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

TL;DR

The paper investigates whether a nonperturbative functional RG with a second-order derivative expansion can describe the droplet-dominated approach to the Ising lower critical dimension. By solving coupled fixed-point equations for the effective potential and field-renormalization function and employing singular perturbation theory, it reveals a boundary-layer structure near the potential minimum as $d\to d_{\rm lc}$, controlled by two nonperturbatively related small parameters. The study finds that, while regulator dependence persists and full analytical control is challenging, the second-order truncation reproduces several droplet-theory features, including tendencies toward $1/\nu\to0$ and a two-parameter scaling scenario, signaling that NPFRG can partly capture nonuniform long-distance physics in low dimensions. These results motivate further refinement of regulators and truncations to align more closely with the exact droplet picture and essential scaling at the 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).

Figures (18)

• Figure 1: Numerical result for the fixed-point effective potential $u(\varphi)$ at the $\partial^2$ order of the derivative expansion when approaching the lower critical dimension. This is illustrated for the Exponential regulator with the prefactor $\alpha=1$.The vertical dashed lines denote the loci of the minimum of the effective potential for the dimensions shown. The function being symmetric under inversion, we only display positive values of the field.
• Figure 2: Numerical result for the fixed-point for the field renormalization function $z(\varphi)$ at the $\partial^2$ order of the derivative expansion when approaching the lower critical dimension. This is illustrated for the Exponential regulator with the prefactor $\alpha=1$. The vertical dashed lines denote the loci of the minimum of the effective potential for the dimensions shown. The function being symmetric under inversion, we only display positive values of the field. The color code is the same as in Fig. \ref{['fig:u_typical']}.
• Figure 3: Dependence on $\tilde{\epsilon}$ of the locus of the minimum of the effective potential, $\varphi_{\rm min}$ rescaled by the expected dependence $\sqrt{\ln(1/\tilde{\epsilon})}$ for several choices of regulators at the $\partial^2$ order of the derivative expansion. All the curves seem to go to nonzero values when $\tilde{\epsilon}\to 0$.
• Figure 4: Dependence on $\tilde{\epsilon}$ of the second (top) and third (bottom) derivatives of the effective potential at the minimum for several choices of regulators (with the same code as in the legend in Fig. \ref{['fig:phim']}) at the $\partial^2$ order of the derivative expansion : $u"(\varphi_{\rm min})$ is multiplied by the expected boundary-layer width, $\delta(\tilde{\epsilon})\sim \tilde{\epsilon} \sqrt{\ln(1/\tilde{\epsilon})}$, and $u"'(\varphi_{\rm min})$ multiplied by the square of the expected boundary-layer width [compare with Eq. (\ref{['uderBL']})]. All the curves seem to go to nonzero values when $\tilde{\epsilon}\to 0$.
• Figure 5: Evidence showing that the maximum of the field renormalization function $z_{\rm max}=z(\varphi_{\rm max})$ grows subdominantly compared to $u"(\varphi_{\rm min})$ when $\tilde{\epsilon} \to 0$ for several regulator choices (with the same code as in the legend in Fig. \ref{['fig:phim']}). Top: Ratio $z_{\rm max}/u"(\varphi_{\rm min})$ versus $\tilde{\epsilon}$; a likely extrapolation is that the ratio goes to zero as $\tilde{\epsilon}\to 0$. Bottom: Same data further divided by a factor $\kappa(\tilde{\epsilon})\to 0$ and illustrated with the functional form $\kappa(\tilde{\epsilon})\propto [\ln(\frac{1}{\tilde{\epsilon}})]^{-\mu}$, with $\mu=2, 2.5, 3$; $\mu\approx 2-2.5$ appears to best collapse the data.