# Critical behavior of the hopping expansion from the Functional Renormalization Group

## TL;DR

This paper develops a lattice-based Functional Renormalization Group approach, solving it exactly via graph rules, to analyze critical behavior and convergence in quantum field theories like  and  models.

## Contribution

It introduces an exact lattice FRG method using graph rules and demonstrates its effectiveness in computing critical points in scalar field theories.

## Key findings

- Successfully computes the critical line in  and  theories.
- Shows the FRG flow's fixed points determine the convergence radius related to criticality.
- Validates the method against known solutions of the Ler-Weisz model.

## Abstract

A lattice version of the widely used Functional Renormalization Group (FRG) for the Legendre effective action is solved - in principle exactly - in terms of graph rules for the linked cluster expansion. Conversely, the FRG induces nonlinear flow equations governing suitable resummations of the graph expansion. The (finite) radius of convergence determining criticality can then be efficiently computed as the unstable manifold of a Gaussian or non-Gaussian fixed point of the FRG flow. The correspondence is tested on the critical line of the L\"{u}scher-Weisz solution of the $\phi^4_4$ theory and its $\phi_3^4$ counterpart.