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arXiv:2602.07818

The 4-εεε Expansion for Long-range Interacting Systems

Zhiyi Li, Kun Chen, Youjin Deng

Published Feb 8, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2602.07818

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The establishment of the Wilson-Fisher fixed point (WFP) for O(n)O(n)O(n) spin models in d=4−εd=4-εd=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 ∝1/rd+σ\propto 1/r^{d+σ}∝1/rd+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 εεε-expansion calculations up to the two-loop level. We show that, as long as σ<2σ<2σ<2, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with d/2<σ<2d/2 < σ< 2d/2<σ<2, the critical exponents, including the anomalous dimension, are functions of εεε, δ=2−σδ=2-σδ=2−σ and nnn, which reduce to the exact results in the limiting cases ε→0ε\to 0ε→0, δ→0δ\to 0δ→0 or n→∞n \to \inftyn→∞. Our (4−ε)(4-ε)(4−ε)-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at σ∗=2σ_*=2σ∗​=2, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.
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