Triviality of hierarchical O(N) spin model in four dimensions with large N

TL;DR

This paper demonstrates that the hierarchical O(N) spin model in four dimensions becomes trivial in the large N limit, with the critical trajectory converging to a Gaussian fixed point through renormalization group analysis.

Contribution

It provides a rigorous analysis of the convergence to Gaussian fixed points for large N in the four-dimensional hierarchical O(N) model.

Findings

Convergence to Gaussian fixed point for large N

Control of the strong coupling regime via O(∞) trajectory

Power decay of effective coupling in weak coupling regime

Abstract

The renormalization group transformation for the hierarchical O(N) spin model in four dimensions is studied by means of characteristic functions of single-site measures, and convergence of the critical trajectory to the Gaussian fixed point is shown for a sufficiently large N. In the strong coupling regime, the trajectory is controlled by the help of the exactly solved O(\infty) trajectory, while, in the weak coupling regime, convergence to the Gaussian fixed point is shown by power decay of the effective coupling constant.