Complex singularities of the critical potential in the large-N limit

TL;DR

This paper investigates the non-convergence of power series expansions for the critical potential in large-N 3D O(N) models, revealing complex singularities that impact approximation accuracy and proposing Pade approximants as a solution.

Contribution

It demonstrates the existence of complex singularities affecting series convergence in large-N models and introduces Pade approximants for improved fixed point calculations.

Findings

Power series in the critical potential do not converge beyond a critical field value.

Conjugated branch points cause non-convergence in the complex plane.

Pade approximants provide accurate high-precision fixed point estimates.

Abstract

We show with two numerical examples that the conventional expansion in powers of the field for the critical potential of 3-dimensional O(N) models in the large-N limit, does not converge for values of phi^2 larger than some critical value. This can be explained by the existence of conjugated branch points in the complex phi^2 plane. Pade approximants [L+3/L] for the critical potential apparently converge at large phi^2. This allows high-precision calculation of the fixed point in a more suitable set of coordinates. We argue that the singularities are generic and not an artifact of the large-N limit. We show that ignoring these singularities may lead to inaccurate approximations.