Critical Phenomena on the Bethe Lattice

TL;DR

This paper studies the critical behavior of scalar field theories on the Bethe lattice, revealing new fixed points and universality classes influenced by the spectral properties of the lattice, using non-perturbative and perturbative methods.

Contribution

It demonstrates the existence of non-trivial fixed points for scalar theories on the Bethe lattice and distinguishes their universality classes from the Ising model.

Findings

Identification of a Wilson-Fisher fixed point for short-range theory.

Discovery of distinct universality classes for scalar and Ising models.

Analysis of spectral gap effects on critical behavior.

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 $\zeta \in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $\zeta$. Specifically, $\zeta=1$ is the short-range theory, while $0<\zeta<1$ 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 $\zeta$ and the spectral dimension $d_s=3$. In particular, our analysis indicates the presence of a Wilson-Fisher fixed point for the short range $\zeta =1$ 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 $\phi^4$ theory and the discrete Ising model on the same underlying lattice may lie in distinct universality classes.