Boundary conditions and universal finite-size scaling for the hierarchical $|\varphi|^4$ model in dimensions 4 and higher

TL;DR

The paper provides a rigorous multiscale renormalisation group analysis of the weakly-coupled $n$-component $|\varphi|^4$ model on a hierarchical lattice in dimensions $d\ge 4$, comparing free and periodic boundary conditions. It establishes a complete finite-size scaling picture: (i) an infinite-volume Gaussian limit at the critical point for $ u=\nu_c$ with boundary-condition dependent scaling of the susceptibility in $d>4$ (and corresponding logarithmic corrections at $d=4$); (ii) a non-Gaussian limiting law inside a universal critical window around an effective finite-volume critical point $\nu_{c,N}^*$ with a universal susceptibility profile; and (iii) a Gaussian limit above the critical window with a distinct scaling, including a mass-generation mechanism for free boundaries. The analysis hinges on a refined RG framework that extends the Bauerschmidt–Brydges–Slade program to $d>4$ and finite volumes, incorporating a large-field regulator and mass-derivative control to handle dangerous irrelevant terms. The results imply universal finite-size scaling functions and provide precise asymptotics for the susceptibility and higher moments, with broad conjectural implications for Euclidean lattice models and SAW in $d\ge 4$. The work also clarifies how boundary conditions shift effective critical points and introduces a mass-generation mechanism tied to FBC, potentially extending to non-hierarchical Euclidean models.

Abstract

We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|\varphi|^4$ model for all integers $n \ge 1$ in all dimensions $d\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that for a volume of size $R^{d}$ with periodic boundary conditions the infinite-volume critical point is an effective finite-volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order $R^{-2}$. For both boundary conditions, the average field has the same non-Gaussian limit within a critical window of width $R^{-d/2}$ around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount $R^{-2}$. In particular, at the infinite-volume critical point the susceptibility scales as $R^{d/2}$ for periodic boundary conditions and as $R^{2}$ for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non-hierarchical) models on $\mathbb{Z}^d$ in dimensions $d \ge 4$. For $d=4$ we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.

Figures (5)

• Figure 1: Effective critical points $\nu_c^{\rm F}$, $\nu_c^{\rm P}$ and scaling windows (red) for FBC and PBC for (a) $d>4$ and (b) $d=4$. The scaling windows for FBC and PBC do not overlap. At the infinite-volume critical point $\nu_c$, the average field has a Gaussian limit with FBC and a non-Gaussian limit with PBC.
• Figure 2: The block $\Lambda_3$ with its blocks in ${\cal B}_0,{\cal B}_1,{\cal B}_2,{\cal B}_3$ for the case $d=L=2$.
• Figure 3: Plots of the universal profile $f_n$
• Figure 4: Plot of $f_1 (s )+s$ illustrating lack of convexity of $f_1(s)+s$ and hence of $f_1(s)$.
• Figure 5: Effective potential $V$ vs $t$ for $g=1$ and $\nu=-1.1, -1.2, -1.3$. The critical value is close to $-1.2$.

Theorems & Definitions (130)

• theorem 1
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• theorem 3
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• proposition 1
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• lemma 1
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• corollary 2