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

TL;DR

This paper investigates the renormalized trajectory of the two-dimensional O(3) non-linear sigma model using Monte Carlo renormalization group methods, revealing how the trajectory deviates from the fixed point at certain correlation lengths.

Contribution

It introduces a detailed analysis of the RT using two Monte Carlo RG techniques and compares the finite N results with large N calculations, highlighting the trajectory's shape and deviations.

Findings

RT breaks away from the FPT at correlation lengths of 3-5.

Large N calculations show similar RT shape with a break at smaller correlation length.

The study demonstrates the RT's behavior in the coupling parameter space.

Abstract

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.