Fermions and the Renormalisation Group at Large N

TL;DR

This paper uses functional renormalisation to solve fermionic quantum field theories at large N, revealing exact solutions, conformal fixed points, and the emergence of higher derivative interactions, with implications for various physical systems.

Contribution

It demonstrates that at large N, fermionic theories have exact, quasi-local solutions with vanishing anomalous dimensions, and explores the emergence of higher derivative interactions and conformal fixed points.

Findings

Exact solutions for effective actions at large N

Identification of conformal fixed points and scaling dimensions

Higher derivative interactions are induced by point-like interactions

Abstract

We investigate fermionic quantum field theories using functional renormalisation. In the limit of many fermion flavours $N$, we demonstrate that theories have exact solutions for their quantum effective actions given by quasi-local interaction functionals of fermion bilinears. The structure implies that local potential approximations are exact, exactly solvable, and that field anomalous dimensions vanish. Theories with non-trivial anomalous dimensions may also arise under conditions that are identified. We further demonstrate that higher derivative interactions are inevitably induced by point-like ones, including at large-$N$. The local potential flows for fermionic theories with the most general $U(N)$ symmetric interactions are provided. For sample theories with scalar, pseudo-scalar, vector, or axial-vector interactions, we identify conformal fixed points, scaling dimensions, conformal manifolds, and quantum-induced shifts in scaling dimensions of higher derivative interactions. We also study fermion mass generation, and subleading modifications due to finite $N$ corrections. Implications for conformal field theories, and applications in condensed matter and particle physics are indicated.