The Renormalized Trajectory of the O(N) Non-linear Sigma Model

The renormalized trajectory (RT) is determined from two different Monte Carlo renormalization group techniques with $\delta$-function block spin transformation in the multi-dimensional coupling parameter space of the two-dimensional non-linear sigma model with O(3) symmetry. At a correlation length $\xi \approx 3$-$5$, the RT is shown to break away from the straight line of the fixed point trajectory (FPT) which is orthogonal to the critical surface and originates from the ultraviolet fixed point (UVFP). The large $N$ calculation of the RT is also presented in the coupling parameter space of the most general bilinear Hamiltonian. The RT in the large $N$ approximation exhibits a similar shape with the sharp break occurring at a somewhat smaller correlation length.