Finite-size Analysis of $O(N)$ Nonlinear $\sigma$ Model on Semi-compact Spaces

TL;DR

This paper applies Fisher's renormalization method to analyze the $O(N)$ nonlinear sigma model on semi-compact spaces, revealing how ultraviolet and infrared cut-offs coexist and calculating key physical quantities.

Contribution

It introduces a finite-size analysis of the $O(N)$ nonlinear sigma model on semi-compact spaces using Fisher's method, highlighting the compatibility of UV and IR cut-offs.

Findings

Mass gap and correlation length calculated for the model.

Ultraviolet cut-off does not conflict with infrared cut-off.

Discussion of the mass gap on $S^{2} imes f R$.

Abstract

Fisher's phenomenological renormalization method is used to calculate the mass gap and the correlation length of the $O(N)$ nonlinear $\sigma$ model on a semi-compact space $S^{1}\times {\bf R}^{2}$. This shows that the ultraviolet momentum cut-off does not conflict with the infrared cut-off along the $S^{1}$ direction. The mass gap on $S^{2}\times {\bf R}$ is also discussed.