The zero-dimensional O(N) vector model as a benchmark for perturbation theory, the large-N expansion and the functional renormalization group

TL;DR

This paper uses the zero-dimensional O(N) vector model to compare perturbation theory, large-N expansion, and functional renormalization group methods, highlighting the strengths and limitations of each in calculating correlation functions.

Contribution

It demonstrates that the FRG with minimal couplings outperforms large-N expansion for small N and shows the derivative expansion becomes exact in this zero-dimensional model.

Findings

Perturbation theory fails at moderate interactions for all N.

Large-N expansion accurately predicts free energy and self-energy even at small N.

FRG with few couplings provides accurate results across N, especially in strong coupling regimes.

Abstract

We consider the zero-dimensional O(N) vector model as a simple example to calculate n-point correlation functions using perturbation theory, the large-N expansion, and the functional renormalization group (FRG). Comparing our findings with exact results, we show that perturbation theory breaks down for moderate interactions for all N, as one should expect. While the interaction-induced shift of the free energy and the self-energy are well described by the large-N expansion even for small N, this is not the case for higher-order correlation functions. However, using the FRG in its one-particle irreducible formalism, we see that very few running couplings suffice to get accurate results for arbitrary N in the strong coupling regime, outperforming the large-N expansion for small N. We further remark on how the derivative expansion, a well-known approximation strategy for the FRG, reduces to an exact method for the zero-dimensional O(N) vector model.