The 4-$ε$ Expansion for Long-range Interacting Systems

TL;DR

The paper investigates long-range $O(n)$ spin systems with algebraically decaying interactions $J(r) \sim 1/r^{d+\sigma}$ using a controlled $d=4-\epsilon$ expansion. It employs two RG frameworks—a standard field-theoretic approach and a perturbative bootstrap scheme—to show that for $\sigma<2$ the short-range Wilson-Fisher fixed point is unstable and a stable long-range Wilson-Fisher fixed point governs critical behavior, with critical exponents depending on $\epsilon$, $\delta=2-\sigma$, and $n$. The results yield explicit expressions for the LR-WFP coupling $u^*$ and the exponents $\eta$ and $\nu$, and demonstrate that the LR-SR crossover occurs strictly at $\sigma_*=2$, in line with recent high-precision numerics and in conflict with Sak's criterion. The work resolves a long-standing controversy and establishes a unified framework linking long-range criticality to nonlocal field theories, with clear directions for higher-loop refinements and a more complete renormalization of competing operators.

Abstract

The establishment of the Wilson-Fisher fixed point (WFP) for $O(n)$ spin models in $d=4-ε$ dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as $\propto 1/r^{d+σ}$, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the $ε$-expansion calculations up to the two-loop level. We show that, as long as $σ<2$, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with $d/2 < σ< 2$, the critical exponents, including the anomalous dimension, are functions of $ε$, $δ=2-σ$ and $n$, which reduce to the exact results in the limiting cases $ε\to 0$, $δ\to 0$ or $n \to \infty$. Our $(4-ε)$-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at $σ_*=2$, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.

Figures (2)

• Figure 1: Feynman diagrams used in the perturbative expansion: (a) one-loop self-energy tadpole, (b) two-loop sunset self-energy, and (c) one-loop interaction bubble.
• Figure 2: Sketch of RG flows for the $O(n)$ spin model in the LR-LD regime with $\sigma < 2$ and $\sigma < d < 2\sigma$. In the $(4-\epsilon)$ dimension, the SR-WFP and the LR-WFP are located at the dimensionless interaction $u \sim O(\epsilon)$.