From local to critical fluctuations in lattice models: a non-perturbative renormalization-group approach

TL;DR

This paper introduces a lattice-specific non-perturbative renormalization-group method that accurately predicts critical properties of lattice models, including phase transitions, with simple approximations and potential for systematic improvements.

Contribution

It develops a lattice adaptation of the NPRG approach using decoupled sites as the initial condition, matching traditional results at long distances and improving critical property predictions.

Findings

Accurately predicts critical temperatures and magnetizations within 1% for 3D Ising, XY, and Heisenberg models.

Provides a framework to include anomalous dimensions and analyze BKT transition in 2D XY model.

Demonstrates the method's effectiveness with simple local potential approximation.

Abstract

We propose a modification of the non-perturbative renormalization-group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a $d$-dimensional hypercubic lattice and classical spin systems. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the 3D Ising, XY and Heisenberg models to an accuracy of the order of one percent. We show how the local potential approximation can be improved to include a non-zero anomalous dimension $\eta$ and discuss the Berezinskii-Kosterlitz-Thouless transition of the 2D XY model on a square lattice.