The effect of droplet configurations within the Functional Renormalization Group of low-dimensional Ising models

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, which 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. [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).