Nonuniformity of the $1/N$ Expansion for Two-Dimensional $O(N)$ Models

TL;DR

The paper demonstrates that the $1/N$ expansion for 2D $O(N)$ models becomes increasingly inaccurate at lower temperatures, limiting its usefulness for studying low-temperature properties, unlike in 1D chains.

Contribution

It reveals the nonuniformity of the $1/N$ expansion in 2D $O(N)$ models across temperatures, challenging previous assumptions about its applicability.

Findings

$1/N$ expansion deviates at low temperatures in 2D models

No nonuniformity observed in 1D $O(N)$ chains

Implications for studying low-temperature properties in 2D models

Abstract

We point out that the $1/N$ expansion, which is widely invoked to infer properties of the $2D$ $O(N)$ models, is nonuniform in the temperature, i.e. with decreasing temperature the $1/N$ expansion truncated at a fixed order deviates more and more from the true answer. This fact precludes the use of the expansion to deduce low temperature properties such as asymptotic scaling for those models. By contrast, in the $1D$ $O(N)$ chains, there are no signs of such a nonuniformity.