. By solving these equations self-consistently, we explore the behavior of $G$, ${D}_{N}$, and ${D}_{S}$ as functions of frequency, temperature, and hole concentration. Our results reveal distinct coherent and incoherent Fermi liquid regimes characterized by the presence or absence of quasiparticle excitations. Additionally, we analyze the intrinsic dc resistivity $\ensuremath{\rho}(T)$, observing a crossover from ${T}^{2}$ to linear behavior with increasing temperature. Our findings delineate Fermi liquid, quantum incoherent, and ``classical'' regimes in strongly correlated systems, emphasizing the importance of quantum diffusive local charge and spin fluctuations.