An exact study of charge-spin separation, pairing fluctuations and pseudogaps in four-site Hubbard Nanoclusters

An exact study is carried out by using the \textit{analytical }eigenvalues of the four-site Hubbard Nanoclusters with the grand canonical and canonical ensemble approaches in a multidimensional parameter space of temperature (T), magnetic field (h), on-site interaction (U), chemical potential ($\mu)$ and number of electrons (N). The electron charge energy gap, with one hole off half filling, corresponds to an excitonic particle-hole pair binding instability with $\Delta^{e-h}>0$ at $U>U_c $ and vanishes at a critical parameter $U_c =4.584$. For $U