Quantum critical phenomena and conformal invariance

We show that the Abelian bosonization of the continuum limit of the one-dimensional (1D) Hubbard model corresponds to the two-dimensional (2D) explicitly conformally invariant Gaussian model in the weak coupling limit. A universality argument is used to extend the equivalence to an entire segment of the critical line of the strongly correlated electron system. An integral equation satisfied by the mapping function between critical lines of the 1D Hubbard model and 2D Gaussian model is obtained and then solved in some limiting cases. By making use of the fact that the free Hubbard system reduces to four fermions and each of them is related to a c=1/2 conformal field theory, we present exactly the partition function of the Hubbard model on a finite 1D lattice.