An O(N) symmetric extension of the Sine-Gordon Equation

TL;DR

This paper introduces an O(N) extension of the Sine-Gordon equation, enabling large-N expansion analysis via Path-Integral methods, revealing significant differences in effective potential behavior compared to the N=1 case.

Contribution

It presents a novel O(N) symmetric extension of the Sine-Gordon model and analyzes its large-N behavior using Path-Integral techniques, highlighting differences from polynomial interaction cases.

Findings

Large-N expansion agrees with variational results

Effective potential differs significantly from N=1 case

Unbroken ground state becomes unstable with increasing coupling

Abstract

We discuss an O(N) exension of the Sine-Gordon (S-G)equation which allows us to perform an expansion around the leading order in large-N result using Path-Integral methods. In leading order we show our methods agree with the results of a variational calculation at large-N. We discuss the striking differences for a non-polynomial interaction between the form for the effective potential in the Gaussian approximation that one obtains at large-N when compared to the N=1 case. This is in contrast to the case when the classical potential is a polynomial in the field and no such drastic differences occur. We find for our large-N extension of the Sine-Gordon model that the unbroken ground state is unstable as one increases the coupling constant (as it is for the original S-G equation) and we determine the stability criteria.