The scaling region of the lattice O(N) sigma model at finite temperature

TL;DR

This paper investigates the finite temperature phase transition of the (3+1)d O(N) sigma model for N=1, 2, 3, revealing how the critical region width varies with N and temperature phases, with implications for understanding phase transition scaling.

Contribution

The study provides new numerical insights into the width of the critical region in the O(N) sigma model at finite temperature, highlighting differences across N values and phases.

Findings

The broken phase scaling region is wider for N=2 and 3 than for N=1.

The critical region in the low T phase of the O(2) model is wider than in the high T phase.

In the (2+1)d Ising model, the correlation length must be about twice the lattice extent to reach the 2d scaling region.

Abstract

We present results from numerical studies of the finite temperature phase transition of the $(3+1)d$ O(N)-symmetric non-linear sigma model for $N=1,2$ and 3. We study the dependence of the width of the 3d critical region on $N$ and we show that the broken phase scaling region is much wider for N=2 and 3 than for N=1. We also compare the widths of the critical region in the low $T$ and high $T$ phases of the O(2) model and we show that the scaling region in the broken phase is much wider than in the symmetric phase. We also report results for the width of the scaling regions in the low $T$ phase$ (2+1)d$ Ising model and we show that the spatial correlation length has to be approximately twice the lattice temporal extent before the 2d scaling region is reached.