The Shape of the Renormalized Trajectory in the Two-dimensional O(N) Non-linear Sigma Model

TL;DR

This paper numerically and analytically investigates the shape of the renormalized trajectory in the two-dimensional O(N) non-linear sigma model, revealing how it deviates from the fixed point trajectory around specific correlation lengths.

Contribution

It provides the first numerical determination of the renormalized trajectory in the 2D O(3) model and compares it with large N analytical results, highlighting the trajectory's shape and deviations.

Findings

Renormalized trajectory deviates from fixed point trajectory around -7 in correlation length.

Trajectory flows into high temperature fixed point at zero correlation length.

Large N analysis shows similar trajectory shape with deviations at -3 correlation length.

Abstract

The renormalized trajectory in the multi-dimensional coupling parameter space of the two-dimensional O(3) non-linear sigma model is determined numerically under \linebreak $\delta$-function block spin transformations using two different Monte Carlo renormalization group techniques. The renormalized trajectory is compared with the straight line of the fixed point trajectory (fixed point action) which leaves the asymptotically free ultraviolet fixed point of the critical surface in the orthogonal direction. Our results show that the renormalized trajectory breaks away from the fixed point trajectory in a range of the correlation length around $\xi \approx 3$-$7$, flowing into the high temperature fixed point at $\xi=0$. The analytic large $N$ calculation of the renormalized trajectory is also presented in the coupling parameter space of the most general bilinear Hamiltonians. The renormalized trajectory in the large $N$ approximation exhibits a similar shape as in the $N=3$ case, with the sharp break occurring at a smaller correlation length of $\xi \approx 2$-$3$.