Exact thermodynamics of electron hole pairing and ferroelectricity in attractive Hubbard nanoclusters.

Using analytical diagonalization and symmetries in the small Hubbard clusters, we demonstrate exact mapping between $U>0$ and $U<0$ models in the ground state and at finite temperatures for general electron concentration and magnetic field. We establish equivalency between corresponding phase diagrams for one hole off half-filling at $U>0$ and in the low spin region for half filled $U<0$ models. The $U<0$ model exhibits Fulde-Ferrell-Larkin-Ovchinnikov phase separation into spin-rich and spin-poor regions analogous to electron pairing for $0U_F$ [1]. The calculated phase diagrams resemble a number of metal dielectric transitions, inhomogeneous paired phases, superconductivity, ferromagnetism and ferroelectricity found recently in transition metal and Nb nanoparticles [3], etc. [1] A.N.Kocharian et. al., Phys. Rev. B{\bf 74}, 024511 (2006); Phys. Lett. A{\bf 364}, 57 (2007). [2] G.~W.~Fernando, et. al., Phys.~Rev. B{\bf 75}, 085109 (2007). [3] X.~Xu et al., Phys. Rev. Lett. {\bf 93}, 086803 (2004).