Effect of droplet configurations within the functional renormalization group of the Ising model approaching the lower critical dimension

TL;DR

This paper investigates the ability of the nonperturbative functional renormalization group (NPFRG) with derivative expansion truncations to capture droplet effects near the lower critical dimension in the Ising model, revealing nonuniform convergence and boundary layer phenomena.

Contribution

It extends previous analysis to second order in the derivative expansion, demonstrating improved robustness and compatibility with droplet theory predictions in the critical behavior near the lower critical dimension.

Findings

Convergence to the lower critical dimension is nonuniform and involves boundary layers.

The second order expansion captures droplet effects and improves agreement with droplet theory.

Emergence of two distinct small parameters controlling critical behavior as dimension approaches lower critical dimension.

Abstract

We explore the application of the nonperturbative functional renormalization group (NPFRG) within its most common approximation scheme based on truncations of the derivative expansion, to the $Z_2$-symmetric scalar $\varphi^4$ theory as the lower critical dimension $d_{\rm lc}$ is approached. We aim to assess whether the NPFRG - a broad, nonspecialized method which is accurate in $d\geq 2$ - can capture the effect of the localized (droplet) excitations that drive the disappearance of the phase transition in $d_{\rm lc}$ and control the critical behavior as $d\to d_{\rm lc}$. We extend a prior analysis to the next (second) order of the derivative expansion to check the convergence of the results and the robustness of the conclusions. The study turns out to be much more involved. Through extensive numerical and analytical work we provide evidence that the convergence to $d_{\rm lc}$ is nonuniform in the field dependence and is characterized by the emergence of a boundary layer near the minima of the fixed-point effective potential. This is the mathematical mechanism through which the NPFRG within the truncated derivative expansion reproduces nontrivial features predicted by the droplet theory of Bruce and Wallace [1,2], namely, the existence of two distinct small parameters as $d\to d_{\rm lc}$ that control different aspects of the critical behavior and that are nonperturbatively related. The second order of the derivative expansion fixes several issues that were encountered at the lower level and improves the compatibility with the droplet-theory predictions. [1] A. D. Bruce and D. J. Wallace, Phys. Rev. Lett. 47, 1743 (1981), [2] A. D. Bruce and D. J. Wallace, Journal of Physics A: Mathematical and General 16, 1721 (1983).