Non-perturbative renormalization-group approach to lattice models

TL;DR

This paper extends the non-perturbative renormalization-group method to lattice models, specifically a $$ theory on a hypercubic lattice, deriving flow equations and renormalized dispersion across the Brillouin zone.

Contribution

It introduces a non-perturbative RG approach to lattice models and demonstrates how to solve the flow equations efficiently using harmonic expansions.

Findings

Reproduces continuum flow equations in the long-distance limit

Derives the renormalized dispersion over the entire Brillouin zone

Simplifies numerical solutions via harmonic expansion

Abstract

The non-perturbative renormalization-group approach is extended to lattice models, considering as an example a $\phi^4$ theory defined on a $d$-dimensional hypercubic lattice. Within a simple approximation for the effective action, we solve the flow equations and obtain the renormalized dispersion $\eps(\q)$ over the whole Brillouin zone of the reciprocal lattice. In the long-distance limit, where the lattice does not matter any more, we reproduce the usual flow equations of the continuum model. We show how the numerical solution of the flow equations can be simplified by expanding the dispersion in a finite number of circular harmonics.