  We study the one-dimensional SU(4) exchange model under magnetic fields,
which is the simplest effective Hamiltonian in order to investigate the quantum
fluctuations concerned with the orbital degrees of freedom in coupled
spin-orbit systems. The Bethe ansatz approaches and numerical calculations
using the density matrix renormalization group method are employed. The main
concern of the paper is how the system changes from the SU(4) to the SU(2)
symmetric limit as the magnetic field is increased. For this model the
conformal field theory predicts an usual behavior: there is a jump of the
critical exponents just before the SU(2) limit. For a finite-size system,
however, the orbital-orbital correlation functions approach continuously to the
SU(2) limit after interesting crossover phenomena. The crossover takes place in
the magnetization range of 1/3 $\sim$ 1/2 for the system with 72 sites studied
in this paper.
