Critical Phenomena on the Bethe Lattice

TL;DR

The paper analyzes critical phenomena of a $\\mathbb{Z}_2$-symmetric scalar field theory on the Bethe lattice, using lattice FRG in the LPA and lattice perturbation theory. By tuning the graph Laplacian to a gapless regime and exploiting a spectral dimension of $d_s=3$, it uncovers non-Gaussian fixed points and shows a Wilson-Fisher fixed point for the short-range case, contrasting with Ising-model mean-field behavior on the same lattice. The study connects autonomous gapless FRG flows to continuum-like $d_s=3$ scalings and supports perturbative results via an $\\epsilon$-expansion, highlighting how curvature and spectral properties shape universality classes on hyperbolic lattices. These results motivate further work on derivative terms, crossover phenomena, and general hyperbolic lattices, with potential implications for non-Euclidean critical phenomena and AdS/CFT-inspired lattice models.

Abstract

We investigate the critical behavior of a family of $\mathbb{Z}_2$-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and lattice perturbation theory. The family is indexed by the parameter $ζ\in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $ζ$. Specifically, $ζ=1$ is the short-range theory, while $0<ζ<1$ defines the long-range model. Due to the hyperbolic nature of Bethe lattices, the Laplacian lacks a zero mode and exhibits a spectral gap. We find that upon closing this spectral gap by a modification of the Laplacian, the scalar field theories exhibit novel critical behavior in the form of non-trivial fixed points with critical exponents governed by $ζ$ and the spectral dimension $d_s=3$. In particular, our analysis indicates the presence of a Wilson-Fisher fixed point for the short range $ζ=1$ theory. In contrast, the nearest-neighbor Ising model on the Bethe lattice is known to exhibit mean-field critical exponents. To the best of our knowledge, this work provides the first evidence that a scalar $φ^4$ theory and the discrete Ising model on the same underlying lattice may lie in distinct universality classes.

Figures (9)

• Figure 1: A truncation of a degree-$3$ Bethe lattice $\mathbb{T}_3$ or $\{\infty,3\}$ tesselation of the hyperbolic plane.
• Figure 2: Local density of states of the ${\mathbb T}_4$ (blue) and ${\mathbb T}_5$ (orange) Bethe lattices.
• Figure 3: Critical flow of couplings $g_2,\,g_4$ and $g_6$ for ${\mathbb T}_{5}$ at $\mathcal{O}(16)$ truncation for short range, $\zeta=1$. For the short-range model, the couplings flow towards the interacting fixed point from critical initial conditions. The fixed point value of the couplings is denoted by dashed lines of the corresponding color in the figure.
• Figure 4: Critical flow of couplings $g_2,\,g_4$ and $g_6$ for ${\mathbb T}_{5}$ at $\mathcal{O}(16)$ truncation for long-range at $\zeta= 3/4$. The couplings flow towards the Gaussian fixed point.
• Figure 5: The scaled Hessian $U^{(2)}(0)/k^2$ for ${\mathbb T}_3$ with interaction $\lambda= 0.06$ in the hopping parameterization. The dashed lines correspond to $\kappa > \kappa_c$ and solid lines correspond to $\kappa < \kappa_c$. Starting from hopping parameter $\kappa = 5.3805$ denoted by the solid blue line, we increase $\kappa$ up to $\kappa = 5.3814$, depicted by dashed brown line. As $k\to 0$, $U^{(2)}(0)/k^2$ changes sign between $\kappa = 5.3809$ to $\kappa = 5.3810$, indicating $5.3809 < \kappa_c(\lambda=0.06) < 5.3810$.