  We present analytical results of fundamental properties of one-dimensional
(1D) Hubbard model with a repulsive interaction, ranging from fractional
excitations to universal thermodynamics, interaction-driven criticality,
correlation functions, Contact susceptibilities and quantum cooling. Using the
exact solutions of the Bethe Ansatz equations of the Hubbard model, we first
rigorously calculate the gapless spin and charge excitations, exhibiting exotic
features of fractionalized spinons and holons. Based on the analysis on the
fractional charge and spin excitations, the spin-incoherent Luttinger liquid
with only the charge propagation mode is elucidated by the asymptotic of the
two-point correlation functions with the help of the conformal field theory.
Near quadruple critical point, we then further analytically obtain the
thermodynamical properties, dimensionless ratios and scaling functions near
quantum phase transitions in terms of chemical potential, magnetic field and
interaction. In particular, we determine additivity rules of spin and charge
susceptibilities, and derive explicit forms of thermodynamics of
spin-incoherent Luttinger liquid. Finally, in order to capture deeper insight
into the Mott insulator and interaction driven criticality, we further study
the double occupancy and its associated Contact and Contact susceptibilities
through which an adiabatic cooling scheme upon the quantum criticality is
introduced.
